Talk:Bell state

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Bell state measurement

Hi, I think that to make the discussion more clear would be important to specify, within this article or within another, how is it actually possible to perform a Bell measurement as this is "It is the crucial part of quantum teleportation". Without a better description of what is a Bell measurement, from my point of view, both the article on quantum teleportation and this one remains too abstract and not accessible. Dave.bradi —Preceding undated comment was added at 18:12, 22 August 2008 (UTC)[reply]

One could replace bell measurement with measurement in the bell basis. Skippydo (talk) 00:10, 23 August 2008 (UTC)[reply]

Can anybody actually describe and explain the Bell state measurement in its eponymous section? --93.138.229.134 (talk) 08:11, 29 May 2013 (UTC)[reply]

The best description of a measurement in the Bell basis is probably framed in terms of quantum computing; it's a CNOT gate applied to qubits A and B, followed by a Hadamard gate on qubit A then a measurement of both qubits in the computational basis. You can see this setup in use in the superdense coding circuit diagram, for example. It's the inverse of the circuit that was used to prepare the Bell state in the first place. A deep explanation of measurement in the bell basis would include a good explanation of how the CNOT gate performs the act of un-entangling two qubits that have been previously entangled. DavidBoden (talk) 09:12, 17 February 2015 (UTC)[reply]

I think the diagram related to this part of the article is not correct. The input-qubits need not be entangled c.q. be in a EPR-state like the diagram suggests. 2A02:A463:2848:1:2D3E:B130:FF82:193F (talk) 13:01, 8 August 2023 (UTC)[reply]

Definition of Bell states

I read: A Bell state is defined as a maximally entangled quantum state of two qubits. This seems wrong: |0>|+> + |1>|-> is also maximally entangled and is not one of the four Bell states. Yves hochedez (talk) 08:36, 6 July 2013 (UTC)[reply]

your state is not normalized correctly (factor 1/sqrt(2) is missing). When you normalize correctly indeed the von neumann entropy is also ln(2) like for the bell states. But the article states (at least now) that the bell states are specific maximally entangled states.--Biggerj1 (talk) 21:11, 10 April 2014 (UTC)[reply]

Biggerj1 (talk) 21:40, 10 April 2014 (UTC)[reply]

Please see "Bell basis" section for details of how all perfect correlations can be written as a sum of Bell basis states. DavidBoden (talk) 22:50, 21 March 2015 (UTC)[reply]

Bell basis

The four Bell states form a Bell basis. A perfect correlation between any two bases on the individual qubits can be described as a sum of Bell states. For example, ${\frac {1}{\sqrt {2}}}(|0+\rangle +|1-\rangle )$ is maximally entangled but not a Bell state; it represents a correlation between the bases $b_{1}$ and $b_{2}=H.b_{1}$ . It can be rewritten as ${\frac {1}{\sqrt {2}}}(|\Phi ^{-}\rangle +|\Psi ^{+}\rangle )$ using Bell basis states.^[a]

Notes

^ ${\frac {1}{\sqrt {2}}}(|0+\rangle +|1-\rangle )$ $={\frac {1}{2}}(|00\rangle +|01\rangle +|10\rangle -|11\rangle )$ $={\frac {1}{\sqrt {2}}}({\frac {1}{\sqrt {2}}}(|00\rangle -|11\rangle )+{\frac {1}{\sqrt {2}}}(|01\rangle )+(|10\rangle )$ $={\frac {1}{\sqrt {2}}}(|\Phi ^{-}\rangle +|\Psi ^{+}\rangle )$

Doubtful statement

Unlike classical phenomena such as the nuclear, electromagnetic, and gravitational fields, entanglement is invariant under distance of separation...

This seems like a category mistake, or just sloppy writing.

If "invariance" refers to the Lorentz group that enforces separability, then indeed, all of the phenomena listed, including all features of quantum mechanics, are invariant under this group.

Indeed, "nuclear, electromagnetic, and gravitational fields" are not "classical phenomena". The first two are quantum phenomena, and electromagnetic fields are the main place where entanglement phenomena can be physically demonstrated.

On the other hand, if the quantum phenomena are modelled by Hilbert spaces and their operators controlled by a dagger category in a kind of "timeless" universe, then they can't be compared very well with such things as nuclear forces with respect to geometric questions.

178.38.97.233 (talk) 12:41, 2 May 2015 (UTC)[reply]

I updated the introduction to be more precise, just now. 2001:2002:51E3:8007:B66D:83FF:FE0E:C298 (talk) 19:33, 23 August 2017 (UTC)[reply]

Bell pair

Bell pair links here, but is not mentioned in this article. Presumably, this is the same as an EPR pair, as mentioned in the article? — Preceding unsigned comment added by 70.247.162.192 (talk) 04:05, 6 May 2016 (UTC)[reply]

"EPR pair" in this article links to the article on entanglement, which however does not contain the term "EPR pair" anywhere. What's the best fix? Vaughan Pratt (talk) 17:22, 30 December 2017 (UTC)[reply]

I am fairly certain that there once was an article named "EPR pair", that was merged into this article (Bell state). A lot of articles reference EPR pair and is redirected to this article, but then the phrase EPR pair does not appear in this article.. I think the two are exactly equivalent. That is, an EPR pair is a Bell state. It seems throughout the literature and random searches on the web, that EPR pair and Bell state is exactly identical. I have not found any definition or anything though! · · · Omnissiahs hierophant (talk) 12:08, 19 April 2021 (UTC)[reply]

Potentially misleading statements

Hello everyone, I think that this sentences: "This perfect correlation at a distance is 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.

Hence, following Einstein, Podolsky, and Rosen in their famous 1935 "EPR paper", there is something missing in the description of the qubit pair given above – namely this "agreement", called more formally a hidden variable."

are not clear and potentially misleading. The affirmations "maybe the two particles "agreed" in advance" and "there is something missing in the description of the qubit pair given above – namely this "agreement", called more formally a hidden variable" have been experimentally falsified by Bell tests as explained in the following sections, but a reader which doesn't read the "Bell basis" section would leave the article with this incorrect idea.

A simple solution would be to start the "Bell basis" section after the sentences: "In his famous paper of 1964, John S. Bell showed by simple probability theory arguments that these correlations (the one for the 0,1 basis and the one for the +,- basis) cannot both be made perfect by the use of any "pre-agreement" stored in some hidden variables – but that quantum mechanics predicts perfect correlations. In a more refined formulation known as the Bell-CHSH inequality, it is shown that a certain correlation measure cannot exceed the value 2 if one assumes that physics respects the constraints of local "hidden variable" theory (a sort of common-sense formulation of how information is conveyed), but certain systems permitted in quantum mechanics can attain values as high as 2 2 {\displaystyle 2{\sqrt {2}}} 2{\sqrt {2}}. Thus, quantum theory violates the Bell inequality and the idea of local "hidden variables.".

Confusing example

When first defining what it means for a state to be a Bell state, the page defines what it means for the states to be entangled as follows.

Their entanglement means the following: The qubit held by Alice (subscript "A") can be in a superposition of 0 and 1. If Alice measured her qubit in the standard basis, the outcome would be either 0 or 1, each with probability 1/2; if Bob (subscript "B") also measured his qubit, the outcome would be the same as for Alice. Thus, Alice and Bob would each seemingly have random outcome. Through communication they would discover that, although their outcomes separately seemed random, these were perfectly correlated.

This seems to specifically exclude the two Ψ Bell states (in which Bob's outcome would always be the exact opposite of Alice's). Perhaps this section could be clarified to indicate that this is only an illustration of what it would mean for two states to be entangled, rather than a definition covering all possible cases. J.Gowers 11:29, 17 October 2023 (UTC)[reply]

[1] ${\frac {1}{\sqrt {2}}}(|0+\rangle +|1-\rangle )$ $={\frac {1}{2}}(|00\rangle +|01\rangle +|10\rangle -|11\rangle )$ $={\frac {1}{\sqrt {2}}}({\frac {1}{\sqrt {2}}}(|00\rangle -|11\rangle )+{\frac {1}{\sqrt {2}}}(|01\rangle )+(|10\rangle )$ $={\frac {1}{\sqrt {2}}}(|\Phi ^{-}\rangle +|\Psi ^{+}\rangle )$

[a]