Bell'sTheorem

1. Bell's Theorem & Bell Inequalities:

John Bell, in the 1960s, formulated a theorem to address the question: "Can the strange predictions of quantum mechanics be explained by some underlying local hidden variable theory?" In other words, can quantum mechanics be explained by some classical (non-quantum) means that respect locality (no faster-than-light influences)?

He derived inequalities (now known as Bell inequalities) that any local hidden variable theory must satisfy. The most famous of these inequalities is the CHSH inequality, named after John Clauser, Michael Horne, Abner Shimony, and Richard Holt.

Consider two entangled quantum particles separated in space. Each particle can be measured along one of two directions, denoted $a$ and $b$, resulting in outcomes $A$ and $B$ which can be either +1 or -1. The expectation value of the product of the outcomes when measurements are made along directions $a$ and $b$ is given by $E(a,b)$.

The CHSH inequality states:

$$|E(a,b) + E(a,b') + E(a',b) - E(a',b')| \leq 2$$

If quantum mechanics were describable by a local hidden variable theory, it would have to respect this inequality.

2. Quantum Mechanics and Violation of Bell Inequalities:

However, when this inequality is tested using the predictions of quantum mechanics (using the state of two entangled particles, like those in the singlet state), it can be shown that:

$$S_{QM} = |E(a,b) + E(a,b') + E(a',b) - E(a',b')| \leq 2\sqrt{2}$$

This $2\sqrt{2}$ value exceeds the 2 from the CHSH inequality, showing that quantum mechanics violates the Bell inequalities. This suggests that no local hidden variable theory can account for the predictions of quantum mechanics.

3. Tsirelson's Bound:

The value $2\sqrt{2}$ mentioned above is also known as Tsirelson's bound. It shows the maximal violation of the CHSH inequality by quantum mechanics. While Bell's inequalities showed that local hidden variable theories can be violated in principle, Tsirelson's bound shows the maximum extent to which quantum mechanics can violate them.

References:

1. Bell, J. S. (1964). On the Einstein Podolsky Rosen Paradox. Physics Physique Физика, 1(3), 195–200.
2. Clauser, J. F., Horne, M. A., Shimony, A., & Holt, R. A. (1969). Proposed Experiment to Test Local Hidden-Variable Theories. Physical Review Letters, 23(15), 880–884.
3. Tsirelson, B. S. (1980). Quantum generalizations of Bell's inequality. Letters in Mathematical Physics, 4(2), 93–100.