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arXiv:quant-ph/0510076 · Physical Review A

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

Stephanie Wehner

Published Feb 14, 2006• 170 citations
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Abstract

Quantum theory imposes a strict limit on the strength of nonlocal correlations. It only allows for a violation of the Clauser, Horne, Shimony, and Holt (CHSH) inequality up to the value $2\sqrt{2}$, known as Tsirelson's bound. In this paper, 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 ${X}_{1},\dots{},{X}_{n}$ and ${Y}_{1},\dots{},{Y}_{n}$ with eigenvalues $\ifmmode\pm\else\textpm\fi{}1$ we have $\ensuremath{\mid}⟨{X}_{1}{Y}_{1}⟩+⟨{X}_{2}{Y}_{1}⟩+⟨{X}_{2}{Y}_{2}⟩+⟨{X}_{3}{Y}_{2}⟩+\ensuremath{\cdots}+⟨{X}_{n}{Y}_{n}⟩\ensuremath{-}⟨{X}_{1}{Y}_{n}⟩\ensuremath{\mid}\ensuremath{\leqslant}2n\phantom{\rule{0.2em}{0ex}}\mathrm{cos}[\ensuremath{\pi}∕(2n)]$. 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.

DOI
10.1103/physreva.73.022110

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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.
Tsirelson's bound is the upper bound for a Bell inequality which is valid for all quantum mechanical systems. We discuss why Tsirelson's bound was developed by looking at some historical arguments in quantum physics, such as the Einstein-Podolsky-Rosen (EPR) paradox, an argument for the quantum mechanical description of physical reality being incomplete, and local hidden variables. We present the counterargument to those theories, Bell's inequality, which later expanded to include any inequality that a local system fulfills, but that an entangled quantum system can violate. We present the proof of two specific Bell inequalities: the Clauser-Horne-Shimony-Holt (CHSH) inequality and the I3322 inequality. Then the Tsirelson's bound for the CHSH inequality is proven with a simple system of two entangled spin-1/2 particles and with a general argument that is valid for all entangled systems. We give the upper quantum bound for the generalized CHSH inequality, which describes the situation that we have more than two measurement options, by using semidefinite programming. We prove the Tsirelson's bound for the I3322 inequality by using maximally entangled systems and semidefinite programming. Finally, we discuss the upper bounds that are obtained from these different methods.
The best upper bound for the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality was first derived by Tsirelson. For increasing number of ±1 valued observables on both sites of the correlation experiment, Tsirelson obtained the Grothendieck's constant as a limit for the maximal violation. In this paper, we construct a generalization of the CHSH inequality with four ±1 valued observables on both sites of a correlation experiment and show that the quantum violation approaches 1.58. Moreover, we estimate the maximal quantum violation of a correlation experiment for large and equal number of ±1 valued observables on both sites. In this case, the maximal quantum violation converges to for very large n , which coincides with the approximate value of Grothendieck's constant.
We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to nnn qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.
We study the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for two-spin systems, of arbitrary spins j1{j}_{1}j1​ and j2{j}_{2}j2​, prepared in an entanglement of spin-coherent states of each of the spins. We show that the Bell-CHSH inequality is quite robustly violated for a wide range of values of the parameters that specify the spin-coherent states, and, for a particular choice of these parameters, maximal violations are obtained. That is, the violations can reach the Tsirelson bound, 222\sqrt{2}22​, for any choices of the spins.
Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.