Email Gateway (May 17 2023 at 10:01): Email Gateway (May 17 2023 at 10:01):From: "Thiemann, René" <Rene.Thiemann@uibk.ac.at>
The CHSH inequality: Tsirelson's upper-bound and other results
by Mnacho Echenim, Mehdi Mhalla, and Coraline Mori
The CHSH inequality, named after Clauser, Horne, Shimony and Holt, was used by
Alain Aspect to prove experimentally that Einstein's hypothesis stating that
quantum mechanics could be defined using local hidden variables was incorrect.
The CHSH inequality is based on a setting in which an experiment consisting of
two separate parties performing joint measurements is run several times, and a
score is derived from these runs. If the local hidden variable hypothesis had
been correct, this score would have been bounded by 2, but a suitable choice of
observables in a quantum setting permits to violate this inequality when
measuring the Bell state; this is the result that Aspect obtained
experimentally. Tsirelson answered the question of how large this violation
could be by proving that in the quantum setting, the highest score that can be
obtained when running this experiment is 2 * sqrt(2). Along with elementary
results on density matrices which represent quantum states in the finite
dimensional setting, we formalize Tsirelson's result and summarize the main
results on the CHSH score:
Under the local hidden variable hypothesis, this score admits 2 as an
upper-bound.
When the density matrix under consideration is separable, the upper-bound
cannot be violated.
When one of the parties in the experiment performs measures using commuting
observables, this upper-bound remains valid.
Otherwise, the upper-bound of this score is 2 * sqrt(2), regardless of the
observables that are used and the quantum state that is measured, and
This upper-bound is reached for a suitable choice of observables when
measuring the Bell state.
Enjoy!
Last updated: Apr 26 2024 at 04:17 UTC