This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations .(November 2011) |

Optical lattices use lasers to separate rubidium atoms (red) for use as information bits in neutral-atom quantum processors—prototype devices which designers are trying to develop into full-fledged quantum computers. Credit: NIST |

**Quantum information** is the information of the state of a quantum system. It is the basic entity of study in **quantum information theory**,^{ [2] } and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.

- History and development
- Development from fundamental quantum mechanics
- Development from information theory
- Qubits and information theory
- Quantum information processing
- Relation to quantum mechanics
- Entropy and information
- Classical information
- Quantum information
- Applications
- Journals
- See also
- Notes
- References

It is an interdisciplinary field that involves quantum mechanics, computer science, information theory, philosophy and cryptography among other fields.^{ [3] }^{ [4] } Its study is also relevant to disciplines such as cognitive science, psychology and neuroscience.^{ [5] } Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, as an eigenstate in one basis is not an eigenstate in the other basis. As both variables are not simultaneously well defined, a quantum state can never contain definitive information about both variables.^{ [5] }

Information is something that is encoded in the state of a quantum system,^{ [6] } it is physical.^{ [6] } While quantum mechanics deals with examining properties of matter at the microscopic level,^{ [7] }^{ [5] } quantum information science focuses on extracting information from those properties,^{ [5] } and quantum computation manipulates and processes information – performs logical operations – using quantum information processing techniques.^{ [8] }

Quantum information, like classical information, can be processed using digital computers, transmitted from one location to another, manipulated with algorithms, and analyzed with computer science and mathematics. Just like the basic unit of classical information is the bit, quantum information deals with qubits. Quantum information can be measured using Von Neumann entropy.

Recently, the field of quantum computing has become an active research area because of the possibility to disrupt modern computation, communication, and cryptography.^{ [8] }

The history of quantum information began at the turn of the 20th century when classical physics was revolutionized into quantum physics. The theories of classical physics were predicting absurdities such as the ultraviolet catastrophe, or electrons spiraling into the nucleus. At first these problems were brushed aside by adding ad hoc hypothesis to classical physics. Soon, it became apparent that a new theory must be created in order to make sense of these absurdities, and the theory of quantum mechanics was born.^{ [2] }

Quantum mechanics was formulated by Schrodinger using wave mechanics and Heisenberg using matrix mechanics.^{ [9] } The equivalence of these methods was proven later.^{ [10] } Their formulations described the dynamics of microscopic systems but had several unsatisfactory aspects in describing measurement processes. Von Neumann formulated quantum theory using operator algebra in a way that it described measurement as well as dynamics.^{ [11] } These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements.

See: Dynamical Pictures

Evolution | Picture ( ) | ||

of: | Heisenberg | Interaction | Schrödinger |

Ket state | constant | ||

Observable | constant | ||

Density matrix | constant |

In 1960s, Stratonovich, Helstrom and Gordon^{ [12] } proposed a formulation of optical communications using quantum mechanics. This was the first historical appearance of quantum information theory. They mainly studied error probabilities and channel capacities for communication.^{ [12] }^{ [13] } Later, Holevo obtained an upper bound of communication speed in the transmission of a classical message via a quantum channel.^{ [14] }^{ [15] }

In the 1970s techniques of manipulating single quantum states such as the atom trap and the scanning tunneling microscope were starting to get developed. Isolating single atoms and moving them around to fashion an array of atoms at will was starting to become a reality. Prior to these developments, complete control over single quantum systems was not possible, and techniques involved a somewhat coarse level of control over a large number of quantum systems, none of which individually were directly accessible.^{ [2] } Extracting and manipulating information stored in individual atoms naturally started to become an interesting avenue, and quantum information and computation was starting to get developed.

In the 1980s, interest arose in whether it might be possible to use quantum effects to signal faster than light, an attempt of disproving Einstein's theory of relativity. If cloning an unknown quantum state were possible then Einstein's theory could be disproved. However, it turns out that quantum states could not, in general, be cloned. The no-cloning theorem is one of the earliest results of quantum information.^{ [2] }

Despite all the excitement and interest over studying isolated quantum systems and trying to find a way to circumvent the theory of relativity, research in quantum information theory became stagnant in the 1980s. However, around the same time another avenue started dabbling into quantum information and computation: Cryptography. In a general sense, *cryptography is the problem of doing communication or computation involving two or more parties who may not trust one another.*^{ [2] }

Bennett and Brassard developed a communication channel on which it is impossible eavesdrop without being detected, a way of communicating secretly at long distances using the BB84 quantum cryptographic protocol.^{ [16] } The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of a eavesdropper in a secure communication line will immediately let the two parties trying to communicate would know of the presence of the eavesdropper.

With the advent of Alan Turing's revolutionary ideas of a programmable computer, or Turing machine, he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine.^{ [17] }^{ [18] } This is known as the Church–Turing thesis.

Soon enough, the first computers were made and computer hardware grew at such a fast pace that the growth, through experience in production, was codified into an empirical relationship called Moore's law. This 'law' is a projective trend that states that the number of transistors in an integrated circuit doubles every two years.^{ [19] } As transistors began to become smaller and smaller in order to pack more power per surface area, quantum effects started to show up in the electronics resulting in inadvertent interference. This led to the advent of quantum computing, which used quantum mechanics to design algorithms.

At this point, quantum computers showed promise of being much faster than classical computers for certain specific problems. One such example problem was developed by David Deutsch and Richard Jozsa, known as the Deutsch–Jozsa algorithm. This problem however held little to no practical applications.^{ [2] } Peter Shor in 1994 came up with a very important and practical problem, one of finding the prime factors of an integer. The discrete logarithm problem as it was called, could be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers are more powerful than Turing machines.

Around the time computer science was making a revolution, so was information theory and communication, through Claude Shannon.^{ [20] }^{ [21] }^{ [22] } Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and noisy channel coding theorem. He also showed that error correcting codes could be used to protect information being sent.

Quantum information theory also followed a similar trajectory, Ben Schumacher in 1995 made an analogue to Shannon's noiseless coding theorem using the qubit. A theory of error-correction also developed, which allows quantum computers to make efficient computations regardless of noise, and make reliable communication over noisy quantum channels.^{ [2] }

Quantum information differs strongly from classical information, epitomized by the bit, in many striking and unfamiliar ways. While the fundamental unit of classical information is the bit, the most basic unit of quantum information is the qubit. Classical information is measured using Shannon entropy, while the quantum mechanical analogue is Von Neumann entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix , it is given by ^{ [2] } Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy^{ [23] } and the conditional quantum entropy.

Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the Bloch sphere. Despite being continuously valued in this way, a qubit is the *smallest* possible unit of quantum information, and despite the qubit state being continuous-valued, it is impossible to measure the value precisely. Five famous theorems describe the limits on manipulation of quantum information.^{ [2] }

- no-teleportation theorem, which states that a qubit cannot be (wholly) converted into classical bits; that is, it cannot be fully "read".
- no-cloning theorem, which prevents an arbitrary qubit from being copied.
- no-deleting theorem, which prevents an arbitrary qubit from being deleted.
- no-broadcast theorem, which prevents an arbitrary qubit from being delivered to multiple recipients, although it can be transported from place to place (
*e.g.*via quantum teleportation). - no-hiding theorem, which demonstrates the conservation of quantum information.

These theorems prove that quantum information within the universe is conserved. They open up possibilities in quantum information processing.

The state of a qubit contains all of its information. This state is frequently expressed as a vector on the Bloch sphere. This state can be changed by applying linear transformations or quantum gates to them. These unitary transformations are described as rotations on the Bloch Sphere. While classical gates correspond to the familiar operations of Boolean logic, quantum gates are physical unitary operators.

- Due to the volatility of quantum systems and the impossibility of copying states, the storing of quantum information is much more difficult than storing classical information. Nevertheless, with the use of quantum error correction quantum information can still be reliably stored in principle. The existence of quantum error correcting codes has also led to the possibility of fault-tolerant quantum computation.
- Classical bits can be encoded into and subsequently retrieved from configurations of qubits, through the use of quantum gates. By itself, a single qubit can convey no more than one bit of accessible classical information about its preparation. This is Holevo's theorem. However, in superdense coding a sender, by acting on one of two entangled qubits, can convey two bits of accessible information about their joint state to a receiver.
- Quantum information can be moved about, in a quantum channel, analogous to the concept of a classical communications channel. Quantum messages have a finite size, measured in qubits; quantum channels have a finite channel capacity, measured in qubits per second.
- Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue of Shannon entropy, called the von Neumann entropy.
- In some cases quantum algorithms can be used to perform computations faster than in any known classical algorithm. The most famous example of this is Shor's algorithm that can factor numbers in polynomial time, compared to the best classical algorithms that take sub-exponential time. As factorization is an important part of the safety of RSA encryption, Shor's algorithm sparked the new field of post-quantum cryptography that tries to find encryption schemes that remain safe even when quantum computers are in play. Other examples of algorithms that demonstrate quantum supremacy include Grover's search algorithm, where the quantum algorithm gives a quadratic speed-up over the best possible classical algorithm. The complexity class of problems efficiently solvable by a quantum computer is known as BQP.
- Quantum key distribution (QKD) allows unconditionally secure transmission of classical information, unlike classical encryption, which can always be broken in principle, if not in practice. Do note that certain subtle points regarding the safety of QKD are still hotly debated.

The study of all of the above topics and differences comprises quantum information theory.

Quantum mechanics is the study of how microscopic physical systems change dynamically in nature. In the field of quantum information theory, the quantum systems studied are abstracted away from any real world counterpart. A qubit might for instance physically be a photon in a linear optical quantum computer, an ion in a trapped ion quantum computer, or it might be a large collection of atoms as in a superconducting quantum computer. Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of density matrices over the complex numbers. Another important difference with quantum mechanics is that, while quantum mechanics often studies infinite-dimensional systems such as a harmonic oscillator, quantum information theory concerns both with continuous-variable systems^{ [24] } and finite-dimensional systems.^{ [25] }^{ [26] }^{ [27] }

Entropy measures the uncertainty in the state of a physical system.^{ [2] } Entropy can be studied from the point of view of both the classical and quantum information theories.

Classical information is based on the concepts of information laid out by Claude Shannon. Classical information, in principle, can be stored in a bit of binary strings. Any system having two states is a capable bit.^{ [28] }

Shannon entropy is the quantification of the information gained by measuring the value of a random variable. Another way of thinking about it is by looking at the uncertainty of a system prior to measurement. As a result, entropy, as pictured by Shannon, can be seen either as a measure of the uncertainty prior to making a measurement or as a measure of information gained after making said measurement.^{ [2] }

Shannon entropy, written as a functional of a discrete probability distribution, associated with events , can be seen as the average information associated with this set of events, in units of bits:

This definition of entropy can be used to quantify the physical resources required to store the output of an information source. The ways of interpreting Shannon entropy discussed above are usually only meaningful when the number of samples of an experiment is large.^{ [29] }

The Rényi entropy is a generalization of Shannon entropy defined above. The Rényi entropy of order r, written as a function of a discrete probability distribution, , associated with events , is defined as:^{ [28] }

for and .

We arrive at the definition of Shannon entropy from Rényi when , of Hartley entropy (or max-entropy) when , and min-entropy when .

Quantum information theory is largely an extension of classical information theory to quantum systems. Classical information is produced when measurements of quantum systems are made.^{ [28] }

One interpretation of Shannon entropy was the uncertainty associated with a probability distribution. When we want to describe the information or the uncertainty of a quantum state, the probability distributions are simply swapped out by density operators .

s are the eigenvalues of .

Von Neumann plays a similar role in quantum information that Shannon entropy does in classical information

**Quantum communication**

Quantum communication is one of the applications of quantum physics and quantum information. There are some famous theorems such as the no-cloning theorem that illustrate some important properties in quantum communication. Dense coding and quantum teleportation are also applications of quantum communication. They are two opposite ways to communicate using qubits. While teleportation transfers one qubit from Alice and Bob by communicating two classical bits under the assumption that Alice and Bob have a pre-shared Bell state, dense coding transfers two classical bits from Alice to Bob by using one qubit, again under the same assumption, that Alice and Bob have a pre-shared Bell state.

**Quantum key distribution**

One of the best known applications of quantum cryptography is quantum key distribution which provide a theoretical solution to the security issue of a classical key. The advantage of quantum key distribution is that it is impossible to copy a quantum key because of the no-cloning theorem. If someone tries to read encoded data, the quantum state being transmitted will change. This could be used to detect eavesdropping.

**BB84**

The first quantum key distribution scheme BB84, developed by Charles Bennett and Gilles Brassard in 1984. It is usually explained as a method of securely communicating a private key from a third party to another for use in one-time pad encryption.^{ [2] }

**E91**

E91 was made by Artur Ekert in 1991. His scheme uses entangled pairs of photons. These two photons can be created by Alice, Bob, or by a third party including eavesdropper Eve. One of the photons is distributed to Alice and the other to Bob so that each one ends up with one photon from the pair.

This scheme relies on two properties of quantum entanglement:

- The entangled states are perfectly correlated which means that if Alice and Bob both measure their particles having either a vertical or horizontal polarization, they always get the same answer with 100% probability. The same is true if they both measure any other pair of complementary (orthogonal) polarizations. This necessitates that the two distant parties have exact directionality synchronization. However, from quantum mechanics theory the quantum state is completely random so that it is impossible for Alice to predict if she will get vertical polarization or horizontal polarization results.
- Any attempt at eavesdropping by Eve destroys this quantum entanglement such that Alice and Bob can detect.

**B92**

B92 is a simpler version of BB84.^{ [30] }

The main difference between B92 and BB84:

- B92 only needs two states
- BB84 needs 4 polarization states

Like the BB84, Alice transmits to Bob a string of photons encoded with randomly chosen bits but this time the bits Alice chooses the bases she must use. Bob still randomly chooses a basis by which to measure but if he chooses the wrong basis, he will not measure anything which is guaranteed by quantum mechanics theories. Bob can simply tell Alice after each bit she sends whether or not he measured it correctly.^{ [31] }

**Quantum computation**

The most widely used model in quantum computation is the quantum circuit, which are based on the quantum bit "qubit". Qubit is somewhat analogous to the bit in classical computation. Qubits can be in a 1 or 0 quantum state, or they can be in a superposition of the 1 and 0 states. However, when qubits are measured the result of the measurement is always either a 0 or a 1; the probabilities of these two outcomes depend on the quantum state that the qubits were in immediately prior to the measurement.

Any quantum computation algorithm can be represented as a network of quantum logic gates.

**Quantum decoherence**

If a quantum system were perfectly isolated, it would maintain coherence perfectly, but it would be impossible to test the entire system. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; this process is called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics.

**Quantum error correction**

**QEC** is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.

Peter Shor first discovered this method of formulating a *quantum error correcting code* by storing the information of one qubit onto a highly entangled state of ancilla qubits. A quantum error correcting code protects quantum information against errors.

Many journals publish research in quantum information science, although only a few are dedicated to this area. Among these are:

*International Journal of Quantum Information**Quantum Information & Computation**Quantum Information Processing**npj Quantum Information*^{ [32] }*Quantum*^{ [33] }*Quantum Science and Technology*^{ [34] }

- ↑ "Free-Images.com – Free Public Domain Images".
*free-images.com*. Retrieved 2020-11-13. - 1 2 3 4 5 6 7 8 9 10 11 12 Nielsen, Michael A. (2010).
*Quantum computation and quantum information*. Chuang, Isaac L. (10th anniversary ed.). Cambridge: Cambridge University Press. ISBN 978-1107002173. OCLC 665137861. - ↑ Bokulich, Alisa; Jaeger, Gregg (2010).
*Philosophy of Quantum Information and Entanglement*. Cambridge University Press. ISBN 978-1-139-48766-5. - ↑ Benatti, Fabio; Fannes, Mark; Floreanini, Roberto; Petritis, Dimitri (2010).
*Quantum Information, Computation and Cryptography: An Introductory Survey of Theory, Technology and Experiments*. Springer Science & Business Media. ISBN 978-3-642-11913-2. - 1 2 3 4 Hayashi, Masahito (2017).
*Quantum Information Theory*. Graduate Texts in Physics. doi:10.1007/978-3-662-49725-8. ISBN 978-3-662-49723-4.^{[ page needed ]} - 1 2 Preskill, John. "Lecture notes for physics 229."
*Quantum information and computation*16 (1998). - ↑ "The Feynman Lectures on Physics Vol. III Ch. 1: Quantum Behavior".
*wayback.archive-it.org*. Retrieved 2020-11-11. - 1 2 Lo, Hoi-Kwong; Spiller, Tim; Popescu, Sandu (1998).
*Introduction to Quantum Computation and Information*. World Scientific. ISBN 978-981-02-4410-1. - ↑ Mahan, Gerald D. (2008-12-29).
*Quantum Mechanics in a Nutshell*. Princeton University Press. doi:10.2307/j.ctt7s8nw. ISBN 978-1-4008-3338-2. JSTOR j.ctt7s8nw. - ↑ Perlman, H. S. (November 1964). "Equivalence of the Schroedinger and Heisenberg Pictures".
*Nature*.**204**(4960): 771–772. Bibcode:1964Natur.204..771P. doi:10.1038/204771b0. S2CID 4194913. - ↑ Neumann, John von (2018-02-27).
*Mathematical Foundations of Quantum Mechanics: New Edition*. Princeton University Press. ISBN 978-0-691-17856-1. - 1 2 Gordon, J. (September 1962). "Quantum Effects in Communications Systems".
*Proceedings of the IRE*.**50**(9): 1898–1908. doi:10.1109/JRPROC.1962.288169. S2CID 51631629. - ↑ Helstrom, Carl Wilhelm (1976).
*Quantum Detection and Estimation Theory*. Academic Press. ISBN 978-0-08-095632-9.^{[ page needed ]} - ↑ A. S. Holevo, “Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel”, Probl. Peredachi Inf., 9:3 (1973); Problems Inform. Transmission, 9:3 (1973), http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=903&option_lang=eng
- ↑ A. S. Holevo, “On Capacity of a Quantum Communications Channel”, Probl. Peredachi Inf., 15:4 (1979); Problems Inform. Transmission, 15:4 (1979), http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=1507&option_lang=eng
- ↑ Bennett, Charles H.; Brassard, Gilles (December 2014). "Quantum cryptography: Public key distribution and coin tossing".
*Theoretical Computer Science*.**560**: 7–11. arXiv: 2003.06557 . doi:10.1016/j.tcs.2014.05.025. S2CID 27022972. - ↑ Weisstein, Eric W. "Church–Turing Thesis".
*mathworld.wolfram.com*. Retrieved 2020-11-13. - ↑ Deutsch, D. (8 July 1985). "Quantum theory, the Church–Turing principle and the universal quantum computer".
*Proceedings of the Royal Society of London A: Mathematical and Physical Sciences*.**400**(1818): 97–117. Bibcode:1985RSPSA.400...97D. doi:10.1098/rspa.1985.0070. S2CID 1438116. - ↑ Moore, G.E. (January 1998). "Cramming More Components Onto Integrated Circuits".
*Proceedings of the IEEE*.**86**(1): 82–85. doi:10.1109/JPROC.1998.658762. S2CID 6519532. - ↑ Shannon, C. E. (October 1948). "A Mathematical Theory of Communication".
*Bell System Technical Journal*.**27**(4): 623–656. doi:10.1002/j.1538-7305.1948.tb00917.x. hdl: 11858/00-001M-0000-002C-4314-2 . - ↑ "A Mathematical Theory of Communication" (PDF). 1998-07-15. Archived (PDF) from the original on 1998-07-15. Retrieved 2020-11-13.
- ↑ Shannon, C. E.; Weaver, W. (1949). "The mathematical theory of communication".Cite journal requires
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(help) - ↑ "Alexandr S. Holevo".
*Mi.ras.ru*. Retrieved 4 December 2018. - ↑ Weedbrook, Christian; Pirandola, Stefano; García-Patrón, Raúl; Cerf, Nicolas J.; Ralph, Timothy C.; Shapiro, Jeffrey H.; Lloyd, Seth (1 May 2012). "Gaussian quantum information".
*Reviews of Modern Physics*.**84**(2): 621–669. arXiv: 1110.3234 . Bibcode:2012RvMP...84..621W. doi:10.1103/RevModPhys.84.621. S2CID 119250535. - ↑ Masahito Hayashi, "Quantum Information Theory: Mathematical Foundation"
^{[ page needed ]} - ↑ J. Watrous, The Theory of Quantum Information (Cambridge Univ. Press, 2018). Freely available at
- ↑ Wilde, Mark M. (2013). "Concepts in Quantum Shannon Theory".
*Quantum Information Theory*. pp. 3–25. doi:10.1017/cbo9781139525343.002. ISBN 9781139525343. - 1 2 3 Jaeger, Gregg (2007-04-03).
*Quantum Information: An Overview*. Springer Science & Business Media. ISBN 978-0-387-36944-0. - ↑ Watrous, John. "CS 766/QIC 820 Theory of Quantum Information (Fall 2011)." (2011).
- ↑ Bennett, Charles H. (1992-05-25). "Quantum cryptography using any two nonorthogonal states".
*Physical Review Letters*.**68**(21): 3121–3124. Bibcode:1992PhRvL..68.3121B. doi:10.1103/PhysRevLett.68.3121. PMID 10045619. - ↑ "Quantum Key Distribution - QKD".
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*IOP Publishing*. Retrieved 12 January 2019.

**Information theory** is the scientific study of the quantification, storage, and communication of digital information. The field was fundamentally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering.

**Quantum computing** is the exploitation of collective properties of quantum states, such as superposition and entanglement, to perform computation. The devices that perform quantum computations are known as **quantum computers**. They are believed to be able to solve certain computational problems, such as integer factorization, substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. Expansion is expected in the next few years as the field shifts toward real-world use in pharmaceutical, data security and other applications.

**Quantum teleportation** is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Moreover, the sender may not know the location of the recipient, and does not know which particular quantum state will be transferred.

In quantum computing, a **qubit** or **quantum bit** is the basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.

**Quantum entanglement** is a physical phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics lacking in classical mechanics.

In quantum mechanics, a **density matrix** is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent *mixed states*. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.

In quantum physics, a **measurement** is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

**Quantum information science** is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in physics. It includes theoretical issues in computational models and more experimental topics in quantum physics, including what can and cannot be done with quantum information. The term **quantum information theory** is also used, but it fails to encompass experimental research, and can be confused with a subfield of quantum information science that addresses the processing of quantum information.

**Quantum error correction** (**QEC**) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.

The **joint quantum entropy** generalizes the classical joint entropy to the context of quantum information theory. Intuitively, given two quantum states and , represented as density operators that are subparts of a quantum system, the joint quantum entropy is a measure of the total uncertainty or entropy of the joint system. It is written or , depending on the notation being used for the von Neumann entropy. Like other entropies, the joint quantum entropy is measured in bits, i.e. the logarithm is taken in base 2.

In quantum information theory, a **quantum channel** is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information is a text document transmitted over the Internet.

In quantum statistical mechanics, the **von Neumann entropy**, named after John von Neumann, is the extension of classical Gibbs entropy concepts to the field of quantum mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is

**Quantum neural networks** are computational neural network models which are based on the principles of quantum mechanics. The first ideas on quantum neural computation were published independently in 1995 by Subhash Kak and Ron Chrisley, engaging with the theory of quantum mind, which posits that quantum effects play a role in cognitive function. However, typical research in quantum neural networks involves combining classical artificial neural network models with the advantages of quantum information in order to develop more efficient algorithms. One important motivation for these investigations is the difficulty to train classical neural networks, especially in big data applications. The hope is that features of quantum computing such as quantum parallelism or the effects of interference and entanglement can be used as resources. Since the technological implementation of a quantum computer is still in a premature stage, such quantum neural network models are mostly theoretical proposals that await their full implementation in physical experiments.

**Holevo's theorem** is an important limitative theorem in quantum computing, an interdisciplinary field of physics and computer science. It is sometimes called **Holevo's bound**, since it establishes an upper bound to the amount of information that can be known about a quantum state. It was published by Alexander Holevo in 1973.

In quantum information theory, the **no-teleportation theorem** states that an arbitrary quantum state cannot be converted into a sequence of classical bits ; nor can such bits be used to reconstruct the original state, thus "teleporting" it by merely moving classical bits around. Put another way, it states that the unit of quantum information, the qubit, cannot be exactly, precisely converted into classical information bits. This should not be confused with quantum teleportation, which does allow a quantum state to be destroyed in one location, and an exact replica to be created at a different location.

Until recently, most studies on time travel are based upon classical general relativity. Coming up with a quantum version of time travel requires physicists to figure out the time evolution equations for density states in the presence of closed timelike curves (CTC).

A **quantum depolarizing channel** is a model for quantum noise in quantum systems. The -dimensional depolarizing channel can be viewed as a completely positive trace-preserving map , depending on one parameter , which maps a state onto a linear combination of itself and the maximally mixed state,

In quantum information theory, **quantum discord** is a measure of nonclassical correlations between two subsystems of a quantum system. It includes correlations that are due to quantum physical effects but do not necessarily involve quantum entanglement.

The **noisy-storage model** refers to a cryptographic model employed in quantum cryptography. It assumes that the quantum memory device of an attacker (adversary) trying to break the protocol is imperfect (noisy). The main goal of this model is to enable the secure implementation of two-party cryptographic primitives, such as bit commitment, oblivious transfer and secure identification.

**Algorithmic cooling** is an algorithmic method for transferring heat from some qubits to others or outside the system and into the environment, which results in a cooling effect. This method uses regular quantum operations on ensembles of qubits, and it can be shown that it can succeed beyond Shannon's bound on data compression. The phenomenon is a result of the connection between thermodynamics and information theory.

- Bennett, C.H.; Shor, P.W. (1998). "Quantum information theory".
*IEEE Transactions on Information Theory*.**44**(6): 2724–2742. doi:10.1109/18.720553. - Gregg Jaeger's book on Quantum Information, Springer, New York, 2007, ISBN 0-387-35725-4
- Lectures at the Institut Henri Poincaré (slides and videos)
- International Journal of Quantum Information World Scientific
- Quantum Information Processing Springer
- Michael A. Nielsen, Isaac L. Chuang, "Quantum Computation and Quantum Information"
- J. Watrous, The Theory of Quantum Information (Cambridge Univ. Press, 2018). Freely available at
- John Preskill, Course Information for Physics 219/Computer Science 219 Quantum Computation, Caltech
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