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Replace raw Quantum Collection backlink with BT backlink template

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{{Short description|Information carried by the state of a quantum system}}
{{Quantum methods backlink|Quantum information methods}}

'''Information theory''' in quantum physics studies how information is represented, processed, transmitted, and measured when encoded in the [[Physics:Quantum state|quantum states]] of physical systems. It is the basic entity of study in quantum information science.<ref>{{cite book
| last1 = Vedral
| first1 = Vlatko
| year = 2006
| title = Introduction to Quantum Information Science
| publisher = Oxford University Press
| location = Oxford
| isbn = 9780199215706
}}</ref><ref>{{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
| isbn = 9780511976667
}}</ref>

Quantum information differs fundamentally from classical information because quantum states may exist in [[quantum superposition]], may exhibit [[Physics:Quantum methods/entanglement|entanglement]], and cannot generally be copied or measured without disturbance. The elementary unit of quantum information is the [[Physics:Quantum methods/qubit|qubit]], which generalizes the classical bit.<ref>{{cite journal
| last1 = Bennett
| first1 = Charles H.
| last2 = Shor
| first2 = Peter
| title = Quantum information theory
| journal = IEEE Transactions on Information Theory
| volume = 44
| issue = 6
| pages = 2724–2742
| year = 1998
| doi = 10.1109/18.720553
}}</ref>

The field combines ideas from [[Physics:Quantum mechanics|quantum mechanics]], [[information theory]], [[computer science]], [[cryptography]], mathematics, and communication theory.<ref>{{cite book
| last1 = Bokulich
| first1 = Alisa
| last2 = Jaeger
| first2 = Gregg
| year = 2010
| title = Philosophy of Quantum Information and Entanglement
| publisher = Cambridge University Press
| location = Cambridge
| isbn = 9780511676550
}}</ref>
[[File:Quantum_information_theory_yellow.jpg|thumb|right|Quantum information theory studies qubits, quantum gates, entropy, channels, and communication protocols.]]

[[File:Qubits_(5940500587).jpg|thumb|right|upright=1|Optical lattices use lasers to separate [[rubidium]] atoms for use as information bits in neutral-atom [[Quantum computing|quantum processors]].]]

== Quantum information ==

Quantum information is information encoded in the state of a quantum system.<ref>{{cite book
| last1 = Preskill
| first1 = John
| title = Quantum Computation
| publisher = California Institute of Technology
| location = Pasadena
| year = 2018
}}</ref>

In classical systems, information is stored in bits that take values 0 or 1. In quantum systems, a qubit may exist in a superposition of basis states, allowing a richer mathematical structure and enabling computational and communication techniques not available in classical physics.

A quantum state may contain correlations between separated systems through entanglement. Because measurements generally disturb quantum states, information extraction is fundamentally constrained by the uncertainty principle and the non-commuting nature of quantum observables.<ref>{{cite book
| last = Hayashi
| first = Masahito
| year = 2017
| title = Quantum Information Theory: Mathematical Foundation
| publisher = Springer
| location = Berlin
| isbn = 978-3-662-49725-8
}}</ref>

== Qubits ==

The qubit is the elementary unit of quantum information. Physically, qubits may be realized using photons, trapped ions, superconducting circuits, neutral atoms, or nuclear spins.<ref>{{cite book
| last1 = Nielsen
| first1 = Michael A.
| last2 = Chuang
| first2 = Isaac L.
| year = 2010
| title = Quantum Computation and Quantum Information
| publisher = Cambridge University Press
| location = Cambridge
| isbn = 9780511976667
}}</ref>

Unlike a classical bit, which can only occupy one of two states, a qubit can exist in a linear combination of basis states. A pure qubit state can be represented geometrically on the Bloch sphere.

Although qubits are continuously parameterized, measurement outcomes are discrete and probabilistic.

== Entropy and information ==

Classical information theory uses Shannon entropy to quantify uncertainty in a probability distribution.<ref>{{cite journal
| last1 = Shannon
| first1 = Claude E.
| title = A mathematical theory of communication
| journal = Bell System Technical Journal
| volume = 27
| issue = 3
| pages = 379–423
| year = 1948
}}</ref>

Quantum information theory generalizes this concept using the von Neumann entropy, defined for a density matrix <math>\rho</math> as:

S(\rho)=-\mathrm{Tr}(\rho\log_2\rho)

Von Neumann entropy measures the uncertainty or mixedness of a quantum state and plays an important role in quantum communication, entanglement theory, quantum compression, and error correction.<ref>{{cite book
| last1 = Watrous
| first1 = John
| year = 2018
| title = The Theory of Quantum Information
| publisher = Cambridge University Press
| location = Cambridge
| isbn = 9781316848142
}}</ref>

== Limits on quantum information ==

Several important theorems distinguish quantum information from classical information.

The '''no-cloning theorem''' states that an arbitrary unknown quantum state cannot be copied.<ref>{{cite book
| last1 = Nielsen
| first1 = Michael A.
| last2 = Chuang
| first2 = Isaac L.
| year = 2010
| title = Quantum Computation and Quantum Information
| publisher = Cambridge University Press
| location = Cambridge
| isbn = 9780511976667
}}</ref>

Related results include no-deleting, no-broadcasting, and no-hiding theorems.

The impossibility of perfectly copying quantum states is essential for quantum cryptography and secure communication.

== Quantum communication ==

Quantum communication studies how quantum states and classical messages encoded in quantum systems are transmitted through quantum channels.<ref>{{cite journal
| last1 = Gordon
| first1 = J. P.
| title = Quantum effects in communications systems
| journal = Proceedings of the IRE
| volume = 50
| issue = 9
| pages = 1898–1908
| year = 1962
}}</ref>

Applications include:

* quantum teleportation
* dense coding
* quantum key distribution
* entanglement distribution
* quantum networks

Quantum teleportation transfers a quantum state between distant systems using entanglement and classical communication.

== Quantum cryptography ==

One of the best known applications of quantum information theory is quantum cryptography. The BB84 protocol, introduced by Bennett and Brassard in 1984, allows secure key distribution using quantum states.<ref>{{cite journal
| last1 = Bennett
| first1 = Charles H.
| last2 = Brassard
| first2 = Gilles
| title = Quantum cryptography: public key distribution and coin tossing
| journal = Theoretical Computer Science
| volume = 560
| pages = 7–11
| year = 2014
}}</ref>

Security arises because measurement disturbs quantum systems. Any eavesdropping attempt changes the transmitted states and can therefore be detected.

== Quantum computation ==

Quantum information processing forms the basis of quantum computing. Quantum algorithms manipulate qubits using quantum gates and unitary transformations.<ref>{{cite journal
| last1 = Deutsch
| first1 = David
| title = Quantum theory, the Church–Turing principle and the universal quantum computer
| journal = Proceedings of the Royal Society A
| volume = 400
| issue = 1818
| pages = 97–117
| year = 1985
}}</ref>

Important quantum algorithms include:

* Shor's factoring algorithm
* Grover's search algorithm
* quantum simulation algorithms

Quantum computation also requires methods for decoherence control and fault-tolerant error correction.

== Quantum error correction ==

Quantum systems are highly sensitive to interactions with their environment. Such interactions lead to quantum decoherence, which destroys coherent quantum behavior.

Quantum error correction protects quantum information against noise and imperfect operations. Error correction is essential for scalable fault-tolerant quantum computers.

== History ==

The origins of quantum information theory lie in the development of quantum mechanics during the early twentieth century.<ref>{{cite book
| last1 = Mahan
| first1 = Gerald D.
| year = 2009
| title = Quantum Mechanics in a Nutshell
| publisher = Princeton University Press
| location = Princeton
| isbn = 978-1-4008-3338-2
}}</ref>

During the 1960s and 1970s, researchers developed quantum communication theory and established limits on information transmission in quantum channels.<ref>{{cite journal
| last = Holevo
| first = Alexander S.
| 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
}}</ref>

In the 1980s and 1990s, rapid developments in cryptography and computer science led to the emergence of quantum computation and quantum information science as major research fields.

== Applications ==

Applications of quantum information theory include:

* quantum computing
* quantum communication
* quantum cryptography
* quantum sensing
* quantum networking
* quantum error correction

The field is considered one of the foundations of modern quantum technologies.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Methods}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum methods/information theory|1}}

Physics:Quantum methods/information theory - Revision history

imported>WikiHarold: Replace raw Quantum Collection backlink with BT backlink template

imported>WikiHarold: Replace raw Quantum Collection backlink with BT backlink template