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arXiv:1409.8400 · Journal of Physics A Mathematical and Theoretical

Local hidden variable theoretic measure of quantumness of mutual information

Ravinder Rupchand Puri

Published Feb 28, 2014• 0 citations
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Abstract

Entanglement, a manifestation of quantumness of correlations between the observables of the subsystems of a composite system, and the quantumness of their mutual information are widely studied characteristics of a system of spin-1/2 particles. The concept of quantumness of correlations between the observables of a system is based on incommensurability of the correlations with the predictions of some local hidden variable (LHV) theory. However, the concept of quantumness of mutual information does not invoke the LHV theory explicitly. In this paper, the concept of quantumness of mutual information for a system of two spin-1/2 particles, named A and B, in the state described by the density matrix is formulated by invoking explicitly the LHV theory. To that end, the classical mutual information I ( a , b ) of the spins is assumed to correspond to the joint probability ( ) for the spin A to have the component in the direction a and the spin B to have the component in the direction b , constructed by invoking the LHV theory. The quantumness of mutual information is then defined as where is the quantum theoretic information content in the state and the LHV theoretic classical information I LHV is defined in terms of I ( a , b ) by choosing the directions a , b as follows. The choice of the directions a , b is made by finding the Bloch vectors and of the spins A and B where ( ) is the spin vector of spin A (spin B) and . The directions a and b are taken to be along the Bloch vector of A and B respectively if those Bloch vectors are non-zero. In that case I LHV = I ( a , b ) and Q LHV turns out to be identical with the measurement induced disturbance. If , then I LHV is defined to be the maximum of I ( a , b ) over a and b . The said optimization in this case can be performed analytically exactly and Q LHV is then found to be the same as the symmetric discord. If , , then I LHV is defined to be the maximum of I ( a , b ) over a with . The Q LHV is then the same as the quantum discord for measurement on A if the eigenstates of are also the eigenstates of the operator on B where a m is the direction of optimization of spin A for evaluation of the quantum discord and | ±, a m 〉 are the eigenstates of .

DOI
10.1088/1751-8113/47/11/115303

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Entanglement, a manifestation of quantumness of correlations between the observables of the subsystems of a composite system, and the quantumness of their mutual information are widely studied characteristics of a system of spin-1/2 particles. The concept of quantumness of correlations between the observables of a system is based on incommensurability of the correlations with the predictions of some local hidden variable (LHV) theory. However, the concept of quantumness of mutual information does not invoke the LHV theory explicitly. In this paper, the concept of quantumness of mutual information for a system of two spin-1/2 particles, named A and B, in the state described by the density matrix is formulated by invoking explicitly the LHV theory. To that end, the classical mutual information I ( a , b ) of the spins is assumed to correspond to the joint probability ( ) for the spin A to have the component in the direction a and the spin B to have the component in the direction b , constructed by invoking the LHV theory. The quantumness of mutual information is then defined as where is the quantum theoretic information content in the state and the LHV theoretic classical information I LHV is defined in terms of I ( a , b ) by choosing the directions a , b as follows. The choice of the directions a , b is made by finding the Bloch vectors and of the spins A and B where ( ) is the spin vector of spin A (spin B) and . The directions a and b are taken to be along the Bloch vector of A and B respectively if those Bloch vectors are non-zero. In that case I LHV = I ( a , b ) and Q LHV turns out to be identical with the measurement induced disturbance. If , then I LHV is defined to be the maximum of I ( a , b ) over a and b . The said optimization in this case can be performed analytically exactly and Q LHV is then found to be the same as the symmetric discord. If , , then I LHV is defined to be the maximum of I ( a , b ) over a with . The Q LHV is then the same as the quantum discord for measurement on A if the eigenstates of are also the eigenstates of the operator on B where a m is the direction of optimization of spin A for evaluation of the quantum discord and | ±, a m 〉 are the eigenstates of .
Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.
In this paper, we consider a hidden variable theoretical description of successive measurements of non-commuting spin observables on an input spin-s state. Although these spin observables are non-commuting, they act on different states, and so the joint probabilities for the outputs of successive measurements are well-defined. We show that, in this scenario, hidden variable theory (HVT) leads to Bell-type inequalities for the correlation between the outputs of successive measurements. We account for the maximum violation of these inequalities by quantum correlations (i.e. the correlations of successive measurements on a quantum state) by varying the spin value and the number of successive measurements. Our approach can be used to obtain a measure of the deviation of Quantum Mechanics from the theory obeying realism and time-locality, in terms of the amount of classical information needed to be transferred between successive measurements in order to simulate the above-mentioned correlations in successive measurements.
Two classically identical expressions for the mutual information generally differ when the systems involved are quantum. This difference defines the quantum discord. It can be used as a measure of the quantumness of correlations. Separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord, thus showing that absence of entanglement does not imply classicality. We relate this to the quantum superposition principle, and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system, i.e., the pointer states.
Quantum mechanics has traditionally relied on entanglement as the primary resource for describing non-classical correlations between distinct physical systems. However, entanglement fails to capture the entirety of mutual information and systemic accessibility, particularly in open, mixed, or high-dimensional environments where local states and shared correlations exhibit distinct scaling behaviors. In this paper, we introduce the concept of Mutual Observability of Quantum States (MOQ), a novel theoretical framework that quantifies how effectively one quantum system can observe or be observed by another, independently of traditional entanglement constraints. We define the mutual observability index V_AB as the ratio of the mutual information I(A:B) to the sum of the local von Neumann entropies S(A) + S(B). Through rigorous mathematical formulation, we establish its strict analytical bounds within the closed interval from 0 to 1, explore its fundamental algebraic properties, and analyze its dynamical behavior in open quantum systems. We demonstrate that MOQ dictates the strength of measurable influence and information transfer between subsystems even when quantum entanglement vanishes entirely. This framework establishes a paradigm shift in understanding quantum correlations, paving the way for advanced applications in quantum communication networks, quantum state estimation, and quantum thermodynamics.
Quantum entanglement and nonlocality are inequivalent notions: There exist entangled states that nevertheless admit local-realistic interpretations. This paper studies a special class of local-hidden-variable theories, in which the linear structure of quantum measurement operators is preserved. It has been proven that a quantum state has such linear hidden-variable representations if, and only if, it is not entangled. Separable states are known to admit nonclassical correlations as well, which are captured by quantum discord and related measures. In the unified framework presented in this paper, zero-discordant states are characterized as the only states that admit fully consistent classical probability representations. Possible generalization of this framework to the quasiprobability representation of multipartite quantum states is also discussed.
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.