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| 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 mechanics|quantum physical]] effects but do not necessarily involve [[quantum entanglement]].
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| The notion of quantum discord was introduced by Harold Ollivier and [[Wojciech H. Zurek]]<ref name="zurek-2000">Wojciech H. Zurek, ''Einselection and decoherence from an information theory perspective'', [[Annalen der Physik]] vol. 9, 855–864 (2000) [http://onlinelibrary.wiley.com/doi/10.1002/1521-3889(200011)9:11/12%3C855::AID-ANDP855%3E3.0.CO;2-K/abstract abstract]</ref><ref name="olliver-zurek-2001">Harold Ollivier and Wojciech H. Zurek, ''Quantum Discord: A Measure of the Quantumness of Correlations'', [[Physics Review Letters]] vol. 88, 017901 (2001) [http://prl.aps.org/abstract/PRL/v88/i1/e017901 abstract]</ref> and, independently by L. Henderson and [[Vlatko Vedral]].<ref>L. Henderson and V. Vedral: ''Classical, quantum and total correlations'', [[Journal of Physics A]] 34, 6899 (2001), {{doi|10.1088/0305-4470/34/35/315}} [http://iopscience.iop.org/0305-4470/34/35/315]</ref> Olliver and Zurek referred to it also as a measure of ''quantumness'' of correlations.<ref name="olliver-zurek-2001"/> From the work of these two research groups it follows that quantum correlations can be present in certain mixed [[separable states]];<ref name="giorda-paris-2010-P1">Paolo Giorda, Matteo G. A. Paris: ''Gaussian quantum discord'', quant-ph arXiv:1003.3207v2 (submitted on 16 Mar 2010, version of 22 March 2010) [http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.3207v2.pdf#page=1 p. 1]</ref> 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.
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| == Definition and mathematical relations ==
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| [[Image:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X; Y).]]
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| In mathematical terms, quantum discord is defined in terms of the [[quantum mutual information]]. More specifically, quantum discord is the difference between two expressions which each, in the [[classical limit]], represent the [[mutual information]]. These two expressions are:
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| :<math>I (A; B) = H (A) + H (B) - H (A,B)</math>
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| :<math>J (A; B) = H (A) - H (A|B)</math>
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| where, in the classical case, <math>H(A)</math> is the [[information entropy]], <math>H(A,B)</math> the [[joint entropy]] and <math>H(A|B)</math> the [[conditional entropy]], and the two expressions yield identical results. In the nonclassical case, the quantum physics analogy for the three terms are used – <math>S (\rho_A)</math> the [[von Neumann entropy]], <math>S(\rho)</math> the [[joint quantum entropy]] and <math>S (\rho_A|\rho_B)</math> the [[conditional quantum entropy]], respectively, for [[probability density function]] <math>\rho</Math>
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| :<math>I (\rho) = S (\rho_A) + S (\rho_B) - S (\rho)</math>
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| :<math>J_A (\rho) = S (\rho_B) - S (\rho_B|\rho_A)</math> | |
| The difference between the two expressions <math>I(\rho) - J_A(\rho)</math> defines the basis-dependent quantum discord, which is asymmetrical in the sense that <math>\mathcal D_A (\rho)</math> can differ from <math>\mathcal D_B (\rho)</math>.<ref name="dakic-vedral-brukner">Borivoje Dakić, Vlatko Vedral, Caslav Brukner: ''Necessary and sufficient condition for nonzero quantum discord'', Phys. Rev. Lett., vol. 105, nr. 19, 190502 (2010), [http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.0190v2.pdf arXiv:1004.0190v2] (submitted 1 April 2010, version of 3 November 2010)</ref><ref>For a succinct overview see for ex [http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.1723v2.pdf arXiv:0809.1723v2]</ref> <math>J</math> represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen [[Matrix factorization#Eigendecomposition|eigenbasis]]; therefore, in order for the quantum discord to reflect the purely nonclassical correlations independently of basis, it is necessary that <math>J</math> first be maximized over the set of all possible [[Projective Hilbert space|projective]] [[quantum measurement|measurements]] onto the eigenbasis:<ref>For a more detailed overview see for ex. ''Signatures of nonclassicality in mixed-state quantum computation'', [[Physical Review A]] vol. 79, 042325 (2009), {{doi|10.1103/PhysRevA.79.042325}} [http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0811.4003 arXiv:0811.4003] and see for ex. Wojciech H. Zurek: ''Decoherence and the transition from quantum to classical - revisited'', [http://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf#page=10 p. 11]</ref>
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| :<math>\mathcal D_A (\rho) = I (\rho) - \max_{\{\Pi_j^A\}} J_{\{\Pi_j^A\}} (\rho) = S (\rho_A) - S(\rho) + \min_{\{\Pi_j^A\}} S (\rho_{B | \{\Pi_j^A\}} ) </math>
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| Nonzero quantum discord indicates the presence of correlations that are due to [[Observable#Incompatibility of observables in quantum mechanics|noncommutativity of quantum operators]].<ref>Shunlong Luo: ''Quantum discord for two-qubit systems'', [[Physical Review A]], vol. 77, 042303 (2008) [http://pra.aps.org/abstract/PRA/v77/i4/e042303 abstract]</ref> For [[pure state]]s, the quantum discord becomes a measure of [[quantum entanglement]],<ref name="datta-et-al-2007-P4">Animesh Datta, Anil Shaji, Carlton M. Caves: ''Quantum discord and the power of one qubit'', arXiv:0709.0548v1 [quant-ph], 4 Sep 2007, [http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.0548v1.pdf#page=4 p. 4]</ref> more specifically, in that case it equals the entropy of entanglement.<ref name="giorda-paris-2010-P1"/>
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| Vanishing quantum discord is a criterion for the [[pointer state]]s, which constitute preferred effectively classical states of a system.<ref name="olliver-zurek-2001"/> 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.<ref>Animesh Datta: ''A condition for the nullity of quantum discord'', [http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.5256v2.pdf arXiv:1003.5256v2]</ref> Other conditions have been identified which can be seen in analogy to the [[Peres–Horodecki criterion]]<ref>Bogna Bylicka, Dariusz Chru´sci´nski: ''Witnessing quantum discord in 2 x N systems'', arXiv:1004.0434v1 [quant-ph], 3 April 2010</ref> and in relation to the [[von Neumann entropy#Properties|strong subadditivity of the von Neumann entropy]].<ref name="madhok-datta-2011">Vaibhav Madhok, Animesh Datta: ''Role of quantum discord in quantum communication'' [http://arxiv.org/abs/1107.0994v1 arXiv:1107.0994v1], (submitted 5 July 2011)</ref>
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| | :<math forcemathmode="mathml">E=mc^2</math> |
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| Efforts have been made to extend the definition of quantum discord to continuous variable systems, in particular to bipartite systems described by Gaussian states.<ref name="giorda-paris-2010-P1"/>
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| == Properties ==
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| 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".<ref name="zurek-2003">W. H. Zurek: ''Quantum discord and Maxwell’s demons", [[Physical Review A]], vol. 67, 012320 (2003), [http://pra.aps.org/abstract/PRA/v67/i1/e012320 abstract]'</ref>
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| Discord can also be viewed In operational terms as an "entanglement consumption in an extended [[State-merging|quantum state merging]] protocol".<ref name="madhok-datta-2011"/><ref>D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, A. Winter: ''[http://arxiv.org/abs/1008.3205 Operational interpretations of quantum discord]'', quant-ph, arXiv:1008.3205</ref> Providing evidence for non-entanglement quantum correlations normally involves elaborate [[quantum tomography]] 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]] quantum system.<ref>R. Auccaise, J. Maziero, L. C. Céleri, D. O. Soares-Pinto, E. R. deAzevedo, T. J. Bonagamba, R. S. Sarthour, I. S. Oliveira, R. M. Serra: ''Experimentally Witnessing the Quantumness of Correlations'', [[Physics Review Letters]], vol. 107, 070501 (2011) [http://prl.aps.org/abstract/PRL/v107/i7/e070501 abstract] ([http://arxiv.org/abs/1104.1596 arXiv:1104.1596])</ref><ref>Miranda Marquit: ''[http://www.physorg.com/news/2011-08-quantum-entanglement.html Quantum correlations – without entanglement]'', [[PhysOrg]], August 24, 2011</ref>
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| Quantum discord has been seen as a possible basis for the performance in terms of [[quantum computation]] ascribed to certain [[Quantum state#Mixed states|mixed-state]] quantum systems,<ref name="datta-et-al-2007-P1">Animesh Datta, Anil Shaji, [[Carlton M. Caves]]: ''Quantum discord and the power of one qubit'', arXiv:0709.0548v1 [quant-ph], 4 Sep 2007, [http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.0548v1.pdf#page=1 p. 1]</ref> with a ''mixed quantum state'' representing a [[statistical ensemble]] of pure states (see [[quantum statistical mechanics]]).
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| 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 [[Quantum decoherence#Dissipation|dissipative environments]]<!--here, this section link is preferable to a link to [[quantum dissipation]] because it provides more succinct information--> 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 (quantum computing)|concurrence]], where discord has proven to be more robust.<ref>See [http://arxiv.org/abs/0911.1096] as well as [http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1845v1.pdf] and citations therein</ref> 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 reservoir|heat bath]], 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.<ref>T. Werlang, G. Rigolin: ''Thermal and magnetic discord in Heisenberg models'', [[Physical Review A]], vol. 81, no. 4 (044101) (2010), {{doi|10.1103/PhysRevA.81.044101}} [http://pra.aps.org/abstract/PRA/v81/i4/e044101 abstract], [http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.3903v2.pdf fulltext (arXiv)]</ref> 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]].<ref>Animesh Datta: ''Quantum discord between relatively accelerated observers'', arXiv:0905.3301v1 [quant-ph] 20 May 2009, [http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.3301v1.pdf]</ref>
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| ==Alternative measures==
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| An operational measure, in terms of distillation of local pure states, the ‘quantum deficit’.<ref>Jonathan Oppenheim, Michał Horodecki, Paweł Horodecki and Ryszard Horodecki:"Thermodynamical Approach to Quantifying Quantum Correlations" [[Physical Review Letters]] 89, 180402 (2002) [http://arxiv.org/abs/quant-ph/0112074]</ref> The one-way and zero-way versions were shown to be equal to the relative entropy of quantumness.<ref> Michał Horodecki, Paweł Horodecki, Ryszard Horodecki, Jonathan Oppenheim, Aditi Sen De, Ujjwal Sen, Barbara Synak-Radtke: "Local versus nonlocal information in quantum-information theory: Formalism and phenomena" [[Physical Review A]] 71, 062307 (2005) [http://arxiv.org/abs/quant-ph/0410090]</ref>
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| Other measures of nonclassical correlations include the measurement induced disturbance (MID) measure and the localized noneffective unitary (LNU) distance<ref>see for ex.: Animesh Datta, Sevag Gharibian: ''Signatures of non-classicality in mixed-state quantum computation'', [[Physical Review A]] vol. 79, 042325 (2009) [http://pra.aps.org/abstract/PRA/v79/i4/e042325 abstract], [http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0811.4003 arXiv:0811.4003]</ref> and various entropy-based measures.<ref>Matthias Lang, Anil Shaji, Carlton Caves: ''Entropic measures of nonclassical correlations'', American Physical Society, APS March Meeting 2011, March 21–25, 2011, [http://adsabs.harvard.edu/abs/2011APS..MARX29007L abstract #X29.007], [http://arxiv.org/abs/1105.4920 arXiv:1105.4920]</ref>
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| There exists a geometric measure of discord,<ref name="dakic-vedral-brukner"/> which obeys a factorization law<ref>Wei Song, Long-Bao Yu, Ping Dong, Da-Chuang Li, Ming Yang, Zhuo-Liang Cao: ''Geometric measure of quantum discord and the geometry of a class of two-qubit states'', [http://arxiv.org/abs/1112.4318v2 arXiv:1112.4318v2] (submitted on 19 December 2011, version of 21 December 2011)</ref>, can be put in relation to von Neumann measurements,<ref>S. Lu, S. Fu: ''Geometric measure of quantum discord'', Phys. Rev. A, vol. 82, no. 3, 034302 (2010)</ref> and a measure of ‘measurement-induced nonlocality’ (MIN).<ref>S. Luo and S. Fu: ''Measurement-Induced Nonlocality''], Phys. Rev. Lett. 106, 120401 (2011) ([http://prl.aps.org/abstract/PRL/v106/i12/e120401 abstract]). Cited after Guo-Feng Zhang, Heng Fan, Ai-Ling Ji, Wu-Ming Liu: ''Dynamics of geometric discord and measurement-induced nonlocality at finite temperature'', [http://arxiv.org/abs/1201.1949 arXiv:1201.1949] (submitted on 10 January 2012)</ref>
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