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Physical Review A

Maximal violations of a Bell inequality by entangled spin-coherent states

Christopher C. Gerry, Adil Benmoussa, Edwin E. Hach, Jaroslav Albert

Published Feb 12, 2009• 16 citations
Publisher

Abstract

We study the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for two-spin systems, of arbitrary spins ${j}_{1}$ and ${j}_{2}$, 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, $2\sqrt{2}$, for any choices of the spins.

DOI
10.1103/physreva.79.022111

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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.
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.
We show that a pair of massive relativistic spin-1/2 particles prepared in a maximally entangled spin state in general is not capable of maximally violating the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's inequalities without a postselection of the particles' momenta, representing a major difference in relation to nonrelativistic systems. This occurs because the quantization axis of the measurements performed on each particle depends on the particle velocity, such that it is not possible to define a reduced density matrix for the particles' spin. We also show that the amount of violation of the CHSH inequality depends on the reference frame and that in some frames the inequality may not be violated.
The Bell--Clauser-Horne-Shimony-Holt (BCHSH) inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that nonmaximally entangled states violate the BCHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac\'{\i}n et al. [Phys. Rev. A 65,65,65, 052325 (2002)] using the Bell--Clauser-Horne inequality (in terms of probabilities).
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.
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.
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 222\sqrt{2}22​, 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 X1,…,Xn{X}_{1},\dots{},{X}_{n}X1​,…,Xn​ and Y1,…,Yn{Y}_{1},\dots{},{Y}_{n}Y1​,…,Yn​ 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.