In
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
, quantum discord is a measure of nonclassical correlations between two subsystems of a
quantum system
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. It includes correlations that are due to
quantum physical effects but do not necessarily involve
quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
.
The notion of quantum discord was introduced by Harold Ollivier and
Wojciech H. Zurek[Wojciech H. Zurek, ''Einselection and decoherence from an information theory perspective'', ]Annalen der Physik
''Annalen der Physik'' (English: ''Annals of Physics'') is one of the oldest scientific journals on physics; it has been published since 1799. The journal publishes original, peer-reviewed papers on experimental, theoretical, applied, and mathe ...
vol. 9, 855–864 (2000
abstract
/ref>[Harold Ollivier and Wojciech H. Zurek, ''Quantum Discord: A Measure of the Quantumness of Correlations'', ]Physical Review Letters
''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the ''Journa ...
vol. 88, 017901 (2001
abstract
/ref> and, independently by Leah Henderson and Vlatko Vedral. 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
In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determinin ...
;[Paolo Giorda, Matteo G. A. Paris: ''Gaussian quantum discord'', quant-ph arXiv:1003.3207v2 (submitted on 16 Mar 2010, version of 22 March 2010]
p. 1
/ref> In other words, separability alone does not imply the absence of quantum correlations. 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
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. It is the quantum mechanical analog of Shannon mutual informati ...
. More specifically, quantum discord is the difference between two expressions which each, in the 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 n ...
, represent the mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
. These two expressions are:
:
:
where, in the classical case, ''H''(''A'') is the information entropy
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
, ''H''(''A'', ''B'') the joint entropy
In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.
Definition
The joint Shannon entropy (in bits) of two discrete random variables X and Y with images \mathcal X and \mathcal Y is defined ...
and ''H''(''A'', ''B'') the conditional entropy
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable Y given that the value of another random variable X is known. Here, information is measured in shannons, na ...
, and the two expressions yield identical results. In the nonclassical case, the quantum physics analogy for the three terms are used – ''S''(''ρA'') the von Neumann entropy
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ...
, ''S''(''ρ'') the joint quantum entropy and ''S''(''ρA'', ''ρB'') a quantum generalization of conditional entropy (not to be confused with 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. For a bipartite state \rho^, the conditional entropy is written S(A, ...
), respectively, for probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
''ρ'';
:
:
The difference between the two expressions defines the basis-dependent quantum discord
:
which is asymmetrical in the sense that can differ from .[Borivoje Dakić, Vlatko Vedral, Caslav Brukner: ''Necessary and sufficient condition for nonzero quantum discord'', Phys. Rev. Lett., vol. 105, nr. 19, 190502 (2010), (submitted 1 April 2010, version of 3 November 2010)] The notation ''J'' represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen eigenbasis
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
; 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 measurements onto the eigenbasis:
:
Nonzero quantum discord indicates the presence of correlations that are due to noncommutativity of quantum operators. For pure state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s, the quantum discord becomes a measure of quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
,[Animesh Datta, Anil Shaji, Carlton M. Caves: ''Quantum discord and the power of one qubit'', uant-ph 4 Sep 2007, p. 4] 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.[Vaibhav Madhok, Animesh Datta: ''Role of quantum discord in quantum communication'' , (submitted 5 July 2011)]
Efforts have been made to extend the definition of quantum discord to continuous variable systems, in particular to bipartite systems described by Gaussian states.[Gerardo Adesso, Animesh Datta: ''Quantum versus classical correlations in Gaussian states'', Phys. Rev. Lett. 105, 030501 (2010), available from arXiv:1003.4979v2 uant-ph 15 July 2010] A very recent work[S. Pirandola, G. Spedalieri, S. L. Braunstein, N. J. Cerf, S. Lloyd: ''Optimality of Gaussian Discord'', Phys. Rev. Lett. 113, 140405 (2014), available from , 26 Nov 2014] has demonstrated that the upper-bound of Gaussian discord indeed coincides with the actual quantum discord of a Gaussian state, when the latter belongs to a suitable large family of Gaussian states.
Computing quantum discord is NP-complete and hence difficult to compute in the general case. For certain classes of two-qubit states, quantum discord can be calculated analytically.
Properties
Zurek provided a physical interpretation for discord by showing that it "determines the difference between the efficiency of quantum and classical Maxwell's demons...in extracting work from collections of correlated quantum systems".[W. H. Zurek: ''Quantum discord and Maxwell's demons", Physical Review A, vol. 67, 012320 (2003)]
abstract
Discord can also be viewed in operational terms as an "entanglement consumption in an extended quantum state merging protocol". Providing evidence for non-entanglement quantum correlations normally involves elaborate quantum tomography
Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states. The source of these states may be any device or system which prepares quantum st ...
methods; however, in 2011, such correlations could be demonstrated experimentally in a room temperature nuclear magnetic resonance system, using chloroform molecules that represent a two-qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
quantum system. Non-linear classicality witnesses have been implemented with Bell-state measurements in photonic systems.
Quantum discord has been seen as a possible basis for the performance in terms of quantum computation ascribed to certain mixed-state quantum systems,[Animesh Datta, Anil Shaji, Carlton M. Caves: ''Quantum discord and the power of one qubit'', arXiv:0709.0548v1 uant-ph 4 Sep 2007]
p. 1
/ref> with a ''mixed quantum state'' representing a statistical ensemble
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
of pure states (see quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
). The view that quantum discord can be a resource for quantum processors was further cemented in 2012, where experiments established that discord between bipartite systems can be consumed to encode information that can only be accessed by coherent quantum interactions.[M. Gu, H. Chrzanowski, S. Assad, T. Symul, K. Modi, T. C.Ralph, V.Vedral, P.K. Lam. "Observing the operational significance of discord consumption", Nature Physics 8, 671–675, 2012]
Quantum discord is an indicator of minimum coherence (physics), coherence in one subsystem of a composite quantum system and as such it plays a resource role in interferometric schemes of phase estimation.[D. Girolami, T. Tufarelli, and G. Adesso, Characterizing Nonclassical Correlations via Local Quantum Uncertainty, Phys. Rev. Lett. 110, 240402 (2013]
/ref>[D. Girolami et al., Quantum Discord Determines the Interferometric Power of Quantum States, Phys. Rev. Lett. 112, 210401 (2014]
/ref> A recent work[S. Pirandola: ''Quantum discord as a resource for quantum cryptography'', Sci. Rep. 4, 6956 (2014), available fro]
/ref> has identified quantum discord as a resource for quantum cryptography, being able to guarantee the security of quantum key distribution in the complete absence of entanglement.
Quantum discord is in some ways different from quantum entanglement. Quantum discord is more resilient to Quantum decoherence#Dissipation, 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
In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liability ...
, 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 in contact with a heat bath
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
, 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
The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...
.
Quantum discord has been studied in quantum many-body systems. Its behavior reflects quantum phase transitions and other properties of quantum spin chains and beyond.
Alternative measures
An operational measure, in terms of distillation of local pure states, is the 'quantum deficit'. The one-way and zero-way versions were shown to be equal to the relative entropy of quantumness.
Other measures of nonclassical correlations include the measurement induced disturbance (MID) measure and the localized noneffective unitary (LNU) distance and various entropy-based measures.
There exists a geometric indicator of discord based on Hilbert-Schmidt distance, which obeys a factorization law, can be put in relation to von Neumann measurements,[S. Lu, S. Fu: ''Geometric measure of quantum discord'', Phys. Rev. A, vol. 82, no. 3, 034302 (2010)] but is not in general a faithful measure.
Faithful, computable and operational measures of discord-type correlations are the local quantum uncertainty and the interferometric power.
References
{{Reflist
Quantum information science