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Sankhya. Series A

REMARKS ON BELL'S INEQUALITY FOR SPIN CORRELATIONS

K. Balasubramanian, J. C. Gupta, Kalyanapuram Rangachari Parthasarathy

Published Jan 1, 1998• 7 citations
OpenAlex

Abstract

where ei,^2,..., en are ?1. It is not difficult to construct real non-negative definite matrices E = ((crij)), 1 5! hj 5: n, satisfying an = 1 for every i but not (1.1). Thus there arises the natural problem of finding simple verifiable conditions on E in order to ensure that it is the correlation matrix of n spin variables. Here we shall find such necessary and sufficient conditions when n = 3 or 4. We also obtain a characterisation of all correlation matrices of n exchangeable spin variables. By using the observables of a fermion field obeying the canonical anticom mutation relations it is possible to realize any non-negative matrix with diagonal entries unity as the correlation matrix of spin observables.

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Russian Journal of Physical Chemistry B•Mar 1, 2017•0 citations
where ei,^2,..., en are ?1. It is not difficult to construct real non-negative definite matrices E = ((crij)), 1 5! hj 5: n, satisfying an = 1 for every i but not (1.1). Thus there arises the natural problem of finding simple verifiable conditions on E in order to ensure that it is the correlation matrix of n spin variables. Here we shall find such necessary and sufficient conditions when n = 3 or 4. We also obtain a characterisation of all correlation matrices of n exchangeable spin variables. By using the observables of a fermion field obeying the canonical anticom mutation relations it is possible to realize any non-negative matrix with diagonal entries unity as the correlation matrix of spin observables.
INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.
It is proven that a necessary and sufficient condition for an nnn-dimensional Hermitian matrix $\ensuremath{\rho}$ to be positive definite is that it be expressible in the form $\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}$, where OOO is a complex orthogonal matrix and EEE is a diagonal matrix with positive elements. This accomplishes a parametrization since OOO has ${n}^{2}\ensuremath{-}n$ real parameters and EEE has nnn of them. The proof is constructive, giving OOO and EEE. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.
For any nnn-partite state ρA1A2⋅⋅⋅Anρ_{A_{1}A_{2}\cdot\cdot\cdot A_{n}}ρA1​A2​⋅⋅⋅An​​, we define its quantum mutual information matrix as an nnn by nnn matrix whose (i,j)(i,j)(i,j)-entry is given by quantum mutual information I(ρAiAj)I(ρ_{A_{i}A_{j}})I(ρAi​Aj​​). Although each entry of quantum mutual information matrix, like its classical counterpart, is also used to measure bipartite correlations, the similarity ends here: quantum mutual information matrices are not always positive semidefinite even for collections of up to 3-partite states. In this work, we obtain necessary and sufficient conditions for the positive semidefinite quantum mutual information matrix. We further define the \emph{genuine} nnn-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states.
The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).