Editing Bell's theorem (section)

===GHZ–Mermin (1990)===
{{main|GHZ experiment}}

[[Daniel Greenberger]], [[Michael Horne (physicist)|Michael A. Horne]], and [[Anton Zeilinger]] presented a four-particle thought experiment in 1990, which [[N. David Mermin|David Mermin]] then simplified to use only three particles.<ref name="GHZ1990">{{cite journal |first1=D. |last1=Greenberger |author-link1=Daniel Greenberger |first2=M. |last2=Horne |author-link2=Michael A. Horne |first3=A. |last3=Shimony |author-link3=Abner Shimony |first4=A. |last4=Zeilinger |author-link4=Anton Zeilinger |title=Bell's theorem without inequalities |journal=[[American Journal of Physics]] |volume=58 |issue=12 |pages=1131 |year=1990|bibcode = 1990AmJPh..58.1131G |doi = 10.1119/1.16243 |doi-access=free }}</ref><ref name="mermin1990">{{cite journal |first=N. David |last=Mermin |author-link=N. David Mermin |title=Quantum mysteries revisited |journal=[[American Journal of Physics]] |volume=58 |issue=8 |pages=731–734 |year=1990|bibcode = 1990AmJPh..58..731M |doi = 10.1119/1.16503}}</ref> In this thought experiment, Victor generates a set of three spin-1/2 particles described by the quantum state<math display="block">|\psi\rangle = \frac{1}{\sqrt{2}}(|000\rangle - |111\rangle) \, , </math>
where as above, <math>|0\rangle</math> and <math>|1\rangle</math> are the eigenvectors of the Pauli matrix <math>\sigma_z</math>. Victor then sends a particle each to Alice, Bob, and Charlie, who wait at widely separated locations. Alice measures either <math>\sigma_x</math> or <math>\sigma_y</math> on her particle, and so do Bob and Charlie. The result of each measurement is either <math>+1</math> or <math>-1</math>. Applying the Born rule to the three-qubit state <math>|\psi\rangle</math>, Victor predicts that whenever the three measurements include one <math>\sigma_x</math> and two <math>\sigma_y</math>'s, the product of the outcomes will always be <math>+1</math>. This follows because <math>|\psi\rangle</math> is an eigenvector of <math>\sigma_x \otimes \sigma_y \otimes \sigma_y</math> with eigenvalue <math>+1</math>, and likewise for <math>\sigma_y \otimes \sigma_x \otimes \sigma_y</math> and <math>\sigma_y \otimes \sigma_y \otimes \sigma_x</math>. Therefore, knowing Alice's result for a <math>\sigma_x</math> measurement and Bob's result for a <math>\sigma_y</math> measurement, Victor can predict with probability 1 what result Charlie will return for a <math>\sigma_y</math> measurement. According to the EPR criterion of reality, there would be an "element of reality" corresponding to the outcome of a <math>\sigma_y</math> measurement upon Charlie's qubit. Indeed, this same logic applies to both measurements and all three qubits. Per the EPR criterion of reality, then, each particle contains an "instruction set" that determines the outcome of a <math>\sigma_x</math> or <math>\sigma_y</math> measurement upon it. The set of all three particles would then be described by the instruction set<math display="block">(a_x,a_y,b_x,b_y,c_x,c_y) \, , </math>
with each entry being either <math>-1</math> or <math>+1</math>, and each <math>\sigma_x</math> or <math>\sigma_y</math> measurement simply returning the appropriate value.

If Alice, Bob, and Charlie all perform the <math>\sigma_x</math> measurement, then the product of their results would be <math>a_x b_x c_x</math>. This value can be deduced from<math display="block">(a_x b_y c_y) (a_y b_x c_y) (a_y b_y c_x) = a_x b_x c_x a_y^2 b_y^2 c_y^2 = a_x b_x c_x \, , </math>
because the square of either <math>-1</math> or <math>+1</math> is <math>1</math>. Each factor in parentheses equals <math>+1</math>, so<math display="block">a_x b_x c_x = +1 \, , </math>
and the product of Alice, Bob, and Charlie's results will be <math>+1</math> with probability unity. But this is inconsistent with quantum physics: Victor can predict using the state <math>|\psi\rangle</math> that the measurement <math>\sigma_x \otimes \sigma_x \otimes \sigma_x</math> will instead yield <math>-1</math> with probability unity.

This thought experiment can also be recast as a traditional Bell inequality or, equivalently, as a nonlocal game in the same spirit as the CHSH game.<ref name="Brassard 2004">{{Cite journal|arxiv = quant-ph/0408052|last1 = Brassard|first1 = Gilles|title = Recasting Mermin's multi-player game into the framework of pseudo-telepathy|last2 = Broadbent|first2 = Anne|last3 = Tapp|first3 = Alain|year = 2005 |journal=Quantum Information and Computation |volume=5 |issue=7 |pages=538–550|doi = 10.26421/QIC5.7-2|bibcode = 2004quant.ph..8052B |author-link1 = Gilles Brassard |author-link2 = Anne Broadbent }}</ref> In it, Alice, Bob, and Charlie receive bits <math>x,y,z</math> from Victor, promised to always have an even number of ones, that is, <math>x\oplus y\oplus z = 0</math>, and send him back bits <math>a,b,c</math>. They win the game if <math>a,b,c</math> have an odd number of ones for all inputs except <math>x=y=z=0</math>, when they need to have an even number of ones. That is, they win the game [[if and only if]] <math>a \oplus b \oplus c = x \lor y \lor z</math>. With local hidden variables the highest probability of victory they can have is 3/4, whereas using the quantum strategy above they win it with certainty. This is an example of [[quantum pseudo-telepathy]].