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 nonclassical 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.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 nonclassical 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.

arXiv:2205.06477May 13, 20220 citations

Measure of quantum correlations that lies approximately between entanglement and discord

Aaron Szasz

When a quantum system is divided into two local subsystems, measurements on the two subsystems can exhibit correlations beyond those possible in a classical joint probability distribution; these are partially explained by entanglement, and more generally by a wider class of measures such as the quantum discord. In this paper, I introduce a simple thought experiment defining a measure of quantum correlations, which I call the accord, and write the result as a minimax optimization over unitary matrices. I find the exact result for pure states as a simple function of the Schmidt coefficients and provide a complete proof, and I likewise provide and prove the result for several classes of mixed states, notably including all states of two qubits and the experimentally relevant case of a pure state mixed with colorless noise. I demonstrate that for two qubit states the accord provides a tight lower bound on the discord; for Bell diagonal states it is also an upper bound on entanglement.When a quantum system is divided into two local subsystems, measurements on the two subsystems can exhibit correlations beyond those possible in a classical joint probability distribution; these are partially explained by entanglement, and more generally by a wider class of measures such as the quantum discord. In this paper, I introduce a simple thought experiment defining a measure of quantum correlations, which I call the accord, and write the result as a minimax optimization over unitary matrices. I find the exact result for pure states as a simple function of the Schmidt coefficients and provide a complete proof, and I likewise provide and prove the result for several classes of mixed states, notably including all states of two qubits and the experimentally relevant case of a pure state mixed with colorless noise. I demonstrate that for two qubit states the accord provides a tight lower bound on the discord; for Bell diagonal states it is also an upper bound on entanglement.

arXiv:1810.12949Physical Review AJun 12, 20195 citations

Quantum discord for qubit–qudit systems

Sai Vinjanampathy, A. R. P. Rau

Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.

arXiv:1106.4488Journal of Physics A Mathematical and TheoreticalFeb 20, 2012112 citations

Quantum discord for two-qubit X states

Mazhar Ali, A. R. P. Rau, G. Alber

Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called $X$ states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called $X$ states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.

arXiv:1002.3429Physical Review AApr 12, 2010835 citations

An introduction to quantum discord and non-classical correlations beyond entanglement

Gerardo Adesso, Marco Cianciaruso, Thomas R. Bromley

In this didactic article we explore the concept of quantum correlations beyond entanglement. We begin by introducing and motivating the classically correlated states and then showing how to quantify the quantum correlations using an entropic approach, arriving at a well known measure called the quantum discord. Quantum correlations and discord are then operationally linked with the task of local broadcasting. We conclude by providing some alternative perspectives on quantum correlations and how to measure them.In this didactic article we explore the concept of quantum correlations beyond entanglement. We begin by introducing and motivating the classically correlated states and then showing how to quantify the quantum correlations using an entropic approach, arriving at a well known measure called the quantum discord. Quantum correlations and discord are then operationally linked with the task of local broadcasting. We conclude by providing some alternative perspectives on quantum correlations and how to measure them.

arXiv:1611.01959Nov 7, 201611 citations

Correlated multipartite quantum states

Josep Amengual i Batle, M. Casas, Angel Ricardo Plastino

We investigate quantum states that possess both maximum entanglement and maximum discord between the pertinent parties. Since entanglement (discord) is defined only for bipartite (two-qubit) systems, we use an appropriate sum over all bipartitions as the associated measure. The ensuing definition---not new for entanglement---is thus extended here to quantum discord. Also, additional dimensions within the parties are considered (qudits). We also discuss quantum correlations that induce Mermin's Bell-inequality violation for all multiqubit systems. One finds some differences when quantum mechanics is defined over the field of real or of complex numbers.We investigate quantum states that possess both maximum entanglement and maximum discord between the pertinent parties. Since entanglement (discord) is defined only for bipartite (two-qubit) systems, we use an appropriate sum over all bipartitions as the associated measure. The ensuing definition---not new for entanglement---is thus extended here to quantum discord. Also, additional dimensions within the parties are considered (qudits). We also discuss quantum correlations that induce Mermin's Bell-inequality violation for all multiqubit systems. One finds some differences when quantum mechanics is defined over the field of real or of complex numbers.

arXiv:1212.0151Physical Review AMar 13, 20133 citations

X

states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.