Editing CHSH inequality (section)

=== Modeling general quantum strategies ===

An arbitrary quantum strategy for the CHSH game can be modeled as a triple <math>\mathcal{S} = \left(|\psi\rangle, (A_{0}, A_1), (B_0, B_1)\right)</math> where 
* <math>|\psi\rangle \in \mathbb{C}^d \otimes \mathbb{C}^d</math> is a bipartite state for some <math>d</math>,
* <math>A_{0}</math> and <math>A_{1}</math> are Alice's [[observable]]s each corresponding to receiving <math>x\in\{0,1\}</math> from the referee, and
* <math>B_{0}</math> and <math>B_{1}</math> are Bob's observables each corresponding to receiving <math>y\in\{0,1\}</math> from the referee.
The optimal quantum strategy described above can be recast in this notation as follows: <math>|\psi\rangle \in\mathbb{C}^2\otimes\mathbb{C}^2</math> is the EPR pair <math display="inline">|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)</math>, the observable <math>A_0 = Z</math> (corresponding to Alice measuring in the <math>\{|0\rangle, |1\rangle\}</math> basis), the observable <math>A_1 = X</math> (corresponding to Alice measuring in the <math>\{|+\rangle, |-\rangle\}</math> basis), where <math>X</math> and <math>Z</math> are [[Pauli matrices]]. The observables <math display="inline">B_{0} = \frac{1}{\sqrt{2}}(X+Z)</math> and <math display="inline">B_{1} = \frac{1}{\sqrt{2}}(Z-X)</math> (corresponding to each of Bob's choice of basis to measure in). 
We will denote the success probability of a strategy <math>\mathcal{S}</math> in the CHSH game by <math>\omega^*_{\text{CHSH}}(\mathcal{S})</math>, and we define the ''bias'' of the strategy <math>\mathcal{S}</math> as <math>\beta^*_{\text{CHSH}}(\mathcal{S}) := 2\omega^*_{\text{CHSH}}(\mathcal{S}) - 1</math>, which is the difference between the winning and losing probabilities of <math>\mathcal{S}</math>.

In particular, we have 
<math display="block">
\beta^*_{\text{CHSH}}(\mathcal{S}) = \frac{1}{4} \sum_{x,y \in \{0,1\}} (-1)^{x\wedge y} \cdot \langle \psi | A_x\otimes B_y |\psi\rangle.
</math>
The bias of the quantum strategy described above is <math display="inline">\frac{1}{\sqrt{2}}</math>.