In quantum information theory, quantum discord is a measure of nonclassical correlations between two subsystems of a quantum system. It includes correlations that are due to quantum physical

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 effects but do not necessarily involve quantum entanglement

Quantum entanglement

Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

. The notion of quantum discord was introduced in 2001 by Harold Ollivier and Wojciech H. Zurek

Wojciech H. Zurek

Wojciech Hubert Zurek is a well-known physicist and a Laboratory Fellow at Los Alamos National Laboratory. He is a leading authority on quantum theory, especially decoherence. His work also has a lot of potential benefit to the emerging field of quantum computing.Zurek earned his M.Sc. in Kraków,...

 and, independently, by L. Henderson and Vlatko Vedral

Vlatko Vedral

Vlatko Vedral is a Serbian born physicist and Professor of Physics at the University of Oxford and CQT at the National University of Singapore and a Fellow of Wolfson College. He is known for his research on the theory of Entanglement and Quantum Information Theory...

. Olliver and Zurek referred to it also as a measure of quantumness of correlations. From the work of these two research groups it follows that quantum correlations can be present in certain mixed separable states

Separable states

In quantum mechanics, separable quantum states are states without quantum entanglement.- Separable pure states :For simplicity, the following assumes all relevant state spaces are finite dimensional...

; In other words, separability alone does not imply the absence of quantum effects. The notion of quantum discord thus goes beyond the distinction which had been made earlier between entangled versus separable (non-entangled) quantum states.

Definition and mathematical relations

In mathematical terms, quantum discord is defined in terms of the quantum mutual information

Quantum mutual information

In quantum information theory, quantum mutual information, or von Neumann mutual information, after John von Neumann, is a measure of correlation between subsystems of quantum state...

. More specifically, quantum discord is the difference between two expressions which each, in the classical limit

Classical limit

The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior...

, represent the mutual information

Mutual information

In probability theory and information theory, the mutual information of two random variables is a quantity that measures the mutual dependence of the two random variables...

. These two expressions are:$I\; (A;\; B)\; =\; H\; (A)\; +\; H\; (B)\; -\; H\; (A,B)$$J\; (A;\; B)\; =\; H\; (A)\; -\; H\; (A|B)$ where, in the classical case, $H(A)$ is the information entropy

Information entropy

In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

, $H(A,B)$ the joint entropy and $H(A|B)$ the conditional entropy

Conditional entropy

In information theory, the conditional entropy quantifies the remaining entropy of a random variable Y given that the value of another random variable X is known. It is referred to as the entropy of Y conditional on X, and is written H...

, and the two expressions yield identical results. In the nonclassical case, the quantum physics analogy for the three terms are used – $S\; (\backslash rho\_A)$ the von Neumann entropy

Von Neumann entropy

In quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics....

, $S(\backslash rho)$ the joint quantum entropy

Joint quantum entropy

The joint quantum entropy generalizes the classical joint entropy to the context of quantum information theory. Intuitively, given two quantum states \rho and \sigma, represented as density operators that are subparts of a quantum system, the joint quantum entropy is a measure of the total...

 and $S\; (\backslash rho\_A|\backslash rho\_B)$ the conditional quantum entropy

Conditional quantum entropy

The conditional quantum entropy is an entropy measure used in quantum information theory. It is a generalization of the conditional entropy of classical information theory...

, respectively, for probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 $\backslash rho$$I\; (\backslash rho)\; =\; S\; (\backslash rho\_A)\; +\; S\; (\backslash rho\_B)\; -\; S\; (\backslash rho)$$J\_A\; (\backslash rho)\; =\; S\; (\backslash rho\_B)\; -\; S\; (\backslash rho\_B|\backslash rho\_A)$ The difference between the two expressions $I(\backslash rho)\; -\; J\_A(\backslash rho)$ defines the basis-dependent quantum discord, which is asymmetrical in the sense that $\backslash mathcal\; D\_A\; (\backslash rho)$ can differ from $\backslash mathcal\; D\_B\; (\backslash rho)$. $J$ represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen eigenbasis; therefore, in order for the quantum discord to reflect the purely nonclassical correlations independently of basis, it is necessary that $J$ first be maximized over the set of all possible projective

Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space P of a complex Hilbert space H is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by...

 measurements onto the eigenbasis:$\backslash mathcal\; D\_A\; (\backslash rho)\; =\; I\; (\backslash rho)\; -\; \backslash max\_\{\backslash \{\backslash Pi\_j^A\backslash \}\}\; J\_\{\backslash \{\backslash Pi\_j^A\backslash \}\}\; (\backslash rho)\; =\; S\; (\backslash rho\_A)\; -\; S(\backslash rho)\; +\; \backslash min\_\{\backslash \{\backslash Pi\_j^A\backslash \}\}\; S\; (\backslash rho\_\{B\; |\; \backslash \{\backslash Pi\_j^A\backslash \}\}\; )$ Nonzero quantum discord indicates the presence of correlations that are due to noncommutativity of quantum operators. For pure states, the quantum discord becomes a measure of quantum entanglement

Quantum entanglement

Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

, more specifically, in that case it equals the entropy of entanglement. Vanishing quantum discord is a criterion for the pointer states, which constitute preferred effectively classical states of a system. It could be shown that quantum discord must be non-negative and that states with vanishing quantum discord can in fact be identified with pointer states. Other conditions have been identified which can be seen in analogy to the Peres–Horodecki criterion and in relation to the strong subadditivity of the von Neumann entropy. Efforts have been made to extend the definition of quantum discord to continuous variable systems, in particular to bipartite systems described by Gaussian states.

Properties

Quantum discord has been given a physical interpretation, in operational terms, as an “entanglement consumption in an extended quantum state merging

State-merging

In quantum information theory, quantum state merging is the transfer of a quantum state when the receiver already has part of the state. The process optimally transfers partial information using entanglement and classical communication. It allows for sending information using an amount of...

 protocol”. Providing evidence for non-entanglement quantum correlations normally involves elaborate quantum tomography

Quantum tomography

Quantum tomography or quantum state tomography is the process of reconstructing the quantum state for a source of quantum systems by measurements on the systems coming from the source. The source may be any device or system which prepares quantum states either consistently into quantum pure states...

 methods; however, in 2011, such correlations could be demonstrated experimentally in a room temperature nuclear magnetic resonance system, using chloroform

Chloroform

Chloroform is an organic compound with formula CHCl3. It is one of the four chloromethanes. The colorless, sweet-smelling, dense liquid is a trihalomethane, and is considered somewhat hazardous...

 molecules that represent a two-qubit

Qubit

In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

 quantum system. Quantum discord has been seen as a possible basis for the performance in terms of quantum computation ascribed to certain mixed-state quantum systems, with a mixed quantum state representing a statistical ensemble of pure states (see quantum statistical mechanics

Quantum statistical mechanics

Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be...

). Evidence has been provided for poignant differences between the properties of quantum entanglement and quantum discord. It has been shown that quantum discord is more resilient to dissipative environments than is quantum entanglement. This has been shown for Markovian environments as well as for non-Markovian environments based on a comparison of the dynamics of discord with that of concurrence, where discord has proven to be more robust. It has been shown that, at least for certain models of a qubit pair which is in thermal equilibrium and form an open quantum system

Open quantum system

In physics, an open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment...

 in contact with a heat bath

Heat reservoir

In thermodynamics, a heat reservoir, thermal reservoir, or heat bath is a system whose heat capacity is so large that when it is in thermal contact with some other system of interest its temperature remains effectively constant. The heat bath is effectively an infinite reservoir of energy and...

, the quantum discord increases with temperature in certain temperature ranges, thus displaying a behaviour that is quite in contrast with that of entanglement, and that furthermore, surprisingly, the classical correlation actually decreases as the quantum discord increases. Nonzero quantum discord can persist even in the limit of one of the subsystems undergoing an infinite acceleration, whereas under this condition the quantum entanglement drops to zero due to the Unruh effect

Unruh effect

The Unruh effect , was first described by Stephen Fulling in 1973, Paul Davies in 1975 and Bill Unruh in 1976. It is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none...

.

Alternative measures

Other measures of nonclassical correlations include the measurement induced disturbance (MID) measure and the localized noneffective unitary (LNU) distance and various entropy-based measures.