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{{Short description|Quantum Collection topic on Quantum Entanglement}}
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'''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
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| title = Quantum entanglement
'''Entanglement''' 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. 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. Consider a bipartite system with subsystems A and B. A pure state is called separable if it can be written as The simplest examples of entangled states are the Bell states: These states exhibit maximal entanglement and perfect correlations between measurement outcomes.
| journal = Reviews of Modern Physics
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| 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
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<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>


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[[File:Quantum_entanglement.jpg|thumb|280px|Quantum Entanglement.]]
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[[File:Quantum_entanglement.jpg|thumb|400px|Entangled quantum states exhibit correlations that cannot be described by independent subsystems.]]
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== Definition ==
== Definition ==
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{{Author|Harold Foppele}}
{{Author|Harold Foppele}}


{{Sourceattribution|Quantum entanglement|1}}
{{Sourceattribution|Physics:Quantum Entanglement|1}}

Latest revision as of 22:21, 21 May 2026



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Entanglement 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. 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. Consider a bipartite system with subsystems A and B. A pure state is called separable if it can be written as The simplest examples of entangled states are the Bell states: These states exhibit maximal entanglement and perfect correlations between measurement outcomes.

Quantum Entanglement.

Definition

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

|ψ⟩=|ψA⟩⊗|ψB⟩.

If no such decomposition exists, the state is said to be entangled.[1][2]

Bell states

The simplest examples of entangled states are the Bell states:

|Φ+⟩=12(|00⟩+|11⟩),

|Φ−⟩=12(|00⟩−|11⟩),

|Ψ+⟩=12(|01⟩+|10⟩),

|Ψ−⟩=12(|01⟩−|10⟩).

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 |Φ+⟩, 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.[3]

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

ρA=TrB(ρAB),

where TrB 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 [4]
  • 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

|0⟩⊗|0⟩

produces the Bell state

12(|00⟩+|11⟩).

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

Table of contents (212 articles)

Index

Full contents

References

  1. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named NielsenChuang2010
  2. ↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-63503-5. 
  3. ↑ Bell, John S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. 
  4. ↑ Francis, Matthew (30 October 2012). "Quantum entanglement shows that reality can't be local". https://arstechnica.com/science/2012/10/quantum-entanglement-shows-that-reality-cant-be-local/. 


Author: Harold Foppele


Source attribution: Physics:Quantum Entanglement

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