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This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
The '''Bell states''' are a concept in [[quantum information science]] and represent the simplest possible examples of [[Quantum entanglement|entanglement]]. They are named after [[John S. Bell]], as they are the subject of his famous [[Bell's theorem|Bell inequality]].  An '''EPR pair''' is a pair of [[qubits]] which jointly are in a Bell state, that is, entangled with each other.  Unlike classical phenomena such as the nuclear, electromagnetic, and gravitational fields, entanglement is invariant under distance of separation and is not subject to relativistic limitations such as [[speed of light]].


==The Bell states==
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A Bell state is defined as a [[Maximally entangled state|maximally entangled]] [[quantum state]] of two [[qubit]]s. The qubits are usually thought to be spatially separated. Nevertheless they exhibit perfect [[correlation]]s which cannot be explained without [[quantum mechanics]].
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To explain, let us first look at the Bell state <math>|\Phi^+\rangle</math>:
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:<math>|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B).</math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


This expression means the following: The qubit held by Alice (subscript "A") can be 0 as well as 1. If Alice measured her qubit the outcome would be perfectly random, either possibility having probability 1/2. But if Bob then measured his qubit, the outcome would be the same as the one Alice got. So, if Bob measured, he would also get a random outcome on first sight, but if Alice and Bob communicated they would find out that, although the outcomes seemed random, they are correlated.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


So far, this is nothing special: Maybe the two particles "agreed" in advance, when the pair was created (before the qubits were separated), which outcome they would show in case of a measurement.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Hence, followed [[Albert Einstein|Einstein]], [[Boris Podolsky|Podolsky]], and [[Nathan Rosen|Rosen]] in 1935 in their famous "[[EPR paradox|EPR]] paper", there is something missing in the description of the qubit pair given above &mdash; namely this "agreement", called more formally a [[hidden variable theory|hidden variable]].  
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But quantum mechanics allows qubits to be in [[quantum superposition]] &mdash; i.e. in 0 and 1 simultaneously, e.g. in either of the states <math>|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)</math> or <math>|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)</math>. If Alice and Bob chose to measure in this [[basis (linear algebra)|basis]], i.e. check whether their qubit were <math>|+\rangle</math> or <math>|-\rangle</math>, they will find the same correlations as above. That is because the Bell state can be formally rewritten as follows:
==Demos==


:<math>|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|+\rangle_A \otimes |+\rangle_B + |-\rangle_A \otimes |-\rangle_B).</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Note that this is still the ''same'' state.


[[John S. Bell]] showed in his famous paper of 1964 by using simple [[probability theory]] arguments that these correlations cannot be perfect in case of "pre-agreement" stored in some hidden variables &mdash; but that quantum mechanics predict perfect correlations. In a more formal and refined formulation known as the [[CHSH Bell test|Bell-CHSH inequality]], this would be stated such that a certain correlation measure cannot exceed the value 2 according to reasoning assuming [[local hidden variable theory|local "hidden variable" theory]] (sort of common-sense) physics, but quantum mechanics predicts <math>2\sqrt{2}</math>.
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There are three other states of two qubits which lead to this maximal value of <math>2\sqrt{2}</math> and the four are known as the four ''maximally entangled two-qubit states'' or ''Bell states'':
==Test pages ==


:<math>|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B)</math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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:<math>|\Phi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B)</math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>|\Psi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B)</math>
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<math>|\Psi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B).</math>
 
==Bell state measurement==
The '''Bell measurement''' is an important concept in [[quantum information science]]: It is a joint quantum-mechanical measurement of two [[qubit]]s that determines in which of the four Bell states the two qubits are in.
 
If the qubits were not in a Bell state before, they get projected into a Bell state (according to the projection rule of [[quantum measurement]]s), and as Bell states are [[Quantum entanglement|entangled]], a Bell measurement is an entangling operation.
 
Bell-state measurement is the crucial step in [[quantum teleportation]]. The result of a Bell-state measurement is used by one's co-conspirator to reconstruct the original state of a teleported particle from half of an entangled pair (the "quantum channel") that was previously shared between the two ends.
 
Experiments which utilize so-called "linear evolution, local measurement" techniques cannot realize a complete Bell state measurement. Linear evolution means that the detection apparatus acts on each particle independently from the state or evolution of the other, and local measurement means that each particle is localized at a particular detector registering a "click" to indicate that a particle has been detected. Such devices can be constructed, for example, from mirrors, beam splitters, and wave plates, and are attractive from an experimental perspective because they are easy to use and have a high measurement cross-section.
 
For entanglement in a single qubit variable, only three distinct classes out of four Bell states are distinguishable using such linear optical techniques. This means two Bell states cannot be distinguished from each other, limiting the efficiency of quantum communication protocols such as [[quantum teleportation|teleportation]]. If a Bell state is measured from this ambiguous class, the teleportation event fails.
 
Entangling particles in multiple qubit variables, such as (for photonic systems) [[polarization (waves)|polarization]] and a two-element subset of [[azimuthal quantum number|orbital angular momentum]] states, allows the experimenter to trace over one variable and achieve a complete Bell state measurement in the other.<ref>Kwiat, Weinfurter. [http://pra.aps.org/abstract/PRA/v58/i4/pR2623_1 "Embedded Bell State Analysis"]</ref> Leveraging so-called hyper-entangled systems thus has an advantage for teleportation. It also has advantages for other protocols such as [[superdense coding]], in which hyper-entanglement increases the channel capacity.
 
In general, for hyper-entanglement in <math>n</math> variables, one can distinguish between at most <math>2^{n+1} - 1</math> classes out of <math>4^n</math> Bell states using linear optical techniques.<ref>Pisenti, Gaebler, Lynn. [http://www.opticsinfobase.org/abstract.cfm?uri=ICQI-2011-QMI25 "Distinguishability of Hyper-Entangled Bell States by Linear Evolution and Local Measurement"]</ref>
 
Bell measurements of [[ion]] qubits in [[ion trap]] experiments, the distinction of all four states is possible.
 
==See also==
*[[Bell test experiments]]
 
==References==
 
* {{citation
  | last1=Nielsen
  | first1=Michael A.
  | author1-link=Michael Nielsen
  | last2=Chuang
  | first2=Isaac L.
  | author2-link=Isaac Chuang
  | title=Quantum computation and quantum information
  | publisher=[[Cambridge University Press]]
  | year=2000
  | isbn=978-0-521-63503-5
}}, [http://books.google.ca/books?id=66TgFp2YqrAC&pg=PA25 pp. 25].
* {{citation
  | last1=Kaye
  | first1=Phillip
  | last2=Laflamme
  | first2=Raymond
  | author2-link=Raymond Laflamme
  | last3=Mosca
  | first3=Michele
  | author3-link=Michele Mosca
  | title=An introduction to quantum computing
  | publisher=[[Oxford University Press]]
  | year=2007
  | isbn=978-0-19-857049-3
}}, [http://books.google.ca/books?id=W0ud06mkPqoC&pg=PA75 pp. 75].
 
==Notes==
{{reflist}}
 
{{DEFAULTSORT:Bell State}}
[[Category:Quantum information science]]
 
[[es:Analizador de Estado de Bell]]
[[he:מצב בל]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
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MathML

E=mc2


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Demos

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Test pages

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