# Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities

Research paper by **Stephanie Wehner**

Indexed on: **14 Jan '06**Published on: **14 Jan '06**Published in: **Quantum Physics**

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#### Abstract

Quantum theory imposes a strict limit on the strength of non-local
correlations. It only allows for a violation of the CHSH inequality up to the
value 2 sqrt(2), known as Tsirelson's bound. In this note, we consider
generalized CHSH inequalities based on many measurement settings with two
possible measurement outcomes each. We demonstrate how to prove Tsirelson
bounds for any such generalized CHSH inequality using semidefinite programming.
As an example, we show that for any shared entangled state and observables
X_1,...,X_n and Y_1,...,Y_n with eigenvalues +/- 1 we have |<X_1 Y_1> + <X_2
Y_1> + <X_2 Y_2> + <X_3 Y_2> + ... + <X_n Y_n> - <X_1 Y_n>| <= 2 n
cos(pi/(2n)). It is well known that there exist observables such that equality
can be achieved. However, we show that these are indeed optimal. Our approach
can easily be generalized to other inequalities for such observables.