## 9.4 Quantum correlations, revisited

In quantum theory the observables A_1, A_2, B_1, B_2 become 2\times 2 Hermitian matrices with two eigenvalues \pm 1, and \langle S\rangle becomes the expected value of the (4\times 4) CHSH matrix S = A_1\otimes(B_1-B_2) + A_2\otimes(B_1+B_2).

We can now evaluate \langle S\rangle using quantum theory. For example, if the two qubits are in the singlet state, then we know that \langle A\otimes B\rangle = -\vec{a}\cdot\vec{b}. We choose vectors \vec{a}_1, \vec{a}_2, \vec{b}_1, and \vec{b}_2 as shown in Figure 9.1, and then, with these choices, \begin{gathered} \langle A_1\otimes B_1\rangle = \langle A_2\otimes B_1\rangle = \langle A_2\otimes B_2\rangle = \frac{1}{\sqrt 2} \\\langle A_1\otimes B_2\rangle = -\frac{1}{\sqrt 2}. \end{gathered}

Thus \langle A_1 B_1\rangle - \langle A_1 B_2\rangle + \langle A_2 B_1\rangle + \langle A_2 B_2\rangle = -2\sqrt{2}, which obviously violates CHSH inequality.

To be clear, this violation has been observed in a number of painstakingly careful experiments. The early efforts were truly heroic, and the experiments had many layers of complexity. Today, however, such experiments are routine.

The behaviour of entangled quantum systems cannot be explained by any local hidden variables.