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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Quantum information and computing}}
'''Quantum entanglement''' is a phenomenon in which the state of a composite quantum system cannot be expressed as a product of the states of its individual subsystems. It is one of the central features distinguishing quantum from classical physics.<ref>{{cite journal
| last1 = Horodecki
| first1 = Ryszard
| last2 = Horodecki
| first2 = Paweł
| last3 = Horodecki
| first3 = Michał
| last4 = Horodecki
| first4 = Karol
| title = Quantum entanglement
| journal = Reviews of Modern Physics
| volume = 81
| issue = 2
| pages = 865–942
| year = 2009
| doi = 10.1103/RevModPhys.81.865
| arxiv = quant-ph/0702225
| bibcode = 2009RvMP...81..865H
| s2cid = 59577352
}}</ref>
<ref name="NielsenChuang2010">{{cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

[[File:Quantum_entanglement.jpg|thumb|400px|Entangled quantum states exhibit correlations that cannot be described by independent subsystems.]]

== Definition ==
Consider a bipartite system with subsystems A and B. A pure state is called '''separable''' if it can be written as

<math>
|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle.
</math>

If no such decomposition exists, the state is said to be '''entangled'''.<ref name="NielsenChuang2010" /><ref>{{cite book
| last1 = Nielsen
| first1 = Michael A.
| last2 = Chuang
| first2 = Isaac L.
| title = Quantum Computation and Quantum Information
| edition = 10th anniversary
| publisher = Cambridge University Press
| location = Cambridge
| year = 2010
| isbn = 978-0-521-63503-5
}}</ref>

== Bell states ==

The simplest examples of entangled states are the '''Bell states''':

<math>
|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle),
</math>

<math>
|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle),
</math>

<math>
|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle),
</math>

<math>
|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle).
</math>

These states exhibit maximal entanglement and perfect correlations between measurement outcomes.

== Measurement correlations ==

In an entangled state, measurement outcomes on one subsystem are correlated with outcomes on the other subsystem.

For example, in the state <math>|\Phi^+\rangle</math>, measuring one qubit determines the outcome probabilities of the other, regardless of spatial separation.

These correlations cannot be explained by classical local hidden-variable theories.<ref>{{cite book |last=Bell |first=John S. |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press |year=1987}}</ref>

== Reduced states ==

Even if a composite system is in a pure state, its subsystems may be described by mixed states.

The reduced density matrix of subsystem A is given by

<math>
\rho_A = \mathrm{Tr}_B(\rho_{AB}),
</math>

where <math>\mathrm{Tr}_B</math> denotes the partial trace over subsystem B.

This reflects the fact that subsystems of entangled systems do not possess independent pure states.

== Entanglement as a resource ==

Entanglement is a key resource in quantum information processing. It enables:

* quantum teleportation <ref>{{cite web
| last = Francis
| first = Matthew
| title = Quantum entanglement shows that reality can't be local
| website = Ars Technica
| date = 30 October 2012
| url = https://arstechnica.com/science/2012/10/quantum-entanglement-shows-that-reality-cant-be-local/
| access-date = 22 August 2023
}}</ref>
* superdense coding
* quantum cryptography
* quantum computation beyond classical limits

== Generation of entanglement ==

Entanglement is typically created by applying quantum operations to multiple qubits.

For example, applying a Hadamard gate followed by a controlled-NOT (CNOT) gate to an initial state

<math>
|0\rangle \otimes |0\rangle
</math>

produces the Bell state

<math>
\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle).
</math>

== Physical significance ==

Quantum entanglement:

* is a fundamental feature of composite quantum systems
* produces correlations beyond classical physics
* underlies many quantum technologies and protocols

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

= References =
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum entanglement|1}}

Physics:Quantum Entanglement - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template