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Journal of Mathematical Physics

Maximal violation of Bell’s inequalities and Pauli spin matrices

Martin Bohata, Jan Hamhalter

Published Aug 1, 2009• 4 citations

Abstract

The paper deals with the structure of Bell’s inequalities (Clauser, Horne, Shimony, and Holt (CHSH) form). It is established that 2 is the universal bound for Bell’s correlations given by a general correlation duality on a complex linear space. Using geometric arguments the maximal violation of Bell’s inequalities is described on this abstract level. It is proved that Bell’s inequalities are maximally violated for general ∗-algebras and a faithful state exactly when the corresponding elements are the Pauli spin matrices. Interesting structural consequences of this result are derived. They demonstrate that the maximal violation requires a very “nonclassical” position of the local systems. Moreover, the maximal strength of Bell’s correlations for a general state with respect to nearly commuting subalgebras is characterized in terms of the spin systems and tracial properties of the local states.

DOI: 10.1063/1.3190118

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The paper deals with the structure of Bell’s inequalities (Clauser, Horne, Shimony, and Holt (CHSH) form). It is established that 2 is the universal bound for Bell’s correlations given by a general correlation duality on a complex linear space. Using geometric arguments the maximal violation of Bell’s inequalities is described on this abstract level. It is proved that Bell’s inequalities are maximally violated for general ∗-algebras and a faithful state exactly when the corresponding elements are the Pauli spin matrices. Interesting structural consequences of this result are derived. They demonstrate that the maximal violation requires a very “nonclassical” position of the local systems. Moreover, the maximal strength of Bell’s correlations for a general state with respect to nearly commuting subalgebras is characterized in terms of the spin systems and tracial properties of the local states.

Maximally entangled states should maximally violate the Bell inequality. It is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of maximally entangled states in both the two-qubit and three-qubit systems in terms of the Bell inequality.

Imagine two parties, Alice and Bob, who share an entangled quantum state. A well-established result that if Alice performs a two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt inequality and show that the maximum violation of these Bell inequalities allowed by quantum theory cannot be attained by a bipartite quantum system having support on a qubit at each site.

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.

We show that it is possible to find maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality using only position measurements on a pair of entangled nonrelativistic free particles. The device settings required in the CHSH inequality are done by choosing one of two times at which position is measured. For different assignments of the “+” outcome to positions, namely, to an interval, to a half-line, or to a periodic set, we determine violations of the inequalities and states where they are attained. These results have consequences for the hidden variable theories of Bohm and Nelson, in which the two-time correlations between distant particle trajectories have a joint distribution, and hence cannot violate any Bell inequality.

We show that it is possible to find maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality using only position measurements on a pair of entangled nonrelativistic free particles. The device settings required in the CHSH inequality are done by choosing one of two times at which position is measured. For different assignments of the “+” outcome to positions, namely, to an interval, to a half-line, or to a periodic set, we determine violations of the inequalities and states where they are attained. These results have consequences for the hidden variable theories of Bohm and Nelson, in which the two-time correlations between distant particle trajectories have a joint distribution, and hence cannot violate any Bell inequality.