What Is Quantum Information?

92 min readApr 20, 2022

Information in a quantum computer may possess nuances that significantly distinguish it from information in a classical computer. This informal paper explores these nuances of distinction. This should provide a sound foundation upon which to build a sound understanding of quantum computing. Attempting to leap too deeply into quantum computing without a firm grasp of the full breadth, depth, scope, and nuance of quantum information would be a huge mistake.

Quantum information applies to more than simply quantum computers, but quantum computers are the focus and context of this paper, although eventually quantum networking and distributed quantum computing will be included as well.

I considered an alternative title for this paper, but decided it was too wordy even though it was more accurate:

Everything You Need to Know About Quantum Information to Understand Quantum Computing

Topics covered in this paper:

In a nutshell
Quantum information as the foundation of quantum computing
Everything You Need to Know About Quantum Information to Understand Quantum Computing
Quantum information processing
The world of quantum bits or qubits
Basis states as the starting point for quantum information
Qubits or quantum states
Scope of quantum information — in this paper
Some tidbits of quantum computing as well, but sparingly
Virtually no math or Greek symbols or obscure or confusing jargon
No confusing diagrams, graphs, or images — just plain and simple text
Simplistic definition for quantum information
Simplistic definition for classical information
Full definition of quantum information
Abbreviated definition for quantum information
Common interpretations of quantum information
Mike and Ike on Quantum Information
My own definition from my Quantum Computing Glossary
What is the unit of quantum information?
What isn’t covered here
Qutrits and qudits — beyond the scope of this paper, but…
Quantum communication and quantum networking — beyond the scope of this paper
Alice and Bob and quantum teleportation — beyond the scope of this paper
Alice and Bob and quantum teleportation for distributed quantum computations — a distant future
Distributed quantum computing — the intersection of quantum computing and quantum networking
Quantum cryptography — beyond the scope of this paper
The overly simplistic answer
Qubit vs. bit is not an accurate comparison
Brief summary of the issues
Information, representations, and storage devices
Yes, a bit is information
Representing, storing, and transmitting classical information
Unlike a classical bit, a qubit is a device
Quantum state represents quantum information
Basis states are the closest things to classic bits on a quantum computer
Vectors, kets, and basis states on a quantum computer
Bra-ket notation for quantum states
Quantum state
Logical quantum state vs. physical quantum state
Probability amplitudes
Phase as information vs. as quantum state
What are all of the two pi’s I see in quantum computing?
Principle of unitarity
The two basis state probability amplitudes are not independent — the probabilities must sum to 1.0
Kets with probability amplitudes
Product states — strings of basis states for entangled qubits
Tensor product states — the fancy name for product states
Exponential product states for entangled qubits and quantum states
Inseparability — Product states cannot be measured or determined from the individual qubits or quantum states
States and quantum systems
State vectors
Dramatic size of state vectors for entangled qubits preclude large classical simulations of quantum computers
Pure and mixed states — the simplistic naive model
Ensembles of qubits vs. collections of qubits or quantum states
Probability density matrix — needed to define pure and mixed states
Pure and mixed states — the technically correct but inscrutable model
No intuitive plain language description for the technically correct definition of pure and mixed states, just the technical mechanics of how to calculate it
Classical states and non-classical states — No name for pure or mixed states in the simple naive case, so maybe just call them classical states and non-classical states
Classical information and non-classical information
Interference
Environmental interference
Interference as a quantum computing technique
Basis states are discrete values while probability amplitudes and phase are continuous values
Beware of quantum algorithms dependent on fine granularity of phase or probability amplitude
Some granularity issues for continuous values of probability amplitudes and phase
Pi is irrational and has an infinity of digits, as do the square roots of 2, 3, and other reasonably small integers
Quantum information represented by the continuous values of probability amplitude and phase vs. probability amplitudes and phase themselves
Ask your quantum hardware vendor how many discrete values they support for phase and probability amplitudes
Quantum uncertainty for continuous values such as probability amplitudes and phase
Linear algebra — beyond the scope of this paper
Hilbert space — beyond the scope of this paper
Vectors — beyond the scope of this paper
Probability and statistics — beyond the scope of this paper
Bloch sphere — beyond the scope of this paper
Wave functions (simplified)
Cat state
|+> and |->
|PHI+>, |PHI->, |PSI+>, and |PSI->
A qubit stores and manipulates quantum state
No, a qubit is not quantum information
Qubits store quantum information but they are not the information itself
But classical bits actually are the information
Yes, it’s odd, but there is no shortened term for quantum information
Quantum information bits?
Registers of qubits or quantum states
Quantum computing — Manipulating quantum state
Quantum computing — Rotations and entanglement
Quantum computing — Rotations of quantum state of a qubit
Quantum state angles — theta and phi
Quantum computing — All operations are relative, no absolute values may be used
Quantum computing — Quantum algorithms, quantum circuits and quantum logic gates
Quantum computing — Unitary transformation matrix
Quantum information science (QIS)
Quantum effects
Classical Information Theory
Quantum Information Theory
Quantum information
Need for the new field of quantum information theory
Quantum information science vs. quantum information theory
Quantum information science and technology (QIST)
Quantum science
Quantum science and technology
Quantum technologies
Measurement of qubits and quantum information — collapse into classical bits
Measurement error
Readout and qubit readout — synonym for qubit measurement
Collapse of the wave function — the inevitable result of measuring qubits
Accessing the quantum state of qubits
Quantum phase estimation (QPE) and quantum amplitude estimation (QAE)
Quantum state tomography
Measurement of qubits is probabilistic, not deterministic
Expectation value — Measurement of qubits as a statistical distribution, not a discrete value
Quantum information is probabilistic by nature rather than deterministic, unlike classical information
Errors in quantum information
Types of errors in quantum information
Undetected errors in quantum information
Response to errors in quantum information
Near-perfect qubits
Quantum Error Correction (QEC)
Error correction for continuous values such as probability amplitudes and phase is an interesting challenge — no solution has been suggested other than redundancy
Coherence time
Computing: Input, processing, and output
Quantum computations cannot read input data — it must be encoded in the gate structure of the quantum circuit
Quantum computations cannot output quantum data — final quantum state must be measured as classical data
No rich data types
No persistent data
No network service access
No Big Data, but a huge solution space
Little data with a big solution space
True random number generation is natural for quantum information
No-cloning theorem
No copying or cloning of quantum information, no read or examine
Photonic quantum information — Continuous-variable (CV) qumodes and squeezed states
Quantum memory and quantum storage
Programming models
Some unresolved or open issues for me
My original proposal for this topic
Summary and conclusions

In a nutshell

Everything You Need to Know About Quantum Information to Understand Quantum Computing. The whole point of this paper as accurately as I can express it.
The real interest is in quantum computing, but quantum information is the necessary foundation which is needed first before we can even begin to discuss quantum computing properly.
Quantum information processing is the broader domain of processing quantum information, including quantum computing, quantum communication, quantum networking, quantum metrology, and quantum sensing.
Focus of this paper is quantum information in the context of two-level quantum systems for quantum computing. The world of quantum bits or qubits.
Starting point for quantum information is the same as classical information: binary 0 and 1. As basis states.
Then add probability amplitude as a continuous value for each basis state. A real number from 0.0 to 1.0 (and the negative values from minus 1.0 to zero as well.) Actually it is a complex number with a real and imaginary part, each from 0.0 to 1.0 (and the negative values from minus 1.0 to zero as well.)
Superposition allows both basis states at the same time, each with its own probability.
Square the probability amplitude to get the probability for a basis state.
Squares of probabilities for each basis state must sum to 1.0.
And add phase as a continuous value to each basis state. Actually, there is a single phase angle between the two basis states. Phase is represented as the complex part of the probability amplitude.
Phase is nominally an angle in radians from 0.0 to two pi (a full circle or a single cycle), but generally it is normalized to the range 0.0 to 1.0 where 1.0 reprepresents a full circle or two pi radians. And the negative values from minus two pi radians to zero or minus 1.0 to zero as well. A phase of 0.5 represents a half circle, pi radians, or 180 degrees. A phase of 0.25 represents a quarter circle, pi over two radians, or 90 degrees.
Entangled quantum states give an exponential number of product states. 2^n product states for n qubits or n quantum states. Each product state corresponds to a bitstring of n classical bits corresponding to the binary representation of its position in the range 0 to 2^n minus one.
Basis states are discrete values while probability amplitudes and phase are continuous values. This enables quantum computing to be a hybrid of the digital world of classical computing and the continuous world of analog computing.
Uncertainty. Quantum information is inherently probabilistic rather than being strictly deterministic. Simultaneously a blessing and a curse, just as with fire. You can’t get superposition, entanglement, and quantum parallelism without accepting the uncertainty.
It is important to distinguish storage devices, representations, and information itself. Information is represented in a representation. Representations are stored in devices.
Qubit vs. bit is not an accurate comparison. A bit is abstract information while a qubit is a storage device which stores quantum state which is a representation of information.
Qubits are storage devices, not quantum information per se. Qubits store quantum state.
Quantum information is represented as quantum state stored in qubits.
There is no support for Big Data but there is support for a huge solution space with 2^k quantum states. I call it little data with a big solution space.
No copying or cloning of quantum information. No read or examine for quantum state or quantum information. There are some exceptions or workarounds.
Any number of qubits or quantum states can be entangled.
Classical and non-classical states for representation of information.
Classical and non-classical information — the information itself, independent from its representation.
Storage and representation are separate from the information itself. How the information is represented and stored.
No rich data types in the quantum world. Just basis states and strings of basis states.
No complex logic supported.
Quantum information is probabilistic by definition and by its very nature rather than deterministic by definition and by nature as is classical information.
Errors are a reality in all information. Resilience, detection, mitigation, and automatic correction are needed.
Near-perfect qubits. Get the error rate as low as possible in the first place so that errors are not a common problem. Facilitates the other responses to errors.
Error correction for continuous values such as probability amplitudes and phase is an interesting challenge. No solution has been suggested other than redundancy.
True random number generation is natural for quantum information. Technically a function of quantum computing, but enabled by the nature of quantum information.
Quantum information is the foundation for quantum computing and quantum information processing in general, including quantum communication and quantum networking.
Programming models are a hybrid between information (or data) and logic (or code.) This paper focuses on the information half of the programming model equation.
Hopefully this paper will provide a sound foundation for those wishing to dive into quantum computing itself.
Attempting to leap too deeply into quantum computing without a firm grasp of the full breadth, depth, scope, and nuance of quantum information would be a huge mistake.

Quantum information as the foundation of quantum computing

The real interest is in quantum computing, but quantum information is the necessary foundation which is needed first before we can even begin to discuss quantum computing properly.

Everything You Need to Know About Quantum Information to Understand Quantum Computing

The title of this section was my best title for this paper, but I decided that it was too wordy even though more accurate.

But it really does accurately convey the point or purpose of this informal paper.

Quantum information processing

Again, the real interest is in quantum computing but it’s only a portion of the larger domain of quantum information processing, which includes:

Quantum computing.
Quantum communication.
Quantum networking.
Quantum metrology.
Quantum sensing.

Generally, quantum information processing refers specifically to quantum computing, but based on context it may refer to the other areas as well.

More recently, quantum information processing has been referred to as quantum information science.

The world of quantum bits or qubits

You can’t get very far in a discussion of either quantum computing or quantum information without running into quantum bits, more commonly referred to as qubits.

The term is somewhat of a misnomer. Quantum bit certainly sounds like a reference to quantum information, but a qubit is a physical storage device for quantum information rather than the information itself which is being stored, as in how many qubits a quantum computer has.

Overall, the focus of this paper is on the information itself rather than the storage device for the information.

The distinction is important, but this is the terminology we are stuck with. More on this distinction in shortly.

Basis states as the starting point for quantum information

Technically, basis state is a technical term of quantum mechanics which is derived from the technical term of basis vector from linear algebra. The basis of a vector space is the set of basis vectors from which all vectors of the vector space can be derived. The details don’t matter — I’m simply confirming the source of this terminology.

In short, the basis states for a quantum system are the elementary states from which all of the quantum states of the quantum system can be derived. They are the starting point for quantum information. And the starting point for quantum computing as well.

In the case of a two-level quantum system such as the qubits of a quantum computer, binary 0 and 1 are the basis states. Ultimately, all information in a quantum computer is based on some combination of 0 and 1.

Commonly 0 and 1 are written as the kets |0> and |1>, which will be described shortly, but they are still just binary 0 and 1.

Qubits or quantum states

Most of the concepts presented and discussed in this paper apply equally to qubits and quantum states — qubits referring to actual quantum hardware devices and quantum state referring to either the quantum state stored in qubits or to quantum information in the abstract which is represented as a quantum state. Hence, you commonly see the phrase “qubits or quantum states” or “qubits and quantum states”, meaning that the antecedent text applies equally to both qubits and quantum states.

In truth, this is somewhat redundant since a qubit stores quantum state and it is the quantum state which is stored in the qubit which is being referenced.

For example, when referring to “entangled qubits or quantum states”, it is actually the quantum state stored within the qubit which is being entangled, so it’s technically redundant since only quantum states are being entangled. It’s simply that the qubit is an indirect reference to the quantum state contained within the qubit.

The main reason I mention this here is that there may be times when I write qubits alone without the qualifier “or quantum states”, and in most such situations the qualifier “or quantum states” is generally implied unless the context makes it clear that the qubits themselves as devices are being referenced, not the quantum state stored in these devices.

In short, most of what is presented and discussed in this paper is intended to focus on quantum information in the abstract, as represented in quantum state, and not so much any actual hardware devices. But of course when the concepts are applied to actual quantum hardware, qubits are the essential mechanism for referring to quantum state — the quantum state held within the qubit.

Scope of quantum information — in this paper

Although quantum information applies to all of the areas of quantum information science, including quantum communication, quantum networking, quantum metrology, and quantum sensing, the focus here is exclusively on its role in quantum computing.

Much of what is discussed here will also apply to quantum networking — distributed quantum computing and sharing quantum information in the form of quantum state between networked quantum computers. But that is all in the distant future, not a current reality, so it’s a bit beyond the scope of this current paper, although this paper should cover much of what is relevant to quantum networking and distributed quantum computing — when it does become a practical reality.

Also, the focus here is on two-level quantum systems for quantum computing — qubits. Quantum systems with more than two levels, such as qutrits and qudits are beyond the scope of this paper, although discussed a little bit since many of the concepts do carry over to at least some degree.

Although there is great interest in storing and manipulating quantum information on the real quantum computers of today, this paper does not presume any of the current limitations of current and near-term quantum computers — other than to focus on two-level quantum systems such as qubits. In fact, it’s a goal to envision and conceptualize quantum information that could be stored and manipulated on quantum computers which might exist two to five to seven to ten years and more in the future from today.

A fair amount of this paper is at least somewhat relevant to quantum communication, but not all, and with added nuances. That would require a separate paper with a separate emphasis.

Quantum metrology and quantum sensing have broader requirements than the simplified two-state quantum systems used by quantum computing, and less of the requirements driven by quantum computation.

I do envision the day when quantum sensors can be directly connected to quantum computers so that quantum information from quantum sensors can be directly processed by a quantum algorithm without intermediate conversion from quantum to analog and digital and then back to quantum again, but that’s a distant, speculative future.

Although the focus here is quantum computing, this paper focuses exclusively on raw quantum information and stops short of the processing of quantum computation, although various forms of quantum information distinctly imply some amount of quantum computation.

Some tidbits of quantum computing as well, but sparingly

Again, although the focus of this informal paper is quantum information alone in the context of quantum computing and not quantum computing itself, there will be occasional references to quantum computing and issues in quantum computing primarily to give the discussion of quantum information enough context to make sense.

It is also hoped that the contextual references will also make it easier for the reader to approach quantum computing after starting out with this foundation of quantum information as a starting point.

In any case, this introduction to quantum information should stand by itself and not require any previous understanding of or exposure to quantum computing — or quantum mechanics or linear algebra or any similar or related topics.

Virtually no math or Greek symbols or obscure or confusing jargon

The goal here in this informal paper is to introduce and explore quantum information to a significant breadth and depth without any need for complex math or a reliance on Greek symbols. The goal is plain language, with a minimum of jargon other than concepts introduced here in this paper. All concepts should be explained without resorting to obscure or confusing jargon.

Now, in truth, there is a tiny amount of math and an occasional Greek symbol, although I use the Roman names for the Greek letters rather than the raw Greek symbols — such as pi, theta, phi, psi, and rho, which are the most common in this paper.

You won’t need to resort to a tutorial in linear algebra or quantum mechanics. A very few concepts from linear algebra and quantum mechanics will be used, with each carefully explained in plain language.

Anything that might qualify as jargon is first explained in plain language.

No confusing diagrams, graphs, or images — just plain and simple text

In theory, a picture should be worth a thousand words, but in my experience when it comes to technical information, all too often a single image, particularly if poorly chosen or poorly presented, raises a thousand questions.

There may be a few places in this paper where a judicious image might clarify a situation, but I will always opt to endeavor to explain all matters plainly and simply in plain language — with no confusing diagrams, graphs, or images.

Generally I will focus on using bullet points and carefully chosen keywords to express concepts as clearly as possible.

Simplistic definition for quantum information

Quantum information is the information that can be stored in qubits and manipulated on a quantum computer.

Simplistic definition for classical information

Classical information is the information that can be stored as bits, the binary values 0 and 1, and manipulated as bits on a classical computer, either as individual bits or as strings of bits (bit strings.)

Full definition of quantum information

Quantum information includes or is based on classical information, the binary bit values of 0 and 1 as well as strings of bits as a starting point. In toto:

Quantum information includes the classical binary bit values of 0 and 1, known as basis states, as well as strings of these binary bit values (bit strings).
Plus the ability to superimpose the bit values of 0 and 1.
And the ability to assign a probability to each of the superimposed bit values. The probabilities may be arbitrary but must sum to 1.0 — this is referred to as the principle of unitarity.
And the ability to associate a phase angle difference between the two superimposed bit values.
And the ability to entangle any number of quantum states of superimposed bit values.
Entangled quantum states give an exponential number of product states. 2^n product states for n qubits or n quantum states. Each product state corresponds to a bitstring of n classical bits corresponding to the binary representation of its position in the range 0 to 2^n minus one.
And the ability to assign a probability to specific bit string sequences of entangled superimposed bit values. The probabilities may be arbitrary but must sum to 1.0.
Quantum information is represented as quantum state, which is stored and manipulated in two-state quantum systems known as qubits. Qubits hold quantum state and hence quantum information, but unlike classical bits which are information, quantum bits (qubits) are storage devices rather than the information being stored.
Uncertainty. Quantum information is inherently probabilistic rather than being strictly deterministic. Simultaneously a blessing and a curse, just as with fire. You can’t get superposition, entanglement, and quantum parallelism without accepting the uncertainty.

To be clear, this definition does reference bit values and bit strings but does not reference either bits or qubits. The reason for excluding those two concepts will become clear shortly.

Also, this definition is limited to two-state quantum systems, such as quantum computers composed of qubits. Quantum systems with more than two states, such as three-state qutrits and ten-state qudits would require modified forms of this definition, but the general form of the definition would be unchanged.

Abbreviated definition for quantum information

This is as abbreviated as I can get for defining quantum information:

Start with classical binary 0 and 1. The quantum basis states. The same basis foundation as classical information.
Optionally assign a probability amplitude to superimpose bit values. Causes superposition of 0 and 1 basis states. The key differentiation from classical information.
Optionally add a phase angle. Useful as a flag in quantum computations.
Optionally entangle any number of quantum states of superimposed bit values. A collection, ensemble, or register of qubits or quantum states, for example. Assemble a team for much higher leverage of information processing prower.
Entanglement results in exponential product states of quantum information. n entangled qubits or quantum states can represent 2^n unique product states, each represented by its position in binary as a bitstring of n bits, from 0 to 2^n minus one. This is what enables quantum parallelism, the turbo power of information processing power and quantum computing.
Uncertainty. Quantum information is inherently probabilistic rather than being strictly deterministic. You can’t get superposition, entanglement, and quantum parallelism without accepting the uncertainty.

Common interpretations of quantum information

To illustrate the very wide range of definitions, interpretations, and formulations of the concept of quantum information, I did a simple Google search on “quantum information is”. I came up with the following formulations in the first 30 search results alone:

Quantum information is the information of the state of a quantum system.
… classical information follows a set of rules. Quantum information breaks those rules, making it at once a powerful basis for computing and an exquisitely fragile beast.
Quantum information is problem solving and data processing using a quantum system as the information carrier, rather than binary ‘1’s and ‘0’s used in conventional computation. Quantum information systems could be able to transmit data that is fundamentally secure and solve problems that are beyond the power of modern computers.
The really novel feature of quantum information technology is that a quantum system can be in a superposition of different states. In a sense, the quantum bit can be in both the |0〉 state and the |1〉 state at the same time.
Most information is stored in relatively large structures — books, text messages, DNA, computers. Quantum information is information stored in very small structures called qubits. Qubits can be made from any quantum system that has two states. In the image in the poster, these states are depicted as electron orbits in an atom. Because of the principle of superposition, qubits, unlike the “classical bits” in your computer, can be in both their possible states at once. This opens up exciting new possibilities in information technology.
In quantum information processing systems, information is stored in the quantum states of a physical system.
Quantum information is extremely fragile, due to interactions between the system and its environment. These interactions cause the system to lose its quantum nature, a process called decoherence.
Quantum information is usually confined to two-level quantum systems, which are referred to as quantum bits, or qubits. The qubits form the register of a quantum computer or carry the information in a quantum communications channel.
quantum information is given by a two–level system and quantum information is considered to be encoded as a superposition of these states
quantum information is a different kind of information than “classical” Shannon information.
all the peculiarities with which quantum information is usually endowed are features of the quantum coding: “the properties [supposedly specific of quantum information] depend on the type of physical system used to store information, not on new properties of information”.
quantum information is a new kind of information, which has an amazing non-classical property: it may flow backwards in time.
quantum information is a new type of information with the striking, and non-classical, property that it may flow backwards in time.
when the discourse about quantum information is properly debugged, the concept of ‘quantum information’ has no different content than that of the concept of quantum state.
at the end of the day, quantum information is not as quantum as originally supposed. In fact, Schumacher formalism could be repeated without using the term ‘quantum’, by talking only about states belonging to a Hilbert space (such as it was introduced in Section 2), with no reference to a specific physical theory. Once this is acknowledged, it is not difficult to conceptually imagine that, in a counterfactual history, “quantum” information could be developed in the nineteenth century, in terms of, say, Gaussian functions with disjoint supports on a phase space.
Quantum information is the information of the state of a quantum system.
Quantum information is exciting and important
Quantum information is a rich theory that seeks to describe and make use of the distinctive possibilities for information processing and communication that quantum systems provide.
quantum information is radically different from classical information. The unit of quantum information is the ‘qubit’, representing the amount of quantum information that can be stored in the state of the simplest quantum system, for example, the polarization state of a photon.
Concepts addressed include entanglement of quantum states, the relation of quantum correlations to quantum information, and the meaning of the informational approach for the foundations of quantum mechanics.
quantum information is physical information that is held in the “state” of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system. However, unlike classical digital states (which are discrete), a two-state quantum system can actually be in a superposition of the two states at any given time.
Quantum information is a fascinating topic precisely because it shows that the laws of information processing are actually dependent on the laws of physics. However, it is also very interesting to see that information theory has something to teach us about physics.
Quantum information is the effort to both understand and use the properties of the quantum world.
a source of quantum information is just a source of sequences of quantum states that, unlike classical physical states, have probability amplitudes that can be first added and then squared to provide physical probabilities, rather than only in being simply added or probabilistically weighted in mixtures of states.
Why Quantum Information is Never Destroyed. Is the entire history of the universe perfectly knowable? Or has information been lost?
Quantum Information is concerned with studying the way in which the laws of quantum mechanics can be used to store and process information and to perform computations. In particular, the possibility of creating superpositions of classical states, and to create correlations without a classical correspondence give rise to a wide range of new phenomena in data processing and computation.
The fundament of quantum information is the concept of ‘quantum bit’, “qubit” definable as the normed superposition of any two orthogonal subspaces of complex Hilbert space…
quantum information is information that is held in the state of a quantum system. Quantum information is the basic entity of study in quantum information theory, and can be manipulated using engineering techniques known as quantum information processing. Much like classical information can be processed with digital computers, transmitted from place to place, manipulated with algorithms, and analyzed with the mathematics of computer science, so also analogous concepts apply to quantum information.
quantum information is the study of how quantum theory affects our ability to conceptualise and process information.
So at the beginning of the twenty first century, quantum information is providing unifying concepts such as entanglement, and unifying techniques such as coherent information processing and quantum error correction, that have the potential to transform and bind together currently disparate fields in science and engineering.

Those were just the first 30 search results — 30 sources and 30 distinctive formulations (okay, maybe there were a couple of repetitions or near repetitions.) That’s not to suggest that those are the best definitions or interpretations, but certainly among the most popular.

This wealth of diversity of views on such a basic and fundamental concept underlying quantum computing, a veritable blind men and the elephant parable, is a large part of what motivated this paper — to come up with a more coherent definition and explanation of quantum information.

Mike and Ike on Quantum Information

Quantum Computation and Quantum Information is a popular textbook on quantum computing by Michael Nielsen and Isaac Chuang AKA “Mike and Ike”. I always found it kind of curious how quantum information figured in the title since the book was primarily about quantum computing.

In actuality, the book is about quantum computing and quantum communication, but I guess back in 2000 that wasn’t such a common term.

Part I of the book is Fundamental concepts, Part II is Quantum computation, and Part III is Quantum information. The latter part is split between quantum information theory and quantum error correction. That former topic of the third part of the book is focused on the topics now considered part of quantum communication, paralleling Shannon’s work on classical information theory.

But the book doesn’t even touch on a lot of the concepts that I consider the focus and nuances of quantum information.

I was unable to quickly identify the book’s essential definition or meaning for quantum information. Looking for a definition, my text search for “quantum information is…” came up empty.

The book might have been better titled Quantum Computation and Quantum Communication.

The book does have its value and I don’t want to detract from it, but it didn’t help me zero in on quantum information itself. This failure in part led to my writing of this informal paper.

For reference, the front matter and first 20 pages of the book:

The book on Amazon:

Although there are some differences in the front matter of the original and 10th Anniversary Edition, the main text of the book is unchanged even after ten years. I didn’t realize this at first until I was informed of this by one of the authors of the book.

My own definition from my Quantum Computing Glossary

I originally came up with this glossary entry during my initial process of trying to make sense of quantum computing four years ago. My thinking has evolved since then, although the essence is unchanged (information held in the quantum state of a quantum system), just more detail and nuance added. In any case, I originally wrote:

quantum information. Data or information which is held in the quantum state of a quantum system — the qubits of a quantum computer. See the Wikipedia Quantum information article. Alternatively, both quantum computing and quantum communication.

From Part 4 of my Quantum Computing Glossary:

What is the unit of quantum information?

This may seem like a very simple question and indeed it has a very simple answer in classical information:

The unit of classical information is the binary bit.

But there is no comparable simple answer in the world of quantum information, other than the wordy definition just given in a preceding section.

As will be seen shortly, the quantum bit or qubit is actually a storage device rather than information per se.

About the closest we can come to a simple statement in the quantum world is:

The unit of quantum information is the basis state.

Unfortunately, that ignores superposition, entanglement, product states, phase, and probability amplitudes, which are where the richness and raw computational power of quantum computing come from, but if you want a simple statement, that’s it.

Actually, there is another truthful statement which does capture all of the richness, but it won’t be either helpful or understandable to most people:

The unit of quantum information is the wave function.

That captures everything, including all basis states, superpositions, probabilities, phase, and entangled product states (strings of basis state bits.)

What isn’t covered here

Quantum information is actually much broader than information that can be stored in qubits on a quantum computer, which is the primary focus of this informal paper. Some related topics not covered by this paper include:

Multi-level quantum computers with more than two levels. Only two-level qubits are covered here.
Qutrits. Three-level qubits.
Qudits. Ten-level qubits.
Continuous-variable (CV) qumodes. As on photonic quantum computers.
Squeezed states. As on photonic quantum computers.
Non-gate quantum computers. Such as quantum annealing from D-Wave Systems. These are quite interesting, but beyond the scope of the information presented in this paper.
Specialized quantum physics simulation hardware. Again, these are very interesting, but beyond the scope of the concept of quantum information presented in this paper.
Black hole information. All possible information about the state of all matter and all energy disappearing into a black hole.
Quantum communication. The world of Alice and Bob and quantum teleportation.
Quantum networking. For now, today, and the near term, but eventually it will become relevant as distributed quantum computing becomes practical.
Quantum Internet. A rather vague and ambiguous concept.
Quantum cryptography. Falls under quantum communication and algorithms and applications of quantum computing (e.g., Shor’s factoring algorithm), but not a function of quantum information itself.
Quantum metrology. Measuring arbitrary physical quantities. More than simple two-level basis states.
Quantum sensing. Sensing arbitrary physical phenomena. More than simple two-level basis states.
Quantum state in general. All possible information about physical reality in general. More than simple two-level basis states.

It’s not that these phenomena are not interesting or relevant in some way, but simply that the focus and purpose of this paper is to explore quantum information of two-level qubit-based quantum computers. The added complexity of qutrits, qudits, quantum metrology, and quantum sensing would make the paper too unwieldy for its immediate intended purpose as a relatively light introduction to quantum information and foundation for an introduction to quantum computing.

Qutrits and qudits — beyond the scope of this paper, but…

This paper focuses exclusively on two-level quantum systems, basically qubits. Quantum systems with more than two levels are beyond the scope of this paper, in general. This includes qutrits with three levels and qudits with ten levels:

Three-state qutrits. The basis states are |0>, |1>, and |2>.
Ten-state qudits. The basis states are |0>, |1>, |2>, |3>, … |8>, and |9>.

That said, a lot of the concepts discussed in this paper actually do apply equally to these higher-order quantum systems, either intact as is, or simply substituting three or ten (or some other level count) for two, such as:

n entangled qutrits represent 3^n quantum states.
n entangled qudits represent 10^n quantum states.
The probability amplitude for a superposition of a qutrit with equal probability for either basis state would be 1/sqrt(3).
The probability amplitude for a superposition of a qudit with equal probability for either basis state would be 1/sqrt(10).
The probabilities (square of the probability amplitudes) still have to add up to 1.0.
The actual probabilities or probability amplitudes can be any values, provided that the sum of the squares of the probability amplitudes is 1.0.

Alas, other concepts get more complicated, which is why arbitrary n-level quantum systems are beyond the scope of this paper.

The intent is not to delve deeply into qutrits and qudits, but simply to highlight how they fit in and extend the model of quantum information for two-level quantum systems of qubits.

Quantum communication and quantum networking — beyond the scope of this paper

Although quantum metrology and quantum sensing are not covered by this paper, the two other categories of quantum information science (besides quantum computing) are partially covered by this paper:

Quantum communication. Transporting classical information in the form of quantum information as quantum state.
Quantum networking. Transporting quantum information as quantum state between two or more quantum computers.

In both cases, the basic quantum information is no different than the quantum information stored in qubits of a quantum computer, although the protocols and processing of that information are somewhat distinct from quantum computing. And the more advanced and complex forms of quantum information needed for quantum computation are not relevant for simple communication.

Alice and Bob and quantum teleportation — beyond the scope of this paper

The notions of Alice and Bob and quantum teleportation relate to quantum communication, which as just noted is beyond the scope of this paper which focuses on quantum computation.

Quantum teleportation is sometimes presented in the context of quantum computing, but that’s more of a misnomer and artifact rather than being either necessary or advantageous since the focus of quantum teleportation is communication at a distance rather than local computing per se.

Alice and Bob and quantum teleportation for distributed quantum computations — a distant future

The lack of a near-term need for Alice and Bob in quantum computing will likely change once we get to quantum networking with distributed quantum computing, where there actually are distinct Alice and Bob quantum computations on geographically separated quantum computers who are communicating using quantum information using teleported quantum state. But it will be a long way in the future before that is a common form of quantum information exchange.

There is likely to be research on this front in the coming years, but it will take years of research, prototyping, and experimentation before distributed quantum computing becomes a practical reality suitable for commercialization.

Distributed quantum computing — the intersection of quantum computing and quantum networking

Distributed quantum computing will be the long-term future of quantum computing, but is still far off in the future, well beyond the horizon. Individual computations which are split over multiple, geographically-distant quantum computers will open up vast opportunities, but even the most basic research has yet to be begun.

Distributed quantum computing will combine remote networked access to quantum state with local quantum computation — — the intersection of quantum computing and quantum networking.

It is mentioned here not because the reader should pursue it in the here and now or even in the relatively near future, but just to give a sense of the scope and potential of the foundation of quantum information for both current and future quantum computation — including distributed quantum computing. The proper foundation of quantum information, independent of the limitations of current quantum computers, will be of timeless value for future quantum technological developments.

Quantum cryptography — beyond the scope of this paper

Quantum cryptography is an interesting topic, but is beyond the scope of this paper.

Interest in quantum cryptography focuses on three areas:

Code breaking. Using a quantum algorithm, such as Shor’s factoring algorithm to crack public key encryption. This is about algorithms and applications rather than quantum information per se.
Secure communication using quantum methods. This is the realm of quantum communication, which is beyond the scope of this paper on quantum information itself.
Secure encryption key generation. An algorithm and application using a quantum computer to generate true random numbers. Again, not a concern of quantum information itself, just an application of it.

All three are very interesting but focus on quantum algorithms, quantum applications, and quantum communication rather than quantum information itself.

The overly simplistic answer

Every puff piece describing quantum computers basically says the following:

A classical bit is either a 0 or a 1.
A quantum bit or qubit can be a 0 and a 1 at the same time.

Presto! That’s it!

But… wait… that’s not really the whole story at all. Not even close.

Qubit vs. bit is not an accurate comparison

Most puff pieces and a lot of even serious treatments either explicitly or implicitly assert the following inaccurate statement:

A quantum bit or qubit is the quantum equivalent of a classical bit.

This statement is inaccurate because:

A classical bit is simply abstract information, separate from how it might be stored or implemented in hardware.
A qubit is a hardware device capable of storing quantum information in the form of quantum state.

Put more simply:

A classical bit is an abstraction while a quantum bit is hardware.

It is quite unfortunate that the quantum folks chose to not choose terms more in alignment with classical computing, but that distinction is now carved in stone.

This distinction will be explored and clarified in subsequent sections.

Brief summary of the issues

Here are some of the issue which will have to be addressed to make the transition from classical information to quantum information:

A qubit is a device, not information per se as with a classical bit.
The information in a qubit is represented as the quantum state of the qubit.
A qubit could still be just a 0 or a 1 as a classical qubit.
A measured or observed qubit is just a classical 0 or 1 bit value.
Probability of 0 vs. 1. Could be exactly equal probability or could be unequal probability.
Probability is actually represented as a probability amplitude which is a complex value not simply a real value between 0.0 and 1.0.
Something called phase or phase angle between the 0 and 1 values of a qubit.
A qubit can be two values, 0 and 1, but quantum values can also have more than two states or values. Such as a qutrit with three values — 0, 1, and 2. Or a qudit with ten values — 0, 1, 2, … 10.
The role of uncertainty in quantum information.
Quantum information and quantum computation are by definition and by nature inherently probabilistic while classical information and classical computation are by definition and by nature inherently deterministic.
No support for Big Data or complex data structures in quantum information.
Support for huge solution spaces for quantum information and computation. n entangled qubits or quantum states enable 2^n product states.

Information, representations, and storage devices

There are three key concepts with information:

The actual information. An abstraction. Distinct from real-world considerations.
How information is represented. The representation. How the information is expressed. There can be any number of representations for the same information, but the information remains the same regardless of the representation.
How information is stored. And retrieved. In a storage device or on or in some form of media. Or simply transmitted from one location to another. Another form of representation. The expression of the information as it exists in the real world. Generally focused on enabling the information to persist for some period of time, ranging from a very short time interval, such as qubit coherence time, to an extended period of time, such as on a mass storage device such as a hard drive or flash drive.

There is technically a fourth concept, metadata, but’s beyond the scope of the focus of this paper. It is additional information associated with the information, either with its actual abstract value, with its representation, or with its storage (or transmission) or possibly even all three may have distinctive metadata. Or the metadata can be represented and stored completely separately from the actual information storage.

Yes, a bit is information

A bit is the smallest unit of classical information.

Information generally consists of a string or sequence of bits.

Representing, storing, and transmitting classical information

A classical bit is the unit of classical information. A bit or sequence or string of bits can be represented and stored in a wide variety of forms, media, and devices.

Some of the forms of representation of a classical bit include:

The literal text 0 or 1.
The literal text zero and one.
The literal text off and on.
A voltage or lack of a voltage.
A current or lack of a current.
A magnetic field or lack of a magnetic field.
An electron or lack of an electron.
A photon or pulse of photons or a lack of photons.
A presence of media or lack of presence of media.

Bits can be represented or stored in media such as:

Holes on punched cards.
Holes on paper tape.
Magnetic tape.
CDs.
DVDs.
Copper wires. Electrons flowing.
Copper traces. Electrons flowing on printed circuit boards.
Aluminum traces. Electrons flowing in integrated circuits.
Fiber optic cables. Photons traveling relatively long distances.
Bar codes. Information stored on paper or plastic.
QR codes. Information stored on paper or plastic.
Radio waves. Information transported as electromagnetic radiation (photons). Includes cell phones and Wi-Fi.
Images on display screens.

Bits can be stored or operated on in devices such as:

Electromagnetic relays.
Digital logic gates. AND, OR, NOT, XOR, etc.
Flip flops. Clear a bit, set a bit, toggle a bit, add two bits.
CPUs. Complex arrangements of digital logic gates and flip flops.
Memory cells. DRAM chips. FLASH memory.
Flash memory.
Disk drives.
Tape drives.

These are not meant to be exhaustive lists, but simply intended to convey a sense of the range of possibilities.

Unlike a classical bit, a qubit is a device

While a classical bit is information, a qubit is a device which can store and manipulate quantum information — represented as quantum state — rather than being the quantum information itself as is the case with classical bits.

Quantum state represents quantum information

In its simplest form, quantum information is represented as the two basis states |0> and |1>.

Those basis states are quantum states. Or at least the simplest form of quantum state.

A superposition of those two basis states might be alpha|0> + beta|1>. Collectively that is a quantum state.

Two possible entanglements of quantum states might be:

alpha|00> + beta|11>
alpha|01> + beta|10>

Those are two different quantum states — technically called Bell states, each with four basis states which are arranged in two pairs.

Basis states are the closest things to classic bits on a quantum computer

The two binary basis states, |0> and |1>, effectively do correspond to the classical binary bit values of 0 and 1.

Vectors, kets, and basis states on a quantum computer

The basic unit of value on a quantum computer is a vector also known as a basis state and written as a ket — |0> and |1> are the kets for the two basis states of 0 and 1. These really do correspond closely to classical bits, 0 and 1.

Basis states are the most basic and fundamental form of quantum state on a quantum computer, and do correspond closely to the binary bit values of classical information, but that is only the starting point.

Bra-ket notation for quantum states

We won’t go into any details here, but the concept of a ket, such as |0> or |1>, is part of a larger notation scheme called bra-ket notation which is used to detail quantum states in quantum mechanics and quantum computing.

Alternatively it is referred to as Dirac notation.

For details, see the Wikipedia:

Quantum state

In addition to the binary basis states, quantum state can include:

Superposition of the basis states. Both a 0 and a 1 at the same time.
Probability for each basis state. The 0 and 1 do not need to have equal probability — the two probabilities just have to add up to 1.0.
Phase. Or phase angle. A phase is simply a fraction of a circle or cycle — half a cycle, quarter of a cycle, or some other fraction of a cycle. There can be a phase difference between the two superimposed basis states. Can be used as a flag or marker to filter out desired data in a parallel computation.
Probability amplitudes. The probability and phase for each basis state is specified as a probability amplitude, which is a complex number where the real part represents the probability and the imaginary part represents the phase, such that the square of the probability amplitude is the actual probability of the basis state. This is why you see 1/sqrt(2) in probability amplitudes — the square of 1/sqrt(2) is 1/2 or 0.5, the actual probability.
Entanglement and product states. This is where it gets really complicated — and where much of the power of a quantum computer comes from to enable quantum parallelism. A collection of entangled qubits can be represented by two or more strings of basis states, comparable to classic bit strings. n qubits or quantum states can be entangled to create 2^n product states. A mere 50 qubits can represent one quadrillion possible values, each a 50-bit bitstring of basis states which represent the binary value of the position of the product state in the sequence of 0 to 2^n minus one.

Logical quantum state vs. physical quantum state

As used in this paper and in most writing about quantum information and quantum computing, quantum state is one level removed from the actual physical state of real quantum systems. In particular, the basis states are abstract symbolic names, |0> and |1>, rather than the names of real physical basis states of real quantum systems, such as spin up and spin down, which are sometimes written as a vertical up arrow and a vertical down arrow.

The literature doesn’t seem to worry about this abstract vs. physical distinction, and the physical distinction isn’t even very relevant to the purpose of this paper, but I feel that the distinction should be made to at least make it clear what we are really talking about.

So I will define that there are two types of quantum state, or maybe I should say two levels of quantum state:

Logical quantum state or abstract quantum state with logical basis states. The names of the basis states are purely abstract and distinct from any actual physical basis states of any real quantum systems, but there will be a direct one to one correspondence between each logical basis state and a corresponding physical basis state. Commonly the logical basis states have names which are sequential numbers from 0 to the number of levels of the quantum system minus one — so for a two-level quantum system there are two logical basis states, named |0> and |1>.
Physical quantum state with physical basis states. The basis states directly refer to physical states of a real quantum system, such as spin up and spin down. The basis state names may be textual names or graphic symbols, but they directly indicate some physical state of a real quantum system.

Just to be clear, this paper generally only uses quantum state to refer to logical quantum state, with basis states |0> and |1> — logical basis states.

Probability amplitudes

This is getting a little too complicated for the intended level of this informal paper, but the probability and phase for a basis state is actually represented as a single complex value called a probability amplitude.

A complex number has a real and an imaginary part. The real part of a probability amplitude is used to represent the probability for the basis state, while the imaginary part of the probability amplitude is used to represent the phase of the basis state.

It’s even a little more complicated than that.

Actually, for a qubit which is nether superimposed or entangled, it’s not more complicated — the probability is 1.0 for one of the two basis states, either |0> or |1>.

If the qubit is superimposed, it is more complicated — the sum of the squares of the two probability amplitudes must equal 1.0, which means that each probability amplitude is roughly the square root of its probability. If the two basis states have equal probability, this means that the probability amplitude is one divided by the square root of two, which when squared is one divided by two or 0.5 for basis states with equal probability.

If the qubit is also entangled with another qubit, the only complication is that there are still two basis states, but they are either the pairs |00> and |11> or |01> and |10>, so-called Bell states.

If more than three qubits are entangled, it gets even more complicated, as discussed in the section on product states. If there are three or more product states with equal probability, their probability amplitudes are generally one divided by the square root of n, where n is the number of entangled qubits or quantum states.

Phase as information vs. as quantum state

The discussion of phase so far has focused on its meaning as quantum information — its logical meaning, but phase is represented somewhat differently when expressed as a complex number in a probability amplitude in quantum state.

That’s the essential point to take away from this section. You can ignore the rest of this section which details the nature of the distinction and is presented for completeness and context, but is not essential to understanding the nature of quantum information per se.

As quantum information, we say:

Phase is nominally an angle in radians from 0.0 to two pi (a full circle or a single cycle), but generally it is normalized to the range 0.0 to 1.0 where 1.0 reprepresents a full circle or two pi radians. And the negative values from minus two pi radians to zero as well, or minus 1.0 to zero for normalized values.) A phase of 0.5 represents a half circle, pi radians, or 180 degrees. A phase of 0.25 represents a quarter circle, pi over two radians, or 90 degrees.

But in the form of the imaginary portion of the complex number representing phase in a probability amplitude, phase is represented as an exponential or a trigonometric equivalent:

EXP(x i)
Where x is the angle in radians from 0.0 to two pi radians (a full circle or a single cycle). And the negative value from minus two pi radians to zero as well.
Or EXP( two pi phi i)
Where phi is the phase angle expressed as a normalized quantity in the range 0.0 to 1.0 where 1.0 reprepresents a full circle or two pi radians. And the negative values from minus 1.0 to zero as well. A phase of 0.5 represents a half circle, pi radians, or 180 degrees. A phase of 0.25 represents a quarter circle, pi over two radians, or 90 degrees.

Euler’s formula can be used to convert between exponential and its trigonometric equivalent:

EXP(x i) = COS(x) + i SIN(x)

Or if using phi as a normalized phase angle in the range of 0.0 to 1.0 (and the negative values from minus 1.0 to zero as well):

EXP(two pi phi i) = COS(two pi phi) + i SIN(two pi phi)

Sometimes phase is expressed in exponential form in unitary transformation matrices, and sometimes in its trigonometric equivalent.

What are all of the two pi’s I see in quantum computing?

You can’t get too deep into quantum computing before you start seeing two pi’s popping up all over the place, particularly in unitary transformation matrices and particularly in imaginary exponentials such as in quantum Fourier transforms. In short, they are simply the conversion factor from normalized phase (0.0 to 1.0) to a phase angle in radians (0.0 to two pi) — a normalized phase of 1.0 corresponds to a phase angle of two pi radians.

So, the following are equivalent:

EXP( x i)
EXP( two pi phi i)

Presuming that x = two pi phi.

Such as for these cases:

x is two pi and phi is 1.0.
x is pi and phi is 0.5.
x is pi over two and phi is 0.25

Principle of unitarity

The principle of unitarity states that the sum of the probabilities of all of the states of a quantum system must always be 1.0 — that all possible quantum states of a quantum system must be accounted for at all times.

This is equivalent to stating that the sum of the squares of the probability amplitudes of the states of a quantum system must always be 1.0.

The two basis state probability amplitudes are not independent — the probabilities must sum to 1.0

This is simply restating the preceding section in more plain language. It might be nice if the probability amplitudes of the two basis states were two separate information variables, but they are linked rather than independent:

The probability of being |0> plus the probability of being |1> is always 1.0. It must be one or the other.
The probability of being |1> is always 1.0 minus the probability of being |0>.
The probability of being |0> is always 1.0 minus the probability of being |1>.

It gets more complicated when there are more than two basis states — three for qutrits and ten for qudits, but the basic principle is the same.

The bottom line is that a quantum algorithm or application can use the probability amplitude as an information variable, but there is only one such variable for each quantum system, not two — not a separate amplitude information variable for each basis state of the quantum system.

Kets with probability amplitudes

Each state of a qubit has a probability and phase that need to be specified for each state.

Probability and amplitude are specified together as a complex number, which is written as a prefix to the ket for the state. If the specific probability amplitude is not known it is usually written as a letter or Greek symbol or name for a Greek symbol.

The probability amplitude can also be factored across multiple basis states using parentheses.

Some examples:

1.0|0> + 0.0|1>.
0.0|0> + 1.0|1>.
1/sqrt(2)|0> + 1/sqrt(2)|1>.
1/sqrt(2)(|0> + |1>).
1/sqrt(3)|0> + 1/sqrt(2/3)|1>.
alpha|0> + beta|1>.
(alpha + beta i)|0 + (delta + gamma i)|1>.

Product states — strings of basis states for entangled qubits

When we entangle qubits we need to concatenate combinations of basis states — concatenating the individual bit values of the basis states to form bitstrings. These concatenated basis states are known as product states. They can be any length, from two. So:

For alpha|00> + beta|11> the product states are |00> and |11>.
For alpha|01> + beta|10> the product states are |01> and |10>.

Three or more qubits can be entangled as well, resulting in two or more product states for each collection of entangled qubits. So:

For the GHZ state of three qubits, the two product states are |000> and |111>.
For the GHZ state of four qubits, the two product states are |0000> and |1111>.
For the W states of three qubits, the three product states are |001>, |010>, and |100>.
For the W states of four qubits, the four product states are |0001>, |0010>, |0100>, and |1000>.

To be clear, when dealing with product states (states of entangled qubits or quantum states), the quantum information consists of the entire sequence of bit strings — the individual qubit states are not separable, by definition, since they are entangled.

Tensor product states — the fancy name for product states

This is deeper than is necessary for the context of this paper, but when you entangle two or more qubits this is technically known as a tensor product, producing tensor product states in a tensor product space. For this paper we simply use the term product state to refer to a tensor product state. You don’t need to know this to understand this paper, but you may run into it when reading other material.

For more on tensor products, see the Wikipedia:

Exponential product states for entangled qubits and quantum states

Although some forms of entangled qubits or quantum states can result in a very modest number of product states, such as for Bell states, and the GHZ and W states, in the general case of entanglement of n qubits or n quantum states there is an exponential number of product states — 2^n product states for n qubits or n quantum states — all possible combinations of 0 and 1. For example:

For 2 qubits there are 2² = 4 product states: a|00> + b|01> + c|10> + d|11>.
For 3 qubits there are 2³ = 8 product states: a|000> + b|001> + c|010> + d|011> + e|100> + f|101> + g|110> + h|111>.
For 4 qubits there are 2⁴ = 16 product states.
For 8 qubits there are 2⁸ = 256 product states.
For 10 qubits there are 2¹⁰ =~ one thousand product states.
For 20 qubits there are 2²⁰ =~ one million product states.
For 30 qubits there are 2³⁰ =~ one billion product states.
For 40 qubits there are 2⁴⁰ =~ one trillion product states.
For 50 qubits there are 2⁵⁰ =~ one quadrillion product states.
For 300 qubits there are 2³⁰⁰ product states or more product states than particles in the universe.

For k-level quantum systems there are k^n product states for n quxits — 3^n for qutrits and 10^n for qudits.

Inseparability — Product states cannot be measured or determined from the individual qubits or quantum states

Product states truly are entangled — they only exist as the entanglement of the quantum state of the collection of qubits or quantum states which are entangled. Measuring the individual qubits or quantum states cannot be used to determine the product state(s) of the collective qubits.

This is referred to as inseparability — the quantum states of the individual qubits or quantum states cannot be separated, they can only be considered collectively.

In some simple cases the full quantum state can be inferred, but not in the general case.

States and quantum systems

In the context of this paper on quantum information, we are focused on quantum state as the form for representation of quantum information.

An isolated, unentangled qubit is a quantum system. It has two states.

Two isolated qubits are two distinct quantum systems. Each has two states.

n isolated, unentangled qubits are n distinct quantum systems. Each has two states.

A quantum computer with n qubits starts out as n unique, isolated, distinct quantum systems.

As you entangle qubits this merges isolated quantum systems into a single quantum system.

The sum of the squares of each of the probability amplitudes for each of the states of a quantum system will be 1.0, by definition — the quantum system has to be in some state at all times.

An unentangled qubit has two possible states.

A collection or ensemble of n entangled qubits has 2^n possible states. For n = 1, 2¹ = 2.

An unentangled qutrit has three possible states.

A collection of n entangled qutrits has 3^n possible states. For n = 1, 3¹ = 3.

An unentangled qudit has ten possible states.

A collection of n entangled qudits has 10^n possible states. For n = 1, 10¹ = 10.

State vectors

The quantum state of a quantum system is commonly represented as a state vector which specifies the probability amplitude for each possible state of the quantum system.

A state vector is sometimes referred to as simply statevector, as one word.

In the context of this paper on quantum information, we are focused on quantum state as the form for representation of quantum information.

This section may seem a little too complicated or deeper than some readers may want to dive at this time,so feel free to skip it for now and simply think of a state vector as a black box representing the quantum state of an isolated quantum system.

A state vector is a matrix with a single column.

The entries in the column of a state vector are the probability amplitudes for the states of the quantum system.

The rows of the column are logically numbered with the binary representations of the integers from 0 to 2^n minus one where n is the number of states in the quantum system.

To be clear, these row numbers are not part of the column or matrix, but simply the logical name for each entry of the state vector.

For an unentangled qubit there are two entries in the state vector, numbered:

0. This entry will represent the probability amplitude for the |0> basis state.
1. This entry will represent the probability amplitude for the |1> basis state.

For two entangled qubits the entries are numbered:

00 = 0. This entry will represent the probability amplitude for the |00> basis state.
01 = 1. This entry will represent the probability amplitude for the |01> basis state.
10 = 2. This entry will represent the probability amplitude for the |10> basis state.
11 = 3. This entry will represent the probability amplitude for the |11> basis state.

For three entangled qubits the entries are numbered:

000 = 0. This entry will represent the probability amplitude for the |000> basis state.
001 = 1. This entry will represent the probability amplitude for the |001> basis state.
010 = 2. This entry will represent the probability amplitude for the |010> basis state.
011 = 3. This entry will represent the probability amplitude for the |011> basis state.
100 = 4. This entry will represent the probability amplitude for the |100> basis state.
101 = 5. This entry will represent the probability amplitude for the |101> basis state.
110 = 6. This entry will represent the probability amplitude for the |110> basis state.
111 = 7. This entry will represent the probability amplitude for the |111> basis state.

For a qubit in the |0> basis state, the state vector entries would be:

|0>. 1.0
|1>. 0.0

For a qubit in the |1> basis state, the state vector entries would be:

|0>. 0.0
|1>. 1.0

For a qubit is an exactly equal superposition of the |0> and |1> basis states, the state vector entries would be:

|0>. 1/sqrt(2)
|1>. 1/sqrt(2)

For two qubits entangled in the Bell state phi+, the state vector entries would be:

|00>. 1/sqrt(2)
|01>. 0.0
|10>. 0.0
|11>. 1/sqrt(2)

For two qubits entangled in the Bell state phi-, the state vector entries would be:

|00>. 1/sqrt(2)
|01>. 0.0
|10>. 0.0
|11>. -1/sqrt(2)

For two qubits entangled in the Bell state psi+, the state vector entries would be:

|00>. 0.0
|01>. 1/sqrt(2)
|10>. 1/sqrt(2)
|11>. 0.0

For two qubits entangled in the Bell state psi-, the state vector entries would be:

|00>. 0.0
|01>. 1/sqrt(2)
|10>. -1/sqrt(2)
|11>. 0.0

For three fully entangled qubits, each state having equal probability, the state vector entries would be:

|000>. 1/sqrt(8)
|001>. 1/sqrt(8)
|010>. 1/sqrt(8)
|011>. 1/sqrt(8)
|100>. 1/sqrt(8)
|101>. 1/sqrt(8)
|110>. 1/sqrt(8)
|111>. 1/sqrt(8)

For more complex entanglements:

For four fully entangled qubits there would be 2⁴ = 16 state vector entries with each entry being 1/sqrt(16).
For eight fully entangled qubits there would be 2⁸ = 256 state vector entries with each entry being 1/sqrt(256).
For 10 fully entangled qubits there would be 2¹⁰ =~ one thousand state vector entries with each entry being 1/sqrt(2¹⁰).
For 20 fully entangled qubits there would be 2²⁰ =~ one million state vector entries with each entry being 1/sqrt(2²⁰).
For 30 fully entangled qubits there would be 2³⁰ =~ one billion state vector entries with each entry being 1/sqrt(2³⁰).
For 40 fully entangled qubits there would be 2⁴⁰ =~ one trillion state vector entries with each entry being 1/sqrt(2⁴⁰).
For 50 fully entangled qubits there would be 2⁵⁰ =~ one quadrillion state vector entries with each entry being 1/sqrt(2⁵⁰).
More than 50 or so fully entangled qubits couldn’t be practically represented on a classical computer.

Again, although the state vector represents the quantum state stored in a collection of qubits, the quantum state also represents some collection of quantum information. In the context of this paper, focused on quantum information alone, a state vector represents the probability amplitudes of the various states of the quantum information, regardless of whether or how it may be stored in physical qubits.

Dramatic size of state vectors for entangled qubits preclude large classical simulations of quantum computers

As just seen in the preceding section, the entanglement of even 40 to 50 qubits can result in very large state vectors — with trillions and even quadrillions of entries, each a pair of multi-byte floating point numbers (a complex number has separate real and imaginary parts, each a real number represented as a floating-point number.)

Entangling 60 or more qubits would result in a state vector which would be larger than any existing or envisioned classical computer, making a classical simulation of such a collection or ensemble of qubits a virtual impossibility.

This is why 50 qubits (or even 40 qubits) is discussed as the upper limit for simulation of a quantum computer on a classical computer.

Pure and mixed states — the simplistic naive model

Some of the common configurations of quantum information are:

Simple quantum state without superposition or entanglement. An isolated quantum system with just a single basis state, |0> or |1>. Essentially a classical bit.
A superposition of basis states. |0> and |1> simultaneously.
An entanglement of quantum states. Implies a superposition as well.

It would be nice to be able to distinguish the first configuration, the simplest one, from the other two configurations, which are more complex.

In fact, it would be nice to refer to the first configuration as a pure state comparable to a classical bit. And to refer to the other two configurations of quantum states as mixed states.

Yes, it would be nice, but… it would not be technically correct. As will be discussed shortly, there is a much more sophisticated and complex rule for distinguishing pure and mixed states.

But, for most of us and for most users, I think most people can be excused if they use this naive and simplistic but technically incorrect terminology:

Pure state. Quantum state consists of a single basis state, |0> or |1>. No superposition and no entanglement. Essentially a classical bit.
Mixed state. Quantum state which consists of a mixture of basis states, both |0> and |1>, either due to superposition alone or due to entanglement as well.

That simplistic naive model certainly works for me, most of the time, even if it’s not absolutely technically correct, as will be discussed shortly.

In a later section I will propose the terms classical state and non-classical state as well as classical information and non-classical information as viable alternative terms. They would be better terms than misusing pure state and mixed state, but I’ll defer to the reader’s best judgment on the choice of terminology.

Ensembles of qubits vs. collections of qubits or quantum states

There’s no great technical distinction between referring to an ensemble of qubits versus a collection of qubits (or quantum states). They’re effectively equivalent, and certainly in the context of this informal paper.

Just as with an ensemble of musicians or other performers, the intention is that the individuals are participating in a joint activity rather than acting completely independently.

I tend to stick to the term collection, but it can be read as ensemble as well.

Generally, it will be true that an ensemble or collection of qubits or quantum states are entangled. If they aren’t entangled then they are independent and act and behave separately, without any coordination.

In some cases, the qubits or quantum states might be entangled, and it is this potential for entanglement which justifies the reference to an ensemble or collection. An example would be a register of qubits or quantum states, which may or may not be entangled based on the flow, sequencing, and timing of information as it is being processed.

Either way, a collection or ensemble of qubits (or quantum states) is simply referring to a selected subset of all of the available qubits on a quantum computer.

Probability density matrix — needed to define pure and mixed states

In order to get technically correct about pure and mixed quantum states we need a variation of the state vector for quantum state called the probability density matrix (or just density matrix) for the quantum state of an ensemble of qubits or quantum states.

Don’t ask or expect me to explain what the probability density matrix is all about or why it is needed, other than that it is needed to determine whether the quantum state for a collection or ensemble of qubits or quantum states is pure or mixed, as discussed in the next section.

Once again, with the section on state vectors, feel free to skip this section, for now. You can learn a lot about quantum information without diving into probability density matrices.

Calculation of the probability density matrix is actually quite simple — unlike trying to understand what it is conceptually all about at any intuitive level:

Start with the state vector for a collection of entangled qubits. This is a matrix consisting of a single column, one entry for each of 2^n product states, where n is the number of entangled qubits.
Create a matrix with a single row which is the conjugate transpose of the state vector. Simply copy the entries from the rows of the state vector to be the columns of the single row of the conjugate transpose but with each entry being the complex conjugate of the original probability amplitude — just flip the sign of the imaginary part of the complex number, if any.
Perform an outer product matrix multiplication of the state vector column and the conjugate transpose row, producing a 2^n by 2^n square matrix. The resulting matrix is the probability density matrix. The individual products of probability amplitudes and complex conjugates are effectively the probability for each entry. This is very similar to simply squaring the probability amplitudes to get probabilities — but don’t ask me to explain the difference!

That’s it! It’s that simple! But, again, don’t ask why this must be done! Just do it!

See the citations at the end of the next section for more details.

Pure and mixed states — the technically correct but inscrutable model

Determining whether a quantum state is pure or mixed is actually quite simple — once you’ve computed the probability density matrix of the quantum state of an ensemble of qubits or quantum states as detailed in the preceding section.

Here it is, the complete process:

Calculate the probability density matrix. As described in the preceding section.
Calculate the trace of the main diagonal of the probability density matrix. This is simply the sum of the probabilities along the main diagonal, from the upper left corner to the lower right corner. This is commonly written as tr(rho²), where Greek rho (which looks like a script p) represents the state vector and rho² represents the probability density matrix entries.
If the trace is 1.0, you have a pure state.
If the trace is less than 1.0, you have a mixed state.

That’s it! It’s that simple! But, again, don’t ask why this must be done! Just do it!

For more on the probability density matrix and pure and mixed states, see the Qiskit textbook:

And see the Wikipedia as well:

No intuitive plain language description for the technically correct definition of pure and mixed states, just the technical mechanics of how to calculate it

The beauty of the naive approach to pure and mixed states is that the naive definition is super-easy to understand intuitively.

The downside of the technically correct definition is that although it is correct, it is completely lacking in intuitive appeal. It’s essentially inscrutable to most people, even those with a fair amount of STEM training.

You can’t explain in plain language alone — free of dense technical jargon — when or how a collection of entangled qubits will be in a pure state or a mixed state.

Classical states and non-classical states — No name for pure or mixed states in the simple naive case, so maybe just call them classical states and non-classical states

Maybe my ultimate compromise between wanting to use the terms pure state and mixed state in a technically incorrect manner is to simply call them classical states, which is not a term used at present by quantum information theorists. In short, I propose:

Classical state. A qubit or quantum state which is simply one of the two basis states with 100% certainty and 0% certainty for the other basis state. There is no superposition and no entanglement.
Non-classical state. A qubit or quantum state which is either in a superposition of the two quantum basis states or is entangled with other qubits or quantum states.

Classical information and non-classical information

Continuing the preceding section since it was focused on states rather than information per se — states are used to store or represent information rather than the information itself. So, we have:

Classical information. Information which can be represented or stored solely using classical states.
Non-classical information. Information which requires non-classical states for its representation and storage — and cannot be represented or stored solely in terms of classical states.

Interference

Interference is a powerful technique in quantum computing, but it is not part of quantum information per se. Actually, there are two very distinct forms of interference:

Environmental interference. Stray electromagnetic radiation or even cosmic rays or natural background radioactive decay can disrupt quantum state and cause errors.
Interference as a quantum computing technique. Waves which are in phase or out of phase can reinforce or cancel. Cancellation can eliminate unwanted data. Reinforcement can highlight desired data. Any further detail will be left to a discussion of quantum computing, which is beyond the scope of this paper.

Environmental interference

It is important to distinguish the computing notion of interference from the operational concept of environmental interference which can disturb or distort the quantum state of a qubit in an unwanted manner, commonly causing errors.

There are several techniques for dealing with such environmental interference:

Accept it and otherwise ignore the issue. Accept that quantum information can be noisy or lossy. For some applications it might not matter or averages out over time.
Repeat operations and look for statistical results. Errors may be less frequent, so merely running a quantum circuit a bunch of times and looking for the most frequent result can eliminate the erroneous results.
Error mitigation. Explicit steps added to a quantum circuit to detect and correct errors manually.
Quantum error correction (QEC). Automatic and transparent detection and correction of errors so that the algorithm only produces correct results — perfect logical qubits — and the application always sees only correct results.

For more detail on quantum error correction and logical qubits, see my paper:

Interference as a quantum computing technique

Interference is a quantum computing technique for filtering out a selected subset of data from a large volume of data, typically a large or even huge solution space.

Waves which are in phase or out of phase can reinforce or cancel:

Cancellation can eliminate unwanted data. When waves are out of phase.
Reinforcement can select desired data. When waves are in phase.

Phase is a part of quantum information when there is a superposition of two basis states.

Phase is the angle between the two basis states.

Phase is an angle from 0.0 to two pi radians, or normalized as a value between 0.0 and 1.0 (and the negative values from minus 1.0 to zero as well), where 0.5 represents half a cycle or 180 degrees or pi radians, and 0.25 represents a quarter of a cycle or 90 degrees or pi over two radians.

To be clear, phase is part of quantum information, but interference is part of quantum computing.

For a little more on interference as a quantum computing technique see the Qiskit introduction to quantum computing:

Basis states are discrete values while probability amplitudes and phase are continuous values

The two basis states of a two-state quantum system are discrete binary values, 0 and 1, or |0> and |1> in ket notation. There are exactly two of them. Or three for three-level quantum systems such as qutrits, or ten for ten-level quantum systems such as qudits.

Probabilities and phase on the other hand are continuous values — real values in the range from 0.0 to 1.0 (and the negative values from minus 1.0 to zero as well for phase.) Theoretically they are each an infinity of possible values. Many more possible values than the two discrete binary basis states. At least in theory.

Granted, with product states there can be many more than two basis states, but far less than the infinity of continuous real values between 0.0 and 1.0.

Beware of quantum algorithms dependent on fine granularity of phase or probability amplitude

As already discussed, phase and probability are nominally continuous values rather than discrete binary values as are the basis states of quantum state, but unfortunately there is no clear guarantee as to how fine-grained those continuous values really are. A free lunch is promised, but there is no promise of the quality, number of courses, or fine details of that free lunch.

They could be fairly fine-grained, very fine-grained, fairly coarse-grained, or very coarse grained. Or anything in between.

At this stage of the evolution of quantum computing we simply have no clarity on this issue. I suspect that over time clarity will develop, at least to some degree, but the timeframe for development of clarity is… unclear.

So, for now, the only guidance must be that users, algorithm designers, and application developers must remain skeptical when it comes to dependence on fine granularity in quantum algorithms.

I don’t recommend trial and error to determine the granularity on a particular computer, but you should insist that the vendor clearly document the granularity in their documentation and technical specifications.

I discuss this issue in greater detail in my paper:

Some granularity issues for continuous values of probability amplitudes and phase

Just to summarize some of the issues for granularity of probability amplitudes and phase:

What is the minimum increment? The granularity in general. Includes both the quantum uncertainty in theory, as well as the limitations of the particular hardware architecture and implementation.
What is the minimum value? What is the smallest non-zero value? Can you successfully decrement to 0.0 from it and increment to it from 0.0?
What is the maximum value? What is the largest value before it properly wraps around to 0.0? Does incrementing that largest value correctly get you to exactly 0.0? Does decrementing from 0.0 get you to exactly this maximum value?
Are absolute 0.0 and 1.0 possible or is there some epsilon for quantum uncertainty? Can a probability amplitude or phase ever be absolutely exactly 0.0 or 1.0? What is the epsilon of uncertainty, both in theory and for the particular hardware architecture and implementation? What is the smallest value achievable if not exactly 0.0? What is the maximum value achievable if not exactly 1.0?

All such issues are discussed in my paper cited in the preceding section.

Pi is irrational and has an infinity of digits, as do the square roots of 2, 3, and other reasonably small integers

The beauty of binary bits is that you can represent arbitrary integer values exactly, perfectly. Just add more bits to represent larger numbers. Each bit you add doubles the number of values you can represent. Each qubit you add doubles the number of quantum states which can be represented.

Real numbers are a different story. Sometimes fractions can be represented exactly as binary bits in floating-point numbers, but frequently that is not the case.

This is a problem that is present for both classical information and quantum information.

But quantum information and quantum computing have another peculiar issue that is not normally an issue for classical information — a distinctive reliance on pi and fractions of pi, and the square root of 2, as well as the square roots of numerous other relatively small integers. The problem is that these are all irrational numbers and the fact that they require an infinity of digits (decimal or binary.)

This means that such values need to be approximated rather than having exact, precise values in quantum information.

It may or may not be a problem in general for most quantum algorithms and applications, but it is certainly an ugly wart on an otherwise elegant theory of physics, information, and computing.

And it does place a burden on quantum algorithm designers and application developers to always be alert to approximations and abandon the hope of exact results in general.

But that’s life in the quantum world — a world constrained and governed by uncertainty.

Quantum information represented by the continuous values of probability amplitude and phase vs. probability amplitudes and phase themselves

It is important to distinguish quantum information at the application level from the actual physical values of probability amplitudes and phase. As already noted, although in theory there may be an infinity of continuous real values for probability amplitude and phase, there may be some practical limit to the number of actual values which are possible on a particular real quantum computer.

So while it may be tempting to casually refer to the continuous real values of probability amplitudes and phase as quantum information — and they actually are at the physical level — they are really the representation of quantum information from an algorithm or application perspective.

To state that more clearly, we have two concepts:

Information itself. As the algorithm or application sees it.
The representation of information. As the underlying hardware and physics sees the information of the algorithm or application.

And it is vitally important that the algorithm or application view of information does not require more values than can be represented by the underlying hardware architecture and implementation.

The algorithm or application is free to use as small a subset of the total possible physical values as it wishes, but not more than that total of possible physical values.

And reliability and fault-tolerance considerations may dictate that a very tiny fraction of the total of possible physical values be used so that redundancy and error correction are supported.

Ask your quantum hardware vendor how many discrete values they support for phase and probability amplitudes

Again, yes, probabilities are in theory continuous real values between 0.0 and 1.0 and phase is also a continuous real value between 0.0 and 1.0, both potentially an infinity of discrete values in theory, but on a real quantum computer (or even a simulator) there will be some maximum number of discrete values or gradations, what I call granularity.

It can vary between different hardware architecture designs and implementations, so the only way to know for sure how many gradations your quantum algorithm will be able to rely upon is to consult your hardware documentation or specifications, or to explicitly ask your quantum hardware vendor.

Sure, you can also conduct some tests yourself, but I am reluctant to encourage people to design, develop, or deploy quantum algorithms or applications which are based on trial and error.

Although it is very reasonable to run a benchmark test to confirm that your quantum hardware actually agrees with its documentation and specifications.

This paper is more focused on quantum information separate from quantum computing per se, but ultimately concerns and questions about quantum information will have to rely on the behavior of the quantum hardware on which the quantum information is accessed.

Quantum uncertainty for continuous values such as probability amplitudes and phase

Separately from the implementation question of what granularity of probability amplitudes and phase are supported on particular hardware, there is also the theoretical issue of quantum uncertainty — how certain or precise a probability amplitude or phase can be even theoretically based on the laws of physics and quantum mechanics.

As best as I can tell at this stage, this is an open question.

It’s not a priority issue today for most quantum algorithm designers and application developers or even hardware designers since they are mostly designing and implementing relatively toy algorithms with minimal requirements for precision of probability amplitudes and phase anway.

But it will gradually become more of an issue as qubit fidelity and qubit connectivity improve to the point where algorithm designers begin to rely on quantum Fourier transforms (QFT) for a nontrivial number of qubits and quantum phase estimation (QPE) or quantum amplitude estimation (QAE) begins to calculate phase to a nontrivial number of bits of precision. Especially for such applications as quantum computational chemistry.

Even then, it could still be a number of years before the hardware implementations advance to the stage where they start bumping into quantum uncertainty itself. Eventually it will happen, even if not within the next few years.

It may not be an urgent issue for near-term quantum algorithm designers, but can and should be a matter of concern for algorithm researchers focused on understanding fundamental algorithm issues for the long term.

Linear algebra — beyond the scope of this paper

Although the mathematics of linear algebra is fundamental to quantum mechanics and quantum computing in general, a full treatment or full understanding of linear algebra is not needed for a basic discussion of quantum information as in this paper, so it is beyond the scope of this paper.

In fact, the few concepts from linear algebra that are needed to understand and work with quantum information have already been introduced and discussed in preceding sections of this paper.

For more on linear algebra, see the Wikipedia:

Also see the IBM Qiskit Textbook appendix on linear algebra:

Hilbert space — beyond the scope of this paper

As with linear algebra in general, the specific notion of a Hilbert space is certainly relevant and fundamental to quantum mechanics and quantum computing in general, but a full treatment or full understanding of Hilbert spaces is not needed for a basic discussion of quantum information as in this paper, so it is beyond the scope of this paper.

For more on Hilbert space, see the Wikipedia:

Vectors — beyond the scope of this paper

As with linear algebra and Hilbert spaces, vectors and vector spaces are certainly relevant and fundamental to quantum mechanics and quantum computing in general, but a full treatment or full understanding of vectors and vector spaces is not needed for a basic discussion of quantum information as in this paper, so it is beyond the scope of this paper.

For the purposes of this paper, the only knowledge of vectors needed is the two basis states (or basis vectors), |0> and |1>.

For more on vectors, see the Wikipedia:

Probability and statistics — beyond the scope of this paper

Although probability and statistics are certainly relevant to quantum mechanics and quantum computing in general, a broad understanding of either is not necessary for a basic comprehension of quantum information.

A few concepts from probability and statistics are needed and relevant to quantum information, but they are introduced as needed in this paper rather than require that the reader demonstrate full competence in the area or even complete introductory tutorials.

If one really wants or feels the need to dive into probability, feel free to consult the Wikipedia:

Ditto for statistics:

Bloch sphere — beyond the scope of this paper

The Bloch sphere is a cute and clever visual aid for understanding the quantum state of a single qubit, but in all honesty it isn’t very useful at all — even though it does look great — so I personally consider it beyond the scope of a serious discussion of quantum information.

The primary value of the Bloch sphere is to visualize the execution of quantum logic gates, which are rotations about the three axes of the Bloch sphere, but this paper focuses on the raw quantum information, not the operations performed on that information during quantum computation, which is beyond the scope of this paper.

The essential limitations of the Bloch sphere for representing and visualizing quantum information are:

Only represents a single qubit.
Doesn’t visualize basis states as orthonormal vectors.
Doesn’t represent or visualize probability amplitudes.
Doesn’t show the two basis states distinctly and separately within a superposition.
Doesn’t represent multiple, entangled qubits.
Doesn’t represent product states for entangled qubits.
Doesn’t represent mixed states or distinguish pure states.
Doesn’t conveniently represent higher-order quantum systems. Such as qutrits or qudits.

For more on the Bloch sphere, see the Wikipedia:

Wave functions (simplified)

Wave functions are an essential part of quantum mechanics, to describe the wave-like behavior of particles.

The essence of a wave function is to list out all of the possible states of an isolated quantum system. These are the quantum states that the quantum system can be in.

Each unentangled qubit is a single isolated quantum system.

Any entangled qubits collectively are a single isolated quantum system.

Each of the possible quantum states has a probability and a phase relative to the other quantum states.

The probabilities of the various possible quantum states must add up to 1.0 since they describe all possible states of the quantum system. This is referred to as the principle of unitarity.

Each probability is normally expressed as a probability amplitude, which is effectively the square root of the probability, so that the sum of the squares of the probability amplitudes must always be 1.0 as per the principle of unitarity.

The essence of the wave function is a sequence of kets, each with a quantum state and a probability amplitude. Each of the quantum states will either be a basis state or a product state — either a single 0 or 1 bit or a bit string of 0’s and 1’s. For example:

1.0|0> + 0.0|1>. Always |0>.
0.0|0> + 1.0|1>. Always |1>.
1/sqrt(2)|0> + 1/sqrt(2)|1>. |0> or |1> with equal probability.
1/sqrt(3)|0> + 1/sqrt(2/3)|1>. Twice as much chance (66.76%) of being |1>.
sqrt(0.25)|0> + sqrt(0.75)|1>. Three times as much chance (75%) of being |1>.
1/sqrt(2)|00> + 1/sqrt(2)|11>. The Bell state, psi+ — |00> or |11>, with equal probability.
1/sqrt(2)|000> + 1/sqrt(2)|111>. The GHZ state.
1/sqrt(3)|001> + 1/sqrt(3)|010> + 1/sqrt(3)|100>. The W state.

There is a lot more to wave functions, but not that is needed to comprehend quantum information in general.

For more depth on wave functions, see the Wikipedia:

Cat state

The so-called cat state is a tribute named after Schrödinger’s cat thought experiment in which an object is in two opposite states at the same time.

Most commonly this is simply a superposition of the |0> and |1> basis states with equal probability, more properly written as:

1/sqrt(2)|0> + 1/sqrt(2)|1>

Or as:

1/sqrt(2)(|0> + |1>)

For more discussion of the thought experiment, see the Wikipedia:

|+> and |->

These are shorthands for the cat states, which are superpositions of the |0> and |1> basis states:

|+>. Equivalent to 1/sqrt(2)|0> + 1/sqrt(2)|1>.
|->. Equivalent to 1/sqrt(2)|0> — 1/sqrt(2)|1>.

|PHI+>, |PHI->, |PSI+>, and |PSI->

Shorthands for the four forms of Bell states:

|PHI+>. Entanglement of 1/SQRT(2)*(|00> + |11>).
|PHI->. Entanglement of 1/SQRT(2)*(|00> — |11>).
|PSI+>. Entanglement of 1/SQRT(2)*(|01> + |10>).
|PSI->. Entanglement of 1/SQRT(2)*(|01> — |10>).

I use the English names for the Greek letters, but the actual Greek letters for phi and psi can or should be used. Aunty eplus and minus symbols should be written as superscripts.

A qubit stores and manipulates quantum state

Quantum state is stored in a qubit — if there is no entanglement.

Quantum state may be stored in a collection or ensemble of qubits — if there is entanglement.

Quantum circuits are sequences of quantum logic gates which operate on qubits. Actually, they are operating on the quantum state stored in qubits.

Operating on the quantum state stored in qubits is effectively operating on the quantum information which is represented as the quantum state stored in the qubits.

So, we can speak of quantum information being stored in qubits in the sense that the qubits store quantum state which represents the quantum information.

No, a qubit is not quantum information

In summary:

A qubit stores and manipulates quantum state.
Quantum information is represented as quantum state. The quantum information is represented as one or more basis states, as individual basis states or strings of basis states (product states) for entangled qubits or quantum states.
Quantum circuits are sequences of quantum logic gates which effectively operate on the quantum information stored in the qubits by operating on the quantum state which represents the quantum information.
But quantum information is an abstraction distinct from qubits and the quantum states that they hold and manipulate.

Qubits store quantum information but they are not the information itself

The actual quantum information is an abstraction which can be represented using basis states arranged in a quantum state or strings of basis states in product states, and it is the quantum state or product state which is actually stored in the qubits.

But classical bits actually are the information

Classical bits are also still an abstraction, which will have to be represented in some form on some media or in some device, but the classical bits themselves are still abstract and distinct from their representations.

Yes, it’s odd, but there is no shortened term for quantum information

These are the three terms we’re stuck with:

Quantum information. The abstract information. Independent of devices and representations. Binary 0 or 1, superposition of 0 and 1, or a string of 0’s and 1’s, as well as probabilities and phase.
Quantum state. The information represented in a form that can be stored and manipulated. A single basis state, superposition of two basis states, or product state — string of basis states.
Qubits. The devices which can store and manipulate quantum state, indirectly allowing us to access and manipulate the information itself.

Quantum information bits?

It might be tempting to call quantum information quantum information bits, but that has some problems:

Too many people would blindly assume that that can be shortened to qubits, which is not correct.
There’s more to quantum information than just the bits, the zeros and ones. We have probability amplitudes, phase, and product states to distinguish.

So, for now at least, we’re stuck just calling it quantum information.

Registers of qubits or quantum states

When reading about quantum algorithms you will frequently read about registers of qubits, as if there was something called a register consisting of some number of qubits, just as classical computers have registers consisting of some number of classical bits (flip flops, actually, each of which can hold one classical bit), but there isn’t really any physical device corresponding to this literary device of a quantum register. Rather, this supposed quantum register is simply so many independent qubits which the algorithm treats as if it were some mythical collective entity.

In short, a register in a quantum computer is simply a logical name for any collection of qubits.

There is no hardware device for a quantum register distinct from the hardware for the individual qubits themselves.

An algorithm may speak about the value or bitstring in a quantum register, but this again is purely artificial and simply consists of the individual measured bit values of each of the separate qubits.

You can’t read or write an entire quantum register separate from accessing each qubit individually.

It’s still reasonable to consider the collective classical bit values as a bitstring, and various software interfaces may make it easy to do so, obviating the need to access each individual qubit explicitly one at a time.

On occasion a published paper will refer to the register of qubits to refer to all of the qubits of a quantum computer. But this is the exception rather than the rule. Usually a qubit register is a selected subset of the qubits intended for a specific purpose in a quantum algorithm.

Quantum computing — Manipulating quantum state

This paper focuses on quantum information alone, in isolation, so it won’t delve very deeply into the manipulation of quantum state, which is quantum computing at its essence.

Of course, what quantum computing is really trying to do is manipulate quantum information, but that is accomplished by manipulating quantum state, which is how quantum information is represented.

Quantum computing — Rotations and entanglement

Not to oversimplify too much and not to delve too deeply into quantum computing, but the essence of the manipulation of quantum state is twofold:

Rotations of the quantum state of a single qubit. In all three dimensions.
Entanglement of two or more qubits.

Quantum computing — Rotations of quantum state of a qubit

Much of quantum computing involves rotations of the quantum state of a single qubit to basically control two aspects of quantum state:

Probability amplitudes of the basis states. Dial probability of the |0> basis state from 1.0 down to 0.0, while simultaneously and synchronously dialing the probability of the |1> basis state up from 0.0 to 1.0. This is rotation about the Y axis of the Bloch sphere.
Phase. The phase angle between the two basis states. Dial the phase angle from 0.0 radians to two pi radians. Phase is commonly referred to as phi and is normalized to the range of 0.0 to 1.0 (and the negative values from minus 1.0 to zero as well), where 1.0 represents two pi radians and 0.5 represents pi radians or 180 degrees and 0.25 represents pi over two radians or 90 degrees. This is rotation about the Z axis of the Bloch sphere.

All rotations are relative — incremental. You never really know where the quantum state is currently at, you only know what incremental change you wish to make, such as to add 45-degrees of rotation of probability amplitude or phase.

Quantum state angles — theta and phi

Since quantum state is manipulated by rotations, it makes sense to visualize or understand quantum state in terms of two angles:

Theta. This angle represents the probability amplitude. From 0.0 to two pi radians (and the negative value from minus two pi radians to zero as well.) Pi represents a switch or flip between the two basis states, |0> to |1> or |1> to |0>. Pi over two represents that the two basis states have equal probability, the cleanest superposition. An angle other than a multiple of pi over two means than the two basis states have unequal probabilities. This represents a rotation about the Y axis of the Bloch sphere.
Phi. This angle represents the phase or phase angle between the two basis states. Although it can range from 0.0 to two pi radians (and the negative values from minus two pi radians to zero as well), it is commonly 0.0 or pi over two radians or minus pi over two radians. This represents a rotation about the Z axis of the Bloch sphere.

Quantum computing — All operations are relative, no absolute values may be used

Unlike in classical computing where a variable can trivially be set to any particular value, all operations on qubits in a quantum computer are relative — all operations are relative rotations about the three axes.

In short, you can’t set quantum state to a particular or desired value. Instead, you can manipulate quantum state via rotation about the three axes.

You can finesse and workaround this limitation to some extent, but the goal is to exploit this capability as a valuable feature, not a limitation.

Quantum algorithm design should be based on what quantum computers can do well, not how classical computers operate.

I won’t go into the workarounds here since they are more properly a function of quantum computing than quantum information itself.

Quantum computing — Quantum algorithms, quantum circuits and quantum logic gates

Again, not to delve too deeply into quantum computing itself, but just to provide some context:

Quantum algorithm. An algorithm intended to be run on a quantum computer rather than a classical computer.
Quantum circuit. The detailed implementation of a quantum algorithm. A sequence or graph of quantum logic gates, each of which performs one step of the algorithm.
Quantum logic gate. The basic operations of a quantum computer. These generally either rotate the quantum state of a single qubit or entangle two or more qubits. The details of how a quantum logic gate does its magic are specified using a unitary transformation matrix.

Quantum computing — Unitary transformation matrix

Again, not to delve too deeply into quantum computing itself, but just to provide some context, the technical details of each quantum logic gate of a quantum algorithm (quantum circuit) is specified using a unitary transformation matrix which essentially controls how the two angles which specify the quantum state of a qubit will be rotated, separately or together.

All of these terms are equivalent:

Unitary transformation matrix.
Unitary transform matrix.
Unitary transformation.
Unitary transform.
Unitary matrix.

Again, oversimplifying and simply for context, there is roughly an equivalence between unitary transformations and quantum logic gates. Both operate on the quantum states of qubits, which represent quantum information.

Quantum information science (QIS)

Quantum information science (QIS) is the umbrella concept covering everything related to quantum information processing:

Quantum computing.
Quantum communication.
Quantum networking.
Quantum metrology.
Quantum sensing.

In addition, there are common elements to all of those areas:

Quantum effects. The essential concepts from quantum mechanics and quantum physics which underpin quantum information science.
Quantum information. Information at the quantum level. From basic concepts to advanced theory. Applies across all of the areas of quantum information science. Binary basis states, continuous values of phase and probability amplitudes and probabilities, and strings of binary basis states for product states of entangled qubits.

For more on quantum information science, see my paper:

Quantum effects

Quantum effects are the essential concepts from quantum mechanics and quantum physics which underpin quantum information science.

The details of quantum effects are beyond the scope of this paper, but details can be found in my paper:

Classical Information Theory

Methods of considering how to represent, code, compress, and detect errors in classical information have been formalized in what is called Information Theory, most formally by Claude Shannon.

Quantum Information Theory

Unfortunately, there is still no quantum equivalent to Shannon’s classical Information Theory. There are plenty of bits and pieces, plenty of isolated papers, but no complete and coherent quantum information theory, yet.

A fair summary of the current state of affairs can be found in this Wikipedia article:

Additional detail can be found in the relevant section on quantum information theory of my research paper:

Quantum information

Quantum information is information at the quantum level, in contrast to classical information.

From basic concepts to advanced theory.

Applies across all of the areas of quantum information science.

Binary basis states, continuous values of phase and probability amplitudes and probabilities, and strings of binary basis states for product states of entangled qubits.

The details of quantum information are discussed throughout this paper, but just to give the flavor, some elements include elements from quantum effects and quantum mechanics:

Quantum system. Isolation and interaction.
Quantum state.
Wavefunctions.
Probability amplitude.
Phase.
Basis states. Typically binary, but can be higher order, such as 3 or 10 states.
Superposition.
Entanglement.
Product states.
Interference.
Measurement. Or observation.

Some additional detail can be found in the relevant section on quantum information theory of my research paper:

Need for the new field of quantum information theory

We need a wholly new field of quantum information theory consisting of:

Some elements of classical information theory.
But not all elements of classical information theory.
Additional elements of information theory which are quantum-specific. Especially from quantum mechanics.

The details of this new field are beyond the scope of this overview paper, but just to give the flavor of this new field, some elements will include elements from quantum mechanics:

Linear algebra. Not clear how much is needed here other than state vectors. Unitary transformation matrices are needed in quantum computer engineering and quantum computer science — execution and specification of quantum logic gates.
Quantum system. Isolation and interaction.
Quantum state.
Wavefunctions.
Probability amplitude.
Phase.
Basis states.
Superposition.
Entanglement.
Product states.
Interference.
Measurement. Or observation.
Errors. Detection. Mitigation. Automatic transparent correction.

Additional detail can be found in the relevant section on quantum information theory of my research paper:

Quantum information science vs. quantum information theory

I would prefer to use the term quantum information science rather than quantum information theory, but unfortunately the term quantum information science is already taken and used as the umbrella covering all of quantum information processing — quantum computing, quantum communication, quantum networking, quantum metrology, and quantum sensing — and quantum information theory.

So, we’re left with the prospect that any formalization of the concepts contained in this paper would be the basis for the new field of quantum information theory. Not that I personally would be involved in such formalization, but somebody would need to do it.

Quantum information science and technology (QIST)

You may see references to QIST — short for quantum information science and technology. It is generally a synonym for QIS — quantum information science. Generally, there is no intention to exclude technology from QIS. Essentially, QIST is redundant.

But, in some contexts QIS may be intended to focus on research and QIST is intended to emphasize the practical applications of the more theoretical and research aspects of QIS. When in doubt, just use QIS and read QIST and QIS as synonyms.

Quantum science

Some people use the term quantum science as a synonym or shorthand for quantum information science (QIS). Generally, it is better to use the full term, and to read the shorthand term as intending to refer to the full term.

Quantum science and technology

Similarly, some people use the term quantum science and technology as a synonym or shorthand for quantum information science and technology (QIST). Generally, it is better to use the full term, and to read the shorthand term as intending to refer to the full term.

Quantum technologies

Some people also use the term quantum technologies as equivalent to any of:

Quantum science.
Quantum information science (QIS).
Quantum information science and technology (QIST).

Generally you will have to look at the context to determine what they really mean. Frequently they won’t actually know and are treating all of these terms and concepts as synonyms — any technology which has a quantum aspect. Vague and ambiguous, and possibly ill-informed, but that’s the nature of a lot of writing anyways.

I would learn towards presuming that quantum technologies is emphasizing practical applications rather than theory and research, but you will have to look at the context to get a sense of whether the writer was intending to exclude theory and research.

Measurement of qubits and quantum information — collapse into classical bits

As most tutorials for quantum computing simply state, measurement of a qubit or any ensemble or collection of qubits causes the wave function to collapse, turning any quantum information into classical information — classical bits.

In the case of qubits in a state of superposition, it is the probabilistic bit value the quantum state will collapse into. It will indeed collapse into one of the two classical bit values, consistent with the probability derived from the probability amplitudes for the basis states.

In the case of product states for entangled qubits, measurement of any of the qubits will cause the collapse of the wave function for the full ensemble or collection of qubits, with each qubit taking on the classical bit value of the corresponding bit position in the product state bit string.

Measurement error

Unfortunately there is always the possibility that attempting to measure a qubit will result in measurement error so that not all of the classical bits will be read correctly. This will be true regardless of whether reading unentangled isolated qubits or entangled qubits in product states.

Readout and qubit readout — synonym for qubit measurement

In some papers or documentation you will encounter the term readout or qubit readout used instead of measurement or qubit measurement. They are generally synonyms.

Ditto for readout errors and measurement errors, and qubit readout errors and qubit measurement errors.

There is no clear sense of which term is preferred, but measurement has been more common in my own personal reading.

Besides, this is more a function of quantum computing itself rather than fundamental quantum information.

Collapse of the wave function — the inevitable result of measuring qubits

Just to restate the preceding section a little more specifically:

Measuring qubits causes the collapse of the wave function for their quantum state.
Collapse of the wave function for quantum state (quantum information) results in classical information (classical bits).

As a general proposition, there is no way to simply and directly access the quantum information (quantum state) of qubits other than through measurement which collapses the wave function(s) of the qubits.

Accessing the quantum state of qubits

That said, there are some technical workarounds for at least partially accessing the quantum state (quantum information) of qubits:

Quantum phase estimation (QPE).
Quantum amplitude estimation (QAE).
Quantum state tomography.

Unfortunately, these are all rather complex, nontrivial operations, left for more advanced quantum computing algorithms, and not for the faint of heart.

In short, yes, technically the details of quantum state (quantum information) can be accessed to some extent in a fashion, but not so easily.

Quantum phase estimation (QPE) and quantum amplitude estimation (QAE)

Quantum phase estimation (QPE) and quantum amplitude estimation (QAE) require the use of quantum Fourier transform and a number of qubits just to access the phase or probability amplitude of the quantum state of a single qubit.

For more on Quantum phase estimation (QPE), see the Qiskit textbook:

Quantum state tomography

Quantum state tomography is even more sophisticated and equally unwieldy to use in any typical quantum computing algorithm.

For more on quantum state tomography, see the Wikipedia:

Measurement of qubits is probabilistic, not deterministic

If you run the same quantum circuit repeatedly, each measurement of the qubits may return different results than for previous runs — at least for any qubit which was in a superposition.

Measurement of qubits which are not in superposition should always return the same classical bit values, but this is not a common use case since the whole point of a quantum algorithm is to perform probabilistic calculations which are not possible on a classical computer using only classical information.

It is also possible that even for a deterministic calculation the results can vary, typically due to errors, such as measurement errors, environmental interference, or crosstalk between qubits, which are quite common for current quantum computers — called NISQ devices because they are noisy.

Expectation value — Measurement of qubits as a statistical distribution, not a discrete value

A quantum circuit will typically be run n times, which is called the shot count (shots) or circuit repetitions. This will result in n arrangements (bit strings) of classical bits for the measured qubits, rather than a single discrete value. Rather than look at all n arrangements separately, some or many of which will be identical, it makes more sense to consider each arrangement of classical bits as a bitstring and to count how many of each unique bitstring occurred over the n runs of the quantum circuit.

The net result is a probability distribution graph for the unique bitstring values.

If you are lucky, there may indeed be only a single unique bitstring in the distribution. But even if your calculation was strictly deterministic there might be multiple unique bitstrings due to errors or even simply due to the probabilistic nature of quantum computing — and of quantum information itself.

This notion of a result value from a quantum computation is technically known as an expectation value. Technically, this is part of quantum computing, but it does relate to the translation between quantum information and classical information.

Under normal circumstances, if you are relatively lucky there will be a single bitstring which has the highest frequency in the distribution, so this can be chosen as the true result.

Unfortunately, there may be multiple bitstrings which have similar frequencies such that none of them has a frequency greater than 50% and none stands out as being clearly more frequent than the rest. Then it’s more of a fielder’s choice — and maybe any of the results is as good as any of the others that has a comparable frequency.

The whole point here is that even though measurement of qubits nominally produces classical bits, it also produces statistical information as well.

Whether you will be able to reduce that statistical information from a frequency distribution to a single collection of classical bits will vary from situation to situation, or even from quantum computer to quantum computer running the same identical quantum circuit.

For more detail on shots and measurements, see my paper:

Quantum information is probabilistic by nature rather than deterministic, unlike classical information

As a general proposition quantum information is probabilistic by nature and by definition rather than deterministic by nature and by definition as is classical information.

All quantum algorithms and applications need to be prepared for the fact that quantum information will always be probabilistic, by definition, by its very nature.

Statistical processing, such as circuit repetitions and other techniques can be used to approximate deterministic information, but in the end it is still only an approximation rather than a certainty.

Errors in quantum information

As with classical information, sometimes errors occur with quantum information.

As with classical information errors, there are several factors to consider:

Type or cause. What type of error is this? What caused it to happen?
Detection. How can we detect that the error occurred, or can we even tell at all?
Response. How can we deal with the error? Can we correct the error? Can we tolerate the error? Can we abort and retry the operation? Can or should we report it?

This paper won’t do a deep dive into errors in quantum information other than to touch on a few highlights.

For more detail on errors in quantum information, see my paper:

Types of errors in quantum information

Just to give a brief highlight of some of the types of errors that can occur in quantum information:

Decoherence. Gradual decay or dissipation of coherence of the quantum state over time.
Gate errors. Execution of quantum logic gates is not always flawless. Sometimes mistakes are made.
Crosstalk. Operations on one qubit may disrupt the quantum state of other nearby qubits.
Measurement errors. Even the simple act of reading a qubit can sometimes fail.
Environmental interference. Stray electromagnetic radiation or even cosmic rays or natural background radioactive decay can disrupt quantum state.

Undetected errors in quantum information

Not all errors in quantum errors in quantum information can be detected.

Sometimes apparent errors are simply natural variability due to the probabilistic nature and uncertainty of quantum mechanics.

One way to deal with undetected errors is to simply run the quantum computation more than once, a bunch of times, to develop a statistical distribution to determine which result is more common.

Response to errors in quantum information

Again, this paper won’t delve deeply into how to deal with errors in quantum information other than offer a few highlights.

Some potential responses to errors in quantum information — if the errors can be detected, otherwise see the preceding section:

Near-perfect qubits. Get the error rate as low as possible in the first place so that errors are not a common problem. Facilitates the other responses to errors.
Simply retry the execution of the quantum circuit. If the circuit is short enough this can be the cheapest, simplest, and easiest approach.
Manual error mitigation. Special operations in the quantum circuit or application to manually detect and correct failures.
Automatic transparent error correction. Called quantum error correction (QEC).The use of redundant hardware and special firmware operations to replicate effects similar to manual error mitigation, but automatically and transparently so that the application sees no errors — perfect logical qubits.

For more detail on errors in quantum information, see my paper:

Near-perfect qubits

The single best approach to errors in quantum information is to develop near-perfect qubits. Get the error rate as low as possible in the first place so that errors are not a common problem. This facilitates the other responses to errors.

If the error rate is low enough, redundant execution is a viable strategy — just execute the quantum circuit a few times and see which result is more common.

Quantum Error Correction (QEC)

Just to emphasize the special nature of automatic and transparent quantum error correction (QEC).

It will play a major role in the future of quantum computing, especially to enable fault-tolerant quantum computing using perfect logical qubits.

For more detail on quantum error correction (QEC), see my paper:

Error correction for continuous values such as probability amplitudes and phase is an interesting challenge — no solution has been suggested other than redundancy

Quantum computing inherits a wealth of automated error correction techniques from classical computing when it comes to errors in the binary values 0 and 1, but errors in continuous values such as probability amplitudes and phase are another matter and a very interesting challenge.

I personally can’t even recall seeing any mention of this issue in the literature, let alone any solutions.

In fact, I’m at a loss to suggest automatic error detection and correction strategies for continuous values other than redundancy and statistical processing — perform the same computation multiple times and see which values occur more frequently.

In any case, for now, I’ll leave it as an unsolved, unaddressed issue. Something for algorithm designers and application developers to be very aware of.

Coherence time

Although decoherence is one of the forms of errors for quantum information, it is more of a natural process than something unexpectedly going wrong as with the other types of errors listed earlier.

It is natural for quantum state to decay or decohere over time. The time it takes is alternatively referred to as the coherence time or the decoherence time — the time interval over which the quantum state remains in coherence or the time until the quantum state is no longer coherent.

The quantum state simply dissipates over time. There are two forms of dissipation:

1 decays to 0. The excited state of a qubit decays to the ground state.
Superposition decays. It’s not definitive what it might decay to, but it would no longer be in a clean and exact superposition of 1 and 0.

That said, some qubit technologies have substantially longer coherence times than others. Some can even maintain their coherence indefinitely, comparable to classical storage devices, or to classical logic devices which can maintain coherence as long as electrical power is maintained.

But a variety of current qubit technologies don’t possess this property of extended or indefinite coherence time, so coherence is a scarce resource.

Computing: Input, processing, and output

The traditional overall model of computing, which remains true even for quantum computing is a simple three stage process:

Input. Receive input data.
Processing. Perform the computation.
Output. Generate the output data or result.

Quantum computations cannot read input data — it must be encoded in the gate structure of the quantum circuit

A classical computation gets its input either as input parameters or by reading input data. Neither approach is available to quantum computations — all input data or input parameters for a quantum computation must be encoded in the gate structure of the quantum circuit.

The quantum circuit must fully contain any desired input data.

Quantum computations cannot output quantum data — final quantum state must be measured as classical data

A classical computation can produce output data in several ways, none of which are available to a quantum computation:

Return a function result.
Return output data.
Write output data.
Store output data in a database.
Send output data to a network service.

Instead, the only way to get data out of a quantum algorithm is to compute the data as quantum state in qubits and then measure those qubits, which collapses the quantum information into classical information.

Measurement of the quantum state in qubits causes the collapse of the wave function of the measured qubits, transforming quantum information (quantum state) into classical information (classical bits).

It is then up to classical software which invoked the quantum circuit to use, store, or output this classical information as it sees fit.

This may be far from an ideal solution, but it’s the best that quantum computing has to offer.

No rich data types

Although quantum computers offer tremendous performance advantages, they do so at the extreme expense of not providing support for rich data types or complex logic.

A quantum computer basically provides two data types:

Individual qubits. Isolated bits.
Entangled bit strings. Bit strings as product states.

That’s it.

Quantum computers lack support for the complex logic needed to support rich data types, such as:

Integers. Of variable sizes, such as 8-bit, 16-bit, 32-bit, and 64-bit.
Floating point real numbers. Of various precisions — single, double, quad.
Text strings. Of fixed or variable length. And of variable character size — such as the Unicode Transformation Formats (UTF): UTF-8, UTF-16, and UTF-32.
Structured data of any of the basic rich types.
Arrays of any of the structured or basic rich types. Fixed or variable length.
Lists of any of the structured or basic rich types.
Trees of any of the structured or basic rich types.
Graphs of any of the structured or basic rich types.
Data structures of arbitrary complexity.
Media. Images, sound, video. Recorded or streamed.

It will be up to classical software which invokes the quantum circuit to translate to or from such rich data types and the raw bits and bit strings of quantum information.

No persistent data

Quantum computers also lack any form of persistent data access from within quantum algorithms, such as:

Files.
File systems.
Databases.

It will be up to classical software which invokes the quantum circuit to access, store, or update persistent data, and any translation to or from rich data types and the raw bits and bit strings of quantum information.

No network service access

Quantum computers also lack any means for quantum algorithms to access network resources and services.

It will be up to classical software which invokes the quantum circuit to access network services and to translate to or from rich data types and the raw bits and bit strings of quantum information.

No Big Data, but a huge solution space

The concept of Big Data is all the rage in classical computing, but just doesn’t fit in quantum computing, where there is no external data access and no mass storage for large volumes of data.

What quantum computing and quantum information offer instead is a huge solution space. n fully entangled qubits offer 2^n quantum states (product states):

A mere 20 qubits offer a million quantum states (product states).
30 qubits offer a billion quantum states.
40 qubits offer a trillion quantum states.
50 qubits offer a quadrillion quantum states.
300 qubits offer 2³⁰⁰ product states or more product states than particles in the universe.

This means that even a relatively simple quantum computation can evaluate the computation over that many possible solutions. That’s a lot of computing power.

The Big Data of classical computing offers its own distinct advantages, but quantum computing also offers its own distinct advantages even if they are not the same as the distinct advantages of classical computing.

Little data with a big solution space

I’ll restate the key point of the preceding section more succinctly as quantum computing offering:

Little data with a big solution space.

Technically, it’s a little more accurate to state that as:

Little data with a little computation over a big solution space with a little output.

The quantum computation must be formulated as a relatively small number of steps and no complex logic, but that small computation is evaluated over the very large solution space. And then, a relatively small portion of the solution space must be selected to be the result of the computation.

For more on this concept of little data with a big solution space, see my paper:

True random number generation is natural for quantum information

True random number generation (TRNG) is a trivial operation on a quantum computer. In fact, there’s no computation which is easier. Technically it is a function of quantum computing, but it is enabled by the nature of quantum information.

To oversimplify, but not by much, if a qubit is placed in an equal superposition of 0 and 1 and then simply measured, it’s a 50/50 coin flip whether the result is a 0 or a 1. That operation can be repeated as many times as desired to generate as many random bits as desired.

For more information on true random number generation on a quantum computer, see my paper:

No-cloning theorem

A powerful constraint in quantum computing is the no-cloning theorem which prevents a quantum circuit from simply copying the quantum state of one qubit and replicating it exactly in another qubit. Or more technically, you can’t replicate the quantum state of one quantum system in another quantum system.

This limitation of quantum computing doesn’t affect quantum information per se, but may need to be taken into account when attempting to use quantum information.

For more on the no cloning theorem, see the Wikipedia:

No copying or cloning of quantum information, no read or examine

Expanding on the no-cloning theorem mentioned in the preceding section, you can’t do any examination of quantum state:

You can’t copy quantum state.
You can’t read quantum state.
You can’t examine quantum state.

The only things you can do with quantum state are:

Store it.
Move it. Intact. But you can’t examine it.
Swap it. Intact. But you can’t examine it.
Manipulate it. Rotations.
Entangle it.
Measure it. This does read it, but at the cost of collapsing the wave function of the quantum state into a strictly classical state.

That said, as stated in a preceding section, there are some technical workarounds for at least partially accessing the quantum state (quantum information) of qubits:

Quantum phase estimation (QPE).
Quantum amplitude estimation (QAE).
Quantum state tomography.

But those are only partial workarounds and don’t permit fully copying or fully examining the full quantum state of a qubit or other quantum system.

Photonic quantum information — Continuous-variable (CV) qumodes and squeezed states

As stated at the outset, continuous-variable (CV) qumodes and squeezed states as found on photonic quantum computers are beyond the scope of this paper.

It’s not that photonic quantum computing is not of interest, but simply that it is completely incompatible with quantum information on non-photonic quantum computers.

For more information, consult the documentation from Xanadu, one of the main proponents and vendors of photonic computing:

Quantum memory and quantum storage

Two interesting areas of research are quantum memory and quantum storage. They are commonly conceptualized as intermediate stages for quantum communication and quantum networking — store and forward buffering of messages. They can also be considered for another research area, persistent quantum state.

The essential concept is some physical mechanism to capture, preserve, and later transfer quantum information (in the form of quantum state) for some extended period of time, considerably longer than the typical lifetime of a qubit in a quantum computer.

This is a nascent research area, so it has no near-term implications for quantum computing, but in the longer term it will become increasingly relevant for working with quantum information, particularly for distributed quantum computing.

All of the concepts of this paper will be relevant for quantum memory, quantum storage, and persistent quantum state.

Programming models

Programming models are a hybrid between information (or data) and logic (or code.) They describe how data or information is stored, organized, and accessed, as well as what operations or logic can be performed on the information and how.

This paper focuses on the information half of the programming model equation.

The code or logic half of the equation is properly the realm of quantum computing.

Some unresolved or open issues for me

Just some issues for which I don’t have the exact correct answer at my fingertips at this time:

What is phase or phase angle when there are more than two basis states or product states? Although each probability amplitude can have its own phase (the imaginary part of the complex number), generally there is a single normalized phase angle between the two basis states. But what happens when phase is nonzero and there are three or more basis states or product states, such as superposition of qutrits or qudits, GHZ and W states, and more than two entangled qubits in general?
What do these symbols stand for? I’ve seen them but can’t recall where and Google search can’t search for non-text symbols: |++> and | →, |+-> and |-+>?

My original proposal for this topic

For reference, here is the original proposal I had for this topic. It may have some value for some people wanting a more concise summary of this paper.

What is quantum information? Tricky. Lots of hype. Qubits vs. quantum state vs. subset of quantum state. No definitive definition. Somewhere between qubits and quantum state. Information vs. representation. Are probabilities “information” per se? Is phase “information” per se? Does phase represent separate information from the information implied by probability amplitude? Basis states are binary, but phase and probability amplitude are continuous values. How much of quantum state is mere representation rather than actual, abstract information? Product states — strings of basis states — vs. individual qubits.

Summary and conclusions

Everything You Need to Know About Quantum Information to Understand Quantum Computing. The whole point of this paper as accurately as I can express it.
The real interest is in quantum computing, but quantum information is the necessary foundation which is needed first before we can even begin to discuss quantum computing properly.
Quantum information processing is the broader domain of processing quantum information, including quantum computing, quantum communication, quantum networking, quantum metrology, and quantum sensing.
Focus of this paper is quantum information in the context of two-level quantum systems for quantum computing. The world of quantum bits or qubits.
Starting point for quantum information is the same as classical information: binary 0 and 1. As basis states.
Then add probability amplitude as a continuous value for each basis state. A real number from 0.0 to 1.0. Actually it is a complex number with a real and imaginary part, each from 0.0 to 1.0 (and the negative values from minus 1.0 to zero as well.)
Superposition allows both basis states at the same time, each with its own probability.
Square the probability amplitude to get the probability for a basis state.
Squares of probabilities for each basis state must sum to 1.0.
And add phase as a continuous value to each basis state. Actually, there is a single phase angle between the two basis states. Phase is represented as the complex part of the probability amplitude.
Phase is nominally an angle in radians from 0.0 to two pi radians (a full circle or a single cycle), but generally it is normalized to the range 0.0 to 1.0 (and the negative values from minus 1.0 to zero as well) where 1.0 represents a full circle or two pi radians. A phase of 0.5 represents a half circle, pi radians, or 180 degrees. A phase of 0.25 represents a quarter circle, pi over two radians, or 90 degrees.
Entangled quantum states give an exponential number of product states. 2^n product states for n qubits or n quantum states. Each product state corresponds to a bitstring of n classical bits corresponding to the binary representation of its position in the range 0 to 2^n minus one.
Basis states are discrete values while probability amplitudes and phase are continuous values. This enables quantum computing to be a hybrid of the digital world of classical computing and the continuous world of analog computing.
Uncertainty. Quantum information is inherently probabilistic rather than being strictly deterministic. Simultaneously a blessing and a curse, just as with fire. You can’t get superposition, entanglement, and quantum parallelism without accepting the uncertainty.
It is important to distinguish storage devices, representations, and information itself. Information is represented in a representation. Representations are stored in devices.
Qubit vs. bit is not an accurate comparison. A bit is abstract information while a qubit is a storage device which stores quantum state which is a representation of information.
Qubits are storage devices, not quantum information per se. Qubits store quantum state.
Quantum information is represented as quantum state stored in qubits.
There is no support for Big Data but there is support for a huge solution space with 2^k quantum states. I call it little data with a big solution space.
No copying or cloning of quantum information. No read or examine for quantum state or quantum information. There are some exceptions or workarounds.
Any number of qubits or quantum states can be entangled.
Classical and non-classical states for representation of information.
Classical and non-classical information — the information itself, independent from its representation.
Storage and representation — separate from the information itself, how the information is represented and stored.
No rich data types in the quantum world. Just basis states and strings of basis states.
No complex logic supported.
Quantum information is probabilistic by definition and by its very nature rather than deterministic by definition and by nature as is classical information.
Errors are a reality in all information. Resilience, detection, mitigation, and automatic correction are needed.
Near-perfect qubits. Get the error rate as low as possible in the first place so that errors are not a common problem. Facilitates the other responses to errors.
Error correction for continuous values such as probability amplitudes and phase is an interesting challenge. No solution has been suggested other than redundancy.
True random number generation is natural for quantum information. Technically a function of quantum computing, but enabled by the nature of quantum information.
Quantum information is the foundation for quantum computing and quantum information processing in general, including quantum communication and quantum networking.
Programming models are a hybrid between information (or data) and logic (or code.) This paper focuses on the information half of the programming model equation.
Hopefully this paper will provide a sound foundation for those wishing to dive into quantum computing itself.
Attempting to leap too deeply into quantum computing without a firm grasp of the full breadth, depth, scope, and nuance of quantum information would be a huge mistake.

For more of my writing: List of My Papers on Quantum Computing.