Editing CHSH inequality (section)

== Optimal violation by a general quantum state ==
In experimental practice, the two particles are not an ideal [[EPR pair]]. There is a necessary and sufficient condition for a two-[[qubit]] [[density matrix]] <math>\rho</math> to violate the CHSH inequality, expressed by the maximum attainable polynomial ''S''<sub>max</sub> defined in {{EquationNote|2|Eq. 2}}.<ref name=Horodecki-1995>{{citation |author1=R. Horodecki |author2=P. Horodecki |author3=M. Horodecki |year=1995 |title=Violating Bell inequality by mixed spin-<math>\tfrac{1}{2}</math> states: Necessary and sufficient condition |journal=Phys.Lett. A |volume=200 |issue=5 |pages=340–344 |doi=10.1016/0375-9601(95)00214-N}}</ref> This is important in entanglement-based [[quantum key distribution]], where the secret key rate depends on the degree of measurement correlations.<ref name=Pironio-2009>{{citation | author1=Stefano Pironio |author2=Antonio Acín |author3=Nicolas Brunner |author4=Nicolas Gisin |author5=Serge Massar |author6=Valerio Scarani | year=2009 | title=Device-independent quantum key distribution secure against collective attacks | journal=New J. Phys. | volume=11 |issue=4 | pages=045021 | doi=10.1088/1367-2630/11/4/045021|arxiv=0903.4460 |bibcode=2009NJPh...11d5021P |s2cid=7971771 |doi-access=free }}</ref>

Let us introduce a 3×3 real matrix <math>T_{\rho}</math> with elements <math>t_{ij} = \operatorname{Tr}[\rho\cdot(\sigma_i \otimes \sigma_j)]</math>, where <math>\sigma_1, \sigma_2, \sigma_3</math> are the [[Pauli matrices]]. Then we find the [[eigenvalues and eigenvectors]] of the real symmetric matrix <math>U_\rho = T_\rho^\text{T} T_\rho</math>,
<math display=block>
U_\rho \boldsymbol{e}_i = \lambda_i \boldsymbol{e}_i, \quad |\boldsymbol{e}_i| = 1, \quad i=1,2,3,
</math>
where the indices are sorted by <math>\lambda_1 \geq \lambda_2 \geq \lambda_3</math>. Then, the maximal CHSH polynomial is determined by the two greatest eigenvalues,<ref name="Horodecki-1995"/>
<math display=block>
S_\text{max}(\rho) = 2\sqrt{\lambda_1 + \lambda_2}.
</math>

=== Optimal measurement bases ===

There exists an optimal configuration of the measurement bases ''a, a', b, b''' for a given <math>\rho</math> that yields ''S''<sub>max</sub> with at least one free parameter.<ref name=Kofman-2012>{{citation |author=A. G. Kofman |year=2012 |title=Optimal conditions for Bell-inequality violation in the presence of decoherence and errors | journal=Quantum Inf. Process. |volume=11 |pages=269–309 |doi=10.1007/s11128-011-0242-1|arxiv=0804.4167 |s2cid=41329613 }}</ref><ref name=Hosak-2021>{{citation | author1=R. Hošák | author2=I. Straka | author3=A. Predojević | author4=R. Filip | author5=M. Ježek | year=2021 | title=Effect of source statistics on utilizing photon entanglement in quantum key distribution | journal=Phys. Rev. A | volume=103 | issue=4 | pages=042411 | doi=10.1103/PhysRevA.103.042411| arxiv=2008.07501 | bibcode=2021PhRvA.103d2411H | s2cid=221140079 }}</ref>

The projective measurement that yields either +1 or −1 for two orthogonal states <math>|\alpha\rangle, |\alpha^\perp\rangle</math> respectively, can be expressed by an operator <math>\Alpha = |\alpha\rangle\langle\alpha| - |\alpha^\perp\rangle\langle\alpha^\perp|</math>. The choice of this measurement basis can be parametrized by a real unit vector <math>\boldsymbol{a} \in \mathbb{R}^3, |\boldsymbol{a}|=1</math> and the [[Pauli vector]] <math>\boldsymbol{\sigma}</math> by expressing <math>\Alpha = \boldsymbol{a} \cdot \boldsymbol{\sigma}</math>. Then, the expected correlation in bases ''a, b'' is
<math display=block>
E(a,b) = \operatorname{Tr}[\rho(\boldsymbol{a} \cdot \boldsymbol{\sigma})\otimes(\boldsymbol{b} \cdot \boldsymbol{\sigma})] = \boldsymbol{a}^\text{T} T_\rho \boldsymbol{b}.
</math>
The numerical values of the basis vectors, when found, can be directly translated to the configuration of the projective measurements.<ref name=Hosak-2021/>

The optimal set of bases for the state <math>\rho</math> is found by taking the two greatest eigenvalues <math>\lambda_{1,2}</math> and the corresponding eigenvectors <math>\boldsymbol{e}_{1,2}</math> of <math>U_\rho</math>, and finding the auxiliary unit vectors
<math display=block>
\begin{align}
\boldsymbol{c} &= \boldsymbol{e}_1 \cos\varphi + \boldsymbol{e}_2 \sin\varphi, \\
\boldsymbol{c}' &= \boldsymbol{e}_1 \sin\varphi - \boldsymbol{e}_2 \cos\varphi,
\end{align}
</math>
where <math>\varphi</math> is a free parameter. We also calculate the acute angle
<math display=block>
\theta = \operatorname{arctan} \sqrt{\frac{\lambda_2 + \lambda_1 \tan^2{\varphi}}{\lambda_1 + \lambda_2 \tan^2{\varphi}}}
</math>
to obtain the bases that maximize {{EquationNote|2|Eq. 2}},
<math display=block>
\begin{align}
\boldsymbol{a} &= T_\rho \boldsymbol{c}' / |T_\rho \boldsymbol{c}'|, \\
\boldsymbol{a}' &= T_\rho \boldsymbol{c} / |T_\rho \boldsymbol{c}|, \\
\boldsymbol{b} &= \boldsymbol{c} \cos\theta + \boldsymbol{c}' \sin\theta, \\
\boldsymbol{b}' &= \boldsymbol{c} \cos\theta - \boldsymbol{c}' \sin\theta.
\end{align}
</math>

In entanglement-based [[quantum key distribution]], there is another measurement basis used to communicate the secret key (<math>\boldsymbol{a}_0</math> assuming Alice uses the side A). The bases <math>\boldsymbol{a}_0, \boldsymbol{b}</math> then need to minimize the quantum bit error rate ''Q'', which is the probability of obtaining different measurement outcomes (+1 on one particle and −1 on the other).<ref name=Pironio-2009/> The corresponding bases are<ref name=Hosak-2021/>
<math display=block>
\begin{align}
\boldsymbol{a}_0 &= T_\rho \boldsymbol{e}_1 / |T_\rho \boldsymbol{e}_1|, \\
\boldsymbol{b} &= \boldsymbol{e}_1.
\end{align}
</math>
The CHSH polynomial ''S'' needs to be maximized as well, which together with the bases above creates the constraint <math>\varphi = \pi/4</math>.<ref name=Hosak-2021/>