Fano type quantum inequalities in terms of q-entropies

Abstract

The information-theoretic approach to Bell's theorem is developed with use of the conditional qentropies. The q-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional q-entropies for all q ≥ 1. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicioǧlu-Shumovsky (KCBS) scenarios. An extension to the case of n-cycle scenario is briefly mentioned. The new inequalities with conditional q-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The q-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the q-entropic formulation.

Contextuality and the probability representation of quantum states

“…The problem of noncontextuality for measuring observables in three-dimensional Hilbert space discussed in [4,7] is studied in spin-tomographic probability representation of quantum mechanics. The example of five observables, which in the case of the projectors are standard probability distribution, are mapped onto the corresponding regions on the two-dimensional simplex.…”

“…The example of five observables, which in the case of the projectors are standard probability distribution, are mapped onto the corresponding regions on the two-dimensional simplex. The probability distributions obtained in [4,7] by using standard Born rule are expressed in terms of the tomographic probability distributions related to the projectors(observables) and the tomogram of the state when the observables are measured. The mutual relation between the different probability distribution involved in the calculating violation of inequalities which characterize existence of a joint probability distribution for all measured variables is discussed.…”

“…For example, for the qubits the violation of Bell inequalities [1] provides the essential difference of the correlations in separable and nonseparable states. For systems which are not composed ones the example of quantum correlations is given by violation of other kinds of inequalities [2][3][4][5][6][7][8].…”

“…We reformulate some of the results [4,7] identifying both the states and normalized projectors with tomographic probability distributions.…”

Correlation Distance and Bounds for Mutual Information

“…Note that the choice of logarithm base is left open throughout this paper, corresponding to a choice of units. There are a number of such bounds [5], all of which easily generalise to the case of quantum probabilities [2,6,7]. However, in a number of applications of the Pinsker inequality and its quantum analog, a lower bound is in fact only needed for the special case that the relative entropy quantifies the mutual information between two systems.…”