9.5 Tsirelson’s inequality

An upper bound on quantum correlations.

One may ask if |\langle S\rangle|= 2\sqrt{2} is the maximal violation of the CHSH inequality, and the answer is “yes, it is”: quantum correlations cannot achieve any value of |\langle S\rangle| larger than 2\sqrt{2}. This is because, for any state |\psi\rangle, the expected value \langle S\rangle = \langle\psi|S|\psi\rangle cannot exceed the largest eigenvalue of S, and we can put an upper bound on the largest eigenvalues of S. To start with, the largest eigenvalue (in absolute value) of a Hermitian matrix M, denoted by \|M\|, is a matrix norm, and it has the following properties: \begin{aligned} \|M\otimes N\| & = \|M\| \|N\| \\\|MN\| & \leqslant\|M\| \|N\| \\\|M+N\| & \leqslant\|M\| + \|N\| \end{aligned} Given that \|A_i\|=1 and \|B_j\|=1 (i,j=1,2), it is easy to show that \|S\| < 4. One can, however, derive a tighter bound. We can show (do it) that S^2 = 4(\mathbf{1}\otimes\mathbf{1}) + [A_1,A_2]\otimes[B_1,B_2]. The norm of each of the commutators (\|[A_1, A_2]\| and \|[B_1, B_2]\|) cannot exceed 2, and \|S^2\|=\|S\|^2, which all together gives \|S\| < 2\sqrt{2} \implies |\langle S\rangle| < 2\sqrt{2}. This result is known as the Tsirelson inequality.