Editing CHSH inequality (section)

=== Tsirelson's inequality and CHSH rigidity ===

{{See also|Tsirelson's bound}}

Tsirelson's inequality, discovered by [[Boris Tsirelson]] in 1980,<ref>{{Cite web|url=https://www.tau.ac.il/~tsirel/download/qbell80.html | title=Quantum generalizations of Bell's inequality|website=www.tau.ac.il}}</ref> states that for any quantum strategy <math>\mathcal{S}</math> for the CHSH game, the bias <math display="inline">\beta^*_{\text{CHSH}}(\mathcal{S}) \leq \frac{1}{\sqrt{2}}</math>. Equivalently, it states that success probability
<math display="block">\omega^*_{\text{CHSH}}(\mathcal{S}) \leq  \cos^2\left(\frac{\pi}{8}\right) = \frac{1}{2} + \frac{1}{2\sqrt{2}}</math>
for any quantum strategy <math>\mathcal{S}</math> for the CHSH game. In particular, this implies the optimality of the quantum strategy described above for the CHSH game.

Tsirelson's inequality establishes that the maximum success probability of ''any'' quantum strategy is <math display="inline">\cos^2\left(\frac{\pi}{8}\right)</math>, and we saw that this maximum success probability is achieved by the quantum strategy described above. In fact, any quantum strategy that achieves this maximum success probability must be isomorphic (in a precise sense) to the canonical quantum strategy described above; this property is called the ''rigidity'' of the CHSH game, first attributed to Summers and Werner.<ref>[https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-110/issue-2/Maximal-violation-of-Bells-inequalities-is-generic-in-quantum-field/cmp/1104159237.full Maximal violation of Bell's inequalities is generic in quantum field theory, Summers and Werner (1987)]</ref>  More formally, we have the following result:

{{math theorem | name = Theorem (Exact CHSH rigidity) | math_statement = Let <math>\mathcal{S} = \left(|\psi\rangle, (A_0, A_1), (B_0, B_1)\right)</math> be a quantum strategy for the CHSH game where <math>|\psi\rangle \in \mathcal{A}\otimes\mathcal{B}</math> such that <math display="inline">\omega_{\text{CHSH}}(\mathcal{S}) = \cos^2\left(\frac{\pi}{8}\right)</math>. Then there exist [[isometries]] <math>V : \mathcal{A}\to\mathcal{A}_1\otimes\mathcal{A}_2</math> and <math>W : \mathcal{B}\to\mathcal{B}_1\otimes\mathcal{B}_2</math> where <math>\mathcal{A}_1,\mathcal{B}_1</math> are isomorphic to <math>\mathbb{C}^2</math> such that letting <math>|\theta\rangle = (V\otimes W)|\psi\rangle</math> we have 
<math display ="block">
|\theta\rangle = |\Phi\rangle_{\mathcal{A}_1, \mathcal{B}_1} \otimes |\phi\rangle_{\mathcal{A}_2,\mathcal{B}_2}
</math>
where <math display="inline">|\Phi\rangle = \frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right)</math> denotes the EPR pair and <math>|\phi\rangle_{\mathcal{A}_2,\mathcal{B}_2}</math> denotes some pure state, and 
<math display ="block">\begin{align}
(V\otimes W)A_0|\psi\rangle = Z_{\mathcal{A}_1}|\theta\rangle, & \qquad (V\otimes W)B_0|\psi\rangle = Z_{\mathcal{B}_1}|\theta\rangle,\\
(V\otimes W)A_1|\psi\rangle = X_{\mathcal{A}_1}|\theta\rangle, & \qquad (V\otimes W)B_1|\psi\rangle = Z_{\mathcal{B}_1}|\theta\rangle.
\end{align}</math>}}

Informally, the above theorem states that given an arbitrary optimal strategy for the CHSH game, there exists a local change-of-basis (given by the isometries <math>V, W</math>) for Alice and Bob such that their shared state <math>|\psi\rangle</math> factors into the tensor of an EPR pair <math>|\Phi\rangle</math> and an additional auxiliary state <math>|\phi\rangle</math>. Furthermore, Alice and Bob's observables <math>(A_0, A_1)</math> and <math>(B_0, B_1)</math> behave, up to unitary transformations, like the <math>Z</math> and <math>X</math> observables on their respective qubits from the EPR pair. An ''approximate'' or ''quantitative'' version of CHSH rigidity was obtained by McKague, et al.<ref>{{Cite journal | url=http://dx.doi.org/10.1088/1751-8113/45/45/455304|title=Robust self-testing of the singlet|first1=M|last1=McKague|first2=T H|last2=Yang|first3=V|last3=Scarani|date=October 19, 2012|journal=Journal of Physics A: Mathematical and Theoretical |volume=45 |issue=45|pages=455304|doi=10.1088/1751-8113/45/45/455304|arxiv=1203.2976|s2cid=118535156 }}</ref> who proved that if you have a quantum strategy <math>\mathcal{S}</math> such that <math display="inline">\omega_{\text{CHSH}}(\mathcal{S}) = \cos^2\left(\frac{\pi}{8}\right) - \epsilon</math> for some <math>\epsilon > 0</math>, then there exist isometries under which the strategy <math>\mathcal{S}</math> is <math>O(\sqrt{\epsilon})</math>-close to the canonical quantum strategy. Representation-theoretic proofs of approximate rigidity are also known.<ref>{{Cite web|url=http://users.cms.caltech.edu/~vidick/ucsd_games.pdf|title=UCSD Summer School Notes: Quantum multiplayer games, testing and rigidity, Thomas Vidick (2018)}}</ref>