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arXiv:1105.4920 · International Journal of Quantum Information

ENTROPIC MEASURES OF NON-CLASSICAL CORRELATIONS

MATTHIAS D. LANG, Carlton M. Caves, Anil Shaji

Published Oct 1, 2011• 51 citations
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Abstract

A framework for categorizing entropic measures of non-classical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of non-classical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit quantum states are presented.

DOI
10.1142/s021974991100826x

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A framework for categorizing entropic measures of non-classical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of non-classical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit quantum states are presented.
In the first part of this dissertation a framework for categorizing entropic measures of nonclassical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of nonclassical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit quantum states are presented. To demonstrate how these measures differ from entanglement we move to the set of Bell-diagonal states for two qubits, which can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and of the correlation measures from our framework for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that all of the proposed correlation measures are neither convex nor concave. In the second part we look at two practical interferometric setups that use nonclassical states of light to enhance their performance. First we consider an interferometer powered by laser light (a coherent state) into one input port and ask the following question: what is the best state to inject into the second input port, given a constraint on the mean number of photons this state can carry, in order to optimize the interferometer's phase sensitivity? This question is the practical question for high-sensitivity interferometry. We answer the question by considering the quantum Cram\'er-Rao bound for such a setup. The answer is squeezed vacuum. Then we analyze the ultimate bounds on the phase sensitivity of an interferometer, given the constraint that the state input to the interferometer's initial 50:50 beam splitter B is a product state of the two input modes. Requiring a product state is a natural restriction: if one were allowed to input an arbitrary, entangled two-mode state ∣Ξ⟩|\Xi \rangle∣Ξ⟩ to the beam splitter, one could generally just as easily input the state B∣Ξ⟩B|\Xi \rangleB∣Ξ⟩ directly into the two modes after the beam splitter, thus rendering the beam splitter unnecessary. We find optimal states for a fixed photon number and for a fixed mean photon number. Our results indicate that entanglement is not a crucial resource for quantum-enhanced interferometry. Initially the analysis for both of these setups is performed for the idealized case of a lossless interferometer. Then the analysis is extended to the more realistic scenario where the interferometer suffers from photon losses.
How can we characterize different types of correlation between quantum systems? Since correlations cannot be generated locally, we take any real function of a multipartite state which cannot increase under local operations to measure a correlation. Correlation measures that can be expressed as an optimization of a linear combination of entropies are particularly useful, since they can often be interpreted operationally. We systematically study such optimized linear entropic functions, and by enforcing monotonicity under local processing we identify four cones of correlation measures for bipartite quantum states. This yields two new optimized measures of bipartite quantum correlation that are particularly simple, which have the additional property of being additive.
The dynamics of three different entropic measures of quantum correlations in mixed bipartite qubit states in the presence of two different classical noises, the static noise (SN) and the random telegraph noise (RTN), are investigated. The three entropic measures of quantum correlations correspond to one-way information deficit, geometric quantum discord and the cubic information. General analytic relations for each quantifier in the two configurations are obtained. In both configurations, the minimized value of each measure of quantum correlations corresponds to the conditional entropy of the same projectors. It is shown that one-way information deficit captures more correlations in highly mixed initial states. On the contrary, in both configurations the cubic information reduces to the geometric quantum discord and captures more correlations for highly pure initial states. The periodic revival of each measure of quantum correlation is more prominent in the case of RTN.
This paper introduces a novel method for assessing the trustworthiness of quantum state measurement results based on non-classical correlation entropy. The core idea revolves around quantifying the correlation within quantum states using this entropy measure. By analyzing the associated entropy, we can evaluate the reliability of measurement outcomes and subsequently propose corrective and optimization strategies. Traditional approaches to quantum measurement uncertainty often rely on probabilistic interpretations and lack a direct quantification of the inherent correlations within the quantum state itself. This work aims to bridge this gap by providing a framework for assessing trustworthiness rooted in the fundamental properties of quantum entanglement and correlation. We present a detailed mathematical formulation, focusing on the calculation of the non-classical correlation entropy and its application to evaluating measurement confidence. The methodology offers a potentially valuable tool for guiding decision-making in quantum information processing, particularly in scenarios where the precise characterization of the quantum state is challenging. Further investigation into the sensitivity of the method to various noise sources and its scalability for complex quantum systems is warranted. ---
Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well-known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these non-classical correlations is discord. Here we introduce a new measure of quantum correlations which we call entropic accord that fits between entanglement and discord. It is defined as the optimised minimax mutual information of the outcome of the projective measurements between two parties. We show a strict hierarchy exists between entanglement, entropic accord and discord for two-qubit states. We study two-qubit states which shows the relationship between the three entropic quantities. In addition to revealing a class of correlations that are distinct from discord and entanglement, the entropic accord measure can be inherently more intuitive in certain contexts.
We propose a general measure of nonclassical correlations for bipartite systems based on generalized entropic functions and majorization properties. Defined as the minimum information loss due to a local measurement, in the case of pure states it reduces to the generalized entanglement entropy, i.e., the generalized entropy of the reduced state. However, in the case of mixed states it can be nonzero in separable states, vanishing just for states diagonal in a general product basis, like the quantum discord. Simple quadratic measures of quantum correlations arise as a particular case of the present formalism. The minimum information loss due to a joint local measurement is also discussed. The evaluation of these measures in simple relevant cases is as well provided, together with comparison with the corresponding entanglement monotones.