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{{short description|Information held in the state of a quantum system}}
{{for|the journal|npj Quantum Information}}
[[File:Qubits_(5940500587).jpg|thumb|upright=1|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.]]
'''Quantum information''' is the [[information]] of the [[quantum state|state]] of a [[quantum system]]. It is the basic entity of study in [[Quantum information science|quantum information theory]],<ref name="Vedral2006"/><ref name="Nielsen2010"/><ref name="Hayashi2006"/> 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.

It is an interdisciplinary field that involves [[quantum mechanics]], [[computer science]], [[information theory]], [[philosophy]] and [[cryptography]] among other fields.<ref name="Bokulich2010"/><ref name="Benatti2010"/><ref name="Benatti2009"/> Its study is also relevant to disciplines such as [[cognitive science]], [[psychology]] and [[neuroscience]].<ref name="Hayashi2015"/><ref name="Hayashi2017"/><ref name="Georgiev2017"/><ref name="Georgiev2020"/> 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]], [[Commutative property|non-commuting]] [[Observable|observables]] cannot be precisely measured simultaneously, as an [[Quantum state|eigenstate]] in one basis is not an eigenstate in the other basis. According to the eigenstate–eigenvalue link, an observable is well-defined (definite) when the state of the system is an eigenstate of the observable.<ref name="Gilton2016">{{cite journal
 | author = Gilton, Marian J. R.
 | title = Whence the eigenstate–eigenvalue link?
 | journal = Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
 | volume = 55
 | pages = 92–100
 | doi = 10.1016/j.shpsb.2016.08.005
 | year = 2016
| bibcode = 2016SHPMP..55...92G
 }}</ref> Since any two non-commuting observables are not simultaneously well-defined, a quantum state can never contain definitive information about both non-commuting observables.<ref name="Hayashi2017" />

Data can be encoded into the [[State (computer science)|quantum state]] of a quantum system as [[quantum information]].<ref name="Preskill2018"/> While quantum mechanics deals with examining properties of matter at the microscopic level,<ref name="Feynman2013"/><ref name="Hayashi2017" /> [[quantum information science]] focuses on extracting information from those properties,<ref name="Hayashi2017" /> and [[Quantum computing|quantum computation]] manipulates and processes information – performs logical operations – using [[quantum information processing]] techniques.<ref name="Lo1998"/>

Quantum information, like classical information, can be processed using [[digital computer]]s, [[communications channel|transmitted]] from one location to another, manipulated with [[algorithm]]s, and analyzed with computer science and [[mathematics]]. Just like the basic unit of classical information is the bit, quantum information deals with [[Qubit|qubits]].<ref name="Bennett1998"/> 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]].<ref name="Lo1998"/><ref name="Garlinghouse2020"/>

== History and development ==

=== Development from fundamental quantum mechanics ===
The history of quantum information theory began at the turn of the 20th century when [[classical physics]] was revolutionized into [[Quantum mechanics|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 hypotheses 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.<ref name="Nielsen2010" />

[[Quantum mechanics]] was formulated by [[Erwin Schrödinger]] using wave mechanics and [[Werner Heisenberg]] using [[matrix mechanics]].<ref name="Mahan2009"/> The equivalence of these methods was proven later.<ref name="Perlman1964"/> 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.<ref>{{Cite book|last=Neumann|first=John von|url=https://books.google.com/books?id=B3OYDwAAQBAJ&q=foundations+of+quantum+mechanics+von+neumann&pg=PR1|title=Mathematical Foundations of Quantum Mechanics: New Edition|date=2018-02-27|publisher=Princeton University Press|isbn=978-0-691-17856-1|language=en}}</ref> These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements.

See: [[Dynamical pictures|Dynamical Pictures]]
{{Pictures in quantum mechanics}}

==== Development from communication ====
In the 1960s, [[Ruslan Stratonovich]], [[Carl W. Helstrom|Carl Helstrom]] and Gordon<ref name="Gordon1962"/> 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.<ref name="Gordon1962"/><ref name="Helstrom1969"/><ref name="Helstrom1976"/> Later, [[Alexander Holevo]] obtained an upper bound of communication speed in the transmission of a classical message via a [[quantum channel]].<ref name="Holevo1973"/><ref name="Holevo1979"/>

==== Development from atomic physics and relativity ====
In the 1970s, techniques for manipulating single-atom quantum states, such as the [[atom trap]] and the [[scanning tunneling microscope]], began to be developed, making it possible to isolate single atoms and arrange them in arrays. Prior to these developments, precise control over single quantum systems was not possible, and experiments used coarser, simultaneous control over a large number of quantum systems.<ref name="Nielsen2010" /> The development of viable single-state manipulation techniques led to increased interest in the field of quantum information and computation.

In the 1980s, interest arose in whether it might be possible to use quantum effects to disprove [[Theory of relativity|Einstein's theory of relativity]]. If it were possible to clone an unknown quantum state, it would be possible to use [[Quantum entanglement|entangled]] quantum states to transmit information faster than the speed of light, disproving Einstein's theory. However, the [[no-cloning theorem]] showed that such cloning is impossible. The theorem was one of the earliest results of quantum information theory.<ref name="Nielsen2010" />

==== Development from cryptography ====
{{See also|Quantum cryptography}}
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.''<ref name="Nielsen2010" />

Bennett and Brassard developed a communication channel on which it is impossible to eavesdrop without being detected, a way of communicating secretly at long distances using the [[BB84]] quantum cryptographic protocol.<ref name="Bennett2014"/> The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of an eavesdropper in a secure communication line will immediately let the two parties trying to communicate know of the presence of the eavesdropper.

==== Development from computer science and mathematics ====
{{See also|Quantum supremacy|Quantum algorithm}}
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.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Church–Turing Thesis|url=https://mathworld.wolfram.com/Church-TuringThesis.html|access-date=2020-11-13|website=mathworld.wolfram.com|language=en}}</ref><ref name="Deutsch1985"/> 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.<ref name="Moore1998"/> 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 uses 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 algorithm|Deutsch–Jozsa algorithm]]. This problem however held little to no practical applications.<ref name="Nielsen2010" /> [[Peter Shor]] in 1994 came up with a very important and practical [[Shor's algorithm|problem]], one of finding the prime factors of an integer. The [[discrete logarithm]] problem as it was called, could theoretically be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers should be more powerful than Turing machines.

==== Development from information theory ====
Around the time computer science was making a revolution, so was information theory and communication, through [[Claude Shannon]].<ref name="Shannon1948a"/><ref name="Shannon1948b"/><ref name="Shannon1964"/> Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and [[Noisy-channel coding theorem|noisy channel coding theorem]]. He also showed that [[Error correction code|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 [[Shannon's source coding theorem|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.<ref name="Nielsen2010" />

==Qubits and information theory==
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 [[Entropy (information theory)|Shannon entropy]], while the quantum mechanical analogue is [[Von Neumann entropy]]. Given a [[statistical ensemble]] of quantum mechanical systems with the [[density matrix]] <math>\rho</math>, it is given by <math> S(\rho) = -\operatorname{Tr}(\rho \ln \rho).</math><ref name="Nielsen2010" /> Many of the same entropy measures in classical [[information theory]] can also be generalized to the quantum case, such as Holevo entropy<ref>{{cite web|url=http://www.mi.ras.ru/~holevo/eindex.html|title=Alexandr S. Holevo|website=Mi.ras.ru|access-date=4 December 2018}}</ref> 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 [[EPR paradox|impossible]] to [[quantum measurement|measure]] the value precisely. Five famous theorems describe the limits on manipulation of quantum information.<ref name="Nielsen2010" /> 
# [[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 are proven from [[Unitarity (physics)|unitarity]], which according to [[Leonard Susskind]] is the technical term for the statement that quantum information within the universe is conserved.{{r|Susskind2014|p=94|quote=The minus first law says that information is never lost. If two identical isolated systems start out in different states, they stay in different states. Moreover, in the past they were also in different states. On the other hand, if two identical systems are in the same state at some point in time, then their histories and their future evolutions must also be identical. Distinctions are conserved. The quantum version of the minus first law has a name — unitarity.}} The five theorems open possibilities in quantum information processing.

== 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 transformation]]s or [[quantum gate]]s to them. These [[unitary transformation (quantum mechanics)|unitary transformations]] are described as rotations on the Bloch sphere. While classical gates correspond to the familiar operations of [[Boolean Logic|Boolean logic]], quantum gates are physical [[unitary operator]]s.

* 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 tolerance|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 [[quantum entanglement|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 [[Claude Shannon|Shannon]] [[information entropy|entropy]], called the von Neumann entropy.
* In some cases, [[quantum algorithm]]s 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 (cryptosystem)|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 algorithm|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|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. Note that certain subtle points regarding the safety of QKD are debated.

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

==Relation to quantum mechanics==
[[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 computing|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 computing|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 matrix|density matrices]] over the [[complex number]]s. Another important difference with quantum mechanics is that while quantum mechanics often studies [[infinite-dimensional]] systems such as a [[Quantum harmonic oscillator|harmonic oscillator]], quantum information theory is concerned with both continuous-variable systems<ref name="Weedbrook2012"/> and finite-dimensional systems.<ref name="Hayashi2017"/><ref name="Watrous2018"/><ref name="Wilde2017"/>

== Entropy and information ==
Entropy measures the uncertainty in the state of a physical system.<ref name="Nielsen2010" /> Entropy can be studied from the point of view of both the classical and quantum information theories.

=== Classical information theory ===
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.<ref name="Jaeger2007"/>

==== Shannon entropy ====
{{Main|Entropy (information theory)}}
{{See also|Shannon's source coding theorem}}

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.<ref name="Nielsen2010" />

Shannon entropy, written as a function of a discrete probability distribution, <math>P(x_1), P(x_2),...,P(x_n)</math> associated with events <math>x_1, ..., x_n</math>, can be seen as the average information associated with this set of events, in units of bits:

<math display="block">H(X) = H[P(x_1), P(x_2),...,P(x_n)]= -\sum_{i=1}^n P(x_i)\log_2P(x_i)</math>

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.<ref name="Watrous2018"/>

==== Rényi entropy ====
{{Main|Rényi entropy}}

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, <math>P(a_1), P(a_2),...,P(a_n)</math>, associated with events <math>a_1, ..., a_n</math>, is defined as:<ref name="Jaeger2007" />

<math display="block">H_r(A) = {1\over1-r} \log_2\sum_{i=1}^n P^r(a_i) 
</math>

for <math> 0 < r <\infty</math> and <math>r\neq1</math>.

We arrive at the definition of Shannon entropy from Rényi when <math>r\rightarrow 1</math>, of [[Hartley entropy]] (or max-entropy) when <math>r\rightarrow 0</math>, and [[min-entropy]] when <math>r\rightarrow \infin</math>.

=== Quantum information theory ===
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.<ref name="Jaeger2007" />

==== Von Neumann entropy ====
{{Main|Von Neumann entropy}}

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 replaced by [[density operator]]s <math>\rho</math>:

<math display="block">S(\rho)\equiv - \mathrm{tr}(\rho\  \log_2\  \rho) = -\sum_{i}\lambda_i \ \log_2\ \lambda_i,</math>

where <math>\lambda_i</math> are the eigenvalues of <math>\rho</math>.

Von Neumann entropy plays a role in quantum information similar to the role Shannon entropy plays in classical information.

== Applications ==
===Quantum communication===

[[Quantum information science|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. [[Superdense coding|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===

{{Main|Quantum key distribution|l1 = 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]], was 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.<ref name="Nielsen2010" />

====E91====

[[E91 protocol|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.<ref name="Bennett1992"/>

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 he measured it correctly.<ref name="Haitjema2007"/>

===Quantum computation===

{{Main|Quantum computing}}

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 [[Quantum superposition|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 [[Probability|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 gate]]s.

===Quantum decoherence===

{{Main|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===

{{Main|Quantum error correction}}

'''QEC''' is used in [[Quantum computer|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 bit|ancilla qubits]]. A quantum error correcting code protects quantum information against errors.

==Journals==
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]]''
* ''[[npj Quantum Information]]''
* ''[[Quantum (journal)|Quantum]]''
* ''[https://www.rintonpress.com/journals/qic/ Quantum Information & Computation]''
* ''[https://www.springer.com/journal/11128 Quantum Information Processing]''
* ''[https://iopscience.iop.org/journal/2058-9565 Quantum Science and Technology]''

==See also==
{{Div col|small=yes}}
*[[Categorical quantum mechanics]]
*[[Einstein's thought experiments]]
*[[Interpretations of quantum mechanics]]
*[[POVM|Positive Operator Valued Measure (POVM)]]
*[[Quantum clock]]
*[[Quantum entanglement]]
*[[Quantum foundations]]
*[[Quantum information science]]
*[[Quantum statistical mechanics]]
*[[Qubit]]
*[[Qutrit]]
*[[Typical subspace]]
{{Div col end}}

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 | doi = 10.1109/jrproc.1962.288169
 | s2cid = 51631629
}}</ref>

<ref name="Haitjema2007">{{cite book
 | last1 = Haitjema
 | first1 = Mart
 | title = A Survey of the Prominent Quantum Key Distribution Protocols
 | publisher = Washington University in St. Louis
 | url = https://www.cse.wustl.edu/~jain/cse571-07/ftp/quantum/#b92
 | year = 2007
 | s2cid = 18346434
}}</ref>

<ref name="Hayashi2006">{{cite book
 | last = Hayashi
 | first = Masahito
 | year = 2006
 | title = Quantum Information: An Introduction
 | publisher = Springer
 | location = Berlin
 | pages = 
 | url = 
 | doi = 10.1007/3-540-30266-2
 | oclc = 68629072
 | isbn = 978-3-540-30266-7
}}</ref>

<ref name="Hayashi2015">{{cite book
 | last1 = Hayashi
 | first1 = Masahito
 | last2 = Ishizaka
 | first2 = Satoshi
 | last3 = Kawachi
 | first3 = Akinori
 | last4 = Kimura
 | first4 = Gen
 | last5 = Ogawa
 | first5 = Tomohiro
 | year = 2015
 | title = Introduction to Quantum Information Science
 | publisher = Springer
 | location = Berlin
 | pages = 
 | url = 
 | doi = 10.1007/978-3-662-43502-1
 | bibcode = 2015iqis.book.....H
 | oclc = 
 | isbn = 978-3-662-43502-1
}}</ref>

<ref name="Hayashi2017">{{cite book
 | last = Hayashi
 | first = Masahito
 | year = 2017
 | title = Quantum Information Theory: Mathematical Foundation
 | series = Graduate Texts in Physics
 | publisher = Springer
 | location = Berlin
 | pages = 
 | url = 
 | doi = 10.1007/978-3-662-49725-8
 | oclc = 
 | isbn = 978-3-662-49725-8
}}</ref>

<ref name="Helstrom1969">{{cite journal
 | last1 = Helstrom
 | first1 = Carl W.
 | title = Quantum detection and estimation theory
 | journal = Journal of Statistical Physics
 | volume = 1
 | issue = 2
 | pages = 231–252
 | year = 1969
 | doi = 10.1007/bf01007479
| bibcode = 1969JSP.....1..231H
 | hdl = 2060/19690016211
 | s2cid = 121571330
 | hdl-access = free
 }}</ref>

<ref name="Helstrom1976">{{cite book
 | last1 = Helstrom
 | first1 = Carl W.
 | year = 1976
 | title = Quantum Detection and Estimation Theory
 | series = Mathematics in Science and Engineering
 | volume = 123
 | publisher = Academic Press
 | location = New York
 | pages = 
 | url = 
 | doi = 10.1016/s0076-5392(08)x6017-5
 | hdl = 2060/19690016211
 | oclc = 2020051
 | isbn = 9780080956329
}}</ref>

<ref name="Holevo1973">{{cite journal
 | last = Holevo
 | first = Alexander S.
 | author-link = Alexander Holevo
 | title = Bounds for the quantity of information transmitted by a quantum communication channel
 | journal = Problems of Information Transmission
 | volume = 9
 | issue = 3
 | pages = 177–183
 | year = 1973
 | url = http://mi.mathnet.ru/ppi903
 | mr = 456936
 | zbl = 0317.94003
}}</ref>

<ref name="Holevo1979">{{cite journal
 | last = Holevo
 | first = Alexander S.
 | author-link = Alexander Holevo
 | title = On capacity of a quantum communications channel
 | journal = Problems of Information Transmission
 | volume = 15
 | issue = 4
 | pages = 247–253
 | year = 1979
 | url = http://mi.mathnet.ru/ppi1507
 | mr = 581651
 | zbl = 0433.94008
}}</ref>

<ref name="Jaeger2007">{{cite book
 | last1 = Jaeger
 | first1 = Gregg
 | year = 2007
 | title = Quantum Information: An Overview
 | publisher = Springer
 | location = New York
 | pages = 
 | doi = 10.1007/978-0-387-36944-0
 | url = https://books.google.com/books?id=E0ho97k7S4oC
 | oclc = 255569451
 | isbn = 978-0-387-36944-0
}}</ref>

<ref name="Lo1998">{{cite book
 | last1 = Lo
 | first1 = Hoi-Kwong
 | last2 = Popescu
 | first2 = Sandu
 | last3 = Spiller
 | first3 = Tim
 | year = 1998
 | title = Introduction to Quantum Computation and Information
 | publisher = World Scientific
 | location = Singapore
 | url = https://books.google.com/books?id=18_-rgkdQGIC
 | doi = 10.1142/3724
 | bibcode = 1998iqci.book.....S
 | oclc = 52859247
 | isbn = 978-981-4496-35-3
}}</ref>

<ref name="Mahan2009">{{cite book
 | last1 = Mahan
 | first1 = Gerald D.
 | year = 2009
 | title = Quantum Mechanics in a Nutshell
 | publisher = Princeton University Press
 | location = Princeton
 | url = 
 | doi = 10.2307/j.ctt7s8nw
 | oclc = 
 | isbn = 978-1-4008-3338-2
 | zbl = 
 | jstor = j.ctt7s8nw
}}</ref>

<ref name="Moore1998">{{cite journal
 | last1 = Moore
 | first1 =  Gordon Earle
 | author-link1 = Gordon Moore
 | title = Cramming more components onto integrated circuits
 | journal = Proceedings of the IEEE
 | volume = 86
 | issue = 1
 | pages = 82–85
 | year = 1998
 | doi = 10.1109/jproc.1998.658762
 | s2cid = 6519532
}}</ref>

<ref name="Nielsen2010">{{cite book
 | last1 = Nielsen
 | first1 = Michael A.
 | last2 = Chuang
 | first2 = Isaac L.
 | year = 2010
 | title = Quantum Computation and Quantum Information
 | edition = 10th anniversary
 | publisher = Cambridge University Press
 | location = Cambridge
 | pages = 
 | url = 
 | doi = 10.1017/cbo9780511976667
 | oclc = 665137861
 | isbn = 9780511976667
 | zbl = 
 | arxiv = 
| s2cid = 59717455
 }}</ref>

<ref name="Perlman1964">{{cite journal
 | last1 = Perlman
 | first1 = H. S.
 | title = Equivalence of the Schroedinger and Heisenberg pictures
 | journal = Nature
 | volume = 204
 | issue = 4960
 | pages = 771–772
 | year = 1964
 | doi = 10.1038/204771b0
 | bibcode = 1964Natur.204..771P 
 | s2cid = 4194913
}}</ref>

<ref name="Preskill2018">{{cite book
 | last1 = Preskill
 | first1 = John
 | title = Quantum Computation (Physics 219/Computer Science 219)
 | publisher = California Institute of Technology
 | location = Pasadena, California
 | url = http://www.theory.caltech.edu/~preskill/ph219/
}}</ref>

<ref name="Shannon1948a">{{cite journal
 | last1 = Shannon
 | first1 = Claude E.
 | author-link1 = Claude Shannon
 | title = A mathematical theory of communication
 | journal = The Bell System Technical Journal
 | volume = 27
 | issue = 3
 | pages = 379–423
 | year = 1948
 | doi = 10.1002/j.1538-7305.1948.tb01338.x
}}</ref>

<ref name="Shannon1948b">{{cite journal
 | last1 = Shannon
 | first1 = Claude E.
 | author-link1 = Claude Shannon
 | title = A mathematical theory of communication
 | journal = The Bell System Technical Journal
 | volume = 27
 | issue = 4
 | pages = 623–656
 | year = 1948
 | doi = 10.1002/j.1538-7305.1948.tb00917.x
}}</ref>

<ref name="Shannon1964">{{cite book
 | last1 = Shannon
 | first1 = Claude E.
 | last2 = Weaver
 | first2 = Warren
 | year = 1964
 | title = The Mathematical Theory of Communication
 | publisher = University of Illinois Press
 | location = Urbana
 | pages = 
 | hdl = 11858/00-001M-0000-002C-4314-2
 | url = http://hdl.handle.net/11858/00-001M-0000-002C-4314-2
}}</ref>

<ref name="Susskind2014">{{cite book
 | last1 = Susskind
 | first1 = Leonard
 | author-link1 = Leonard Susskind
 | last2 = Friedman
 | first2 = Art
 | author-link2 =
 | year = 2014
 | title = Quantum Mechanics: The Theoretical Minimum. What You Need to Know to Start Doing Physics
 | publisher = Basic Books
 | location = New York
 | pages = 
 | url = https://books.google.com/books?id=iQE4DgAAQBAJ&pg=PT83
 | doi = 
 | oclc = 1038428525
 | isbn = 978-0-465-08061-8
}}</ref>

<ref name="Vedral2006">{{cite book
 | last1 = Vedral
 | first1 = Vlatko
 | author-link1 = Vlatko Vedral
 | year = 2006
 | title = Introduction to Quantum Information Science
 | publisher = Oxford University Press
 | location = Oxford
 | pages = 
 | url = 
 | doi = 10.1093/acprof:oso/9780199215706.001.0001
 | oclc = 822959053
 | isbn = 9780199215706
 | zbl = 
 | arxiv = 
}}</ref>

<ref name="Watrous2018">{{cite book
 | last1 = Watrous
 | first1 = John
 | year = 2018
 | title = The Theory of Quantum Information
 | publisher = Cambridge University Press
 | location = Cambridge
 | pages = 
 | url = https://cs.uwaterloo.ca/~watrous/TQI/
 | doi = 10.1017/9781316848142
 | oclc = 1034577167
 | isbn = 9781316848142
 | zbl = 
 | arxiv = 
}}</ref>

<ref name="Weedbrook2012">{{cite journal
 | last1 = Weedbrook
 | first1 = Christian
 | author-link1 =
 | last2 = Pirandola
 | first2 = Stefano
 | author-link2 =
 | last3 = García-Patrón
 | first3 = Raúl
 | author-link3 =
 | last4 = Cerf
 | first4 = Nicolas J.
 | author-link4 = Nicolas J. Cerf
 | last5 = Ralph
 | first5 = Timothy C.
 | author-link5 = Tim C. Ralph
 | last6 = Shapiro
 | first6 = Jeffrey H.
 | author-link6 = Jeffrey Shapiro
 | last7 = Lloyd
 | first7 = Seth
 | author-link7 = Seth Lloyd
 | title = Gaussian quantum information
 | journal = Reviews of Modern Physics
 | volume = 84
 | issue = 2
 | pages = 621–669
 | year = 2012
 | doi = 10.1103/RevModPhys.84.621
 | arxiv = 1110.3234
 | bibcode = 2012RvMP...84..621W 
 | s2cid = 119250535
}}</ref>

<ref name="Wilde2017">{{cite book
 | last1 = Wilde
 | first1 = Mark M.
 | year = 2017
 | title = Quantum Information Theory
 | publisher = Cambridge University Press
 | location = Cambridge
 | edition = 2nd
 | url = 
 | doi = 10.1017/9781316809976
 | oclc = 
 | isbn = 9781316809976
 | zbl = 
 | arxiv = 1106.1445
}}</ref>
}}

{{Quantum computing}}
{{Quantum mechanics topics}}
{{emerging technologies|quantum=yes|other=yes}}

{{Authority control}}

[[Category:Quantum information theory]]

[[it:Informazione quantistica]]