# Quantum discord

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 effects but do not necessarily involve quantum entanglement.

The notion of quantum discord was introduced by Harold Ollivier and Wojciech H. Zurek[1][2] and, independently by L. Henderson and Vlatko Vedral.[3] Olliver and Zurek referred to it also as a measure of quantumness of correlations.[2] From the work of these two research groups it follows that quantum correlations can be present in certain mixed separable states;[4] 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

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).

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:

${\displaystyle I(A;B)=H(A)+H(B)-H(A,B)}$
${\displaystyle J(A;B)=H(A)-H(A|B)}$

where, in the classical case, H(A) is the information entropy, H(A, B) the joint entropy and H(A|B) the conditional entropy, 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, S(ρ) the joint quantum entropy and S(ρA|ρB) a quantum generalization of conditional entropy (not to be confused with conditional quantum entropy), respectively, for probability density function ρ;

${\displaystyle I(\rho )=S(\rho _{A})+S(\rho _{B})-S(\rho )}$
${\displaystyle J_{A}(\rho )=S(\rho _{B})-S(\rho _{B}|\rho _{A})}$

The difference between the two expressions [I(ρ) − JA(ρ)] defines the basis-dependent quantum discord, which is asymmetrical in the sense that ${\displaystyle {\mathcal {D}}_{A}(\rho )}$ can differ from ${\displaystyle {\mathcal {D}}_{B}(\rho )}$.[5][6] The notation 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 measurements onto the eigenbasis:[7]

${\displaystyle {\mathcal {D}}_{A}(\rho )=I(\rho )-\max _{\{\Pi _{j}^{A}\}}J_{\{\Pi _{j}^{A}\}}(\rho )=S(\rho _{A})-S(\rho )+\min _{\{\Pi _{j}^{A}\}}S(\rho _{B|\{\Pi _{j}^{A}\}})}$

Nonzero quantum discord indicates the presence of correlations that are due to noncommutativity of quantum operators.[8] For pure states, the quantum discord becomes a measure of quantum entanglement,[9] more specifically, in that case it equals the entropy of entanglement.[4]

Vanishing quantum discord is a criterion for the pointer states, which constitute preferred effectively classical states of a system.[2] 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.[10] Other conditions have been identified which can be seen in analogy to the Peres–Horodecki criterion[11] and in relation to the strong subadditivity of the von Neumann entropy.[12]

Efforts have been made to extend the definition of quantum discord to continuous variable systems,[13] in particular to bipartite systems described by Gaussian states.[4][14] A very recent work[15] has demonstrated that the upper-bound of Gaussian discord[4][14] 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.[16] For certain classes of two-qubit states, quantum discord can be calculated analytically.[8][17][18]

## 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".[19]

Discord can also be viewed in operational terms as an "entanglement consumption in an extended quantum state merging protocol".[12][20] 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.[21][22]

Quantum discord has been seen as a possible basis for the performance in terms of quantum computation ascribed to certain mixed-state quantum systems,[23] with a mixed quantum state representing a statistical ensemble of pure states (see quantum statistical mechanics). 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.[24] Quantum discord is an indicator of minimum coherence in one subsystem of a composite quantum system and as such it plays a resource role in interferometric schemes of phase estimation.[25][26] A recent work[27] 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 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.[28] 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, 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.[29] 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.[30]

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.[31][32][33][34]

## Alternative measures

An operational measure, in terms of distillation of local pure states, is the ‘quantum deficit’.[35] The one-way and zero-way versions were shown to be equal to the relative entropy of quantumness.[36]

Other measures of nonclassical correlations include the measurement induced disturbance (MID) measure and the localized noneffective unitary (LNU) distance[37] and various entropy-based measures.[38]

There exists a geometric indicator of discord based on Hilbert-Schmidt distance,[5] which obeys a factorization law,[39] can be put in relation to von Neumann measurements,[40] but is not in general a faithful measure.

Faithful, computable and operational measures of discord-type correlations are the local quantum uncertainty[25] and the interferometric power.[26]

## References

1. ^ Wojciech H. Zurek, Einselection and decoherence from an information theory perspective, Annalen der Physik vol. 9, 855–864 (2000) abstract
2. ^ a b c Harold Ollivier and Wojciech H. Zurek, Quantum Discord: A Measure of the Quantumness of Correlations, Physical Review Letters vol. 88, 017901 (2001) abstract
3. ^ 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 [1]
4. ^ a b c d 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
5. ^ a b Borivoje Dakić, Vlatko Vedral, Caslav Brukner: Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett., vol. 105, nr. 19, 190502 (2010), arXiv:1004.0190v2 (submitted 1 April 2010, version of 3 November 2010)
6. ^ For a succinct overview see for ex arXiv:0809.1723v2
7. ^ 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 arXiv:0811.4003 and see for ex. Wojciech H. Zurek: Decoherence and the transition from quantum to classical - revisited, p. 11
8. ^ a b Luo, Shunlong (3 April 2008). "Quantum discord for two-qubit systems". Physical Review A. 77 (4): 042303. doi:10.1103/PhysRevA.77.042303.
9. ^ Animesh Datta, Anil Shaji, Carlton M. Caves: Quantum discord and the power of one qubit, arXiv:0709.0548v1 [quant-ph], 4 Sep 2007, p. 4
10. ^ Animesh Datta: A condition for the nullity of quantum discord, arXiv:1003.5256v2
11. ^ Bogna Bylicka, Dariusz Chru´sci´nski: Witnessing quantum discord in 2 x N systems, arXiv:1004.0434v1 [quant-ph], 3 April 2010
12. ^ a b Vaibhav Madhok, Animesh Datta: Role of quantum discord in quantum communication arXiv:1107.0994v1, (submitted 5 July 2011)
13. ^ C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, S. Lloyd: Gaussian Quantum Information, Reviews of Modern Physics 84, 621 (2012), available from arXiv:1110.3234v1 [2]
14. ^ a b Gerardo Adesso, Animesh Datta: Quantum versus classical correlations in Gaussian states, Phys. Rev. Lett. 105, 030501 (2010), available from arXiv:1003.4979v2 [quant-ph], 15 July 2010
15. ^ S. Pirandola, G. Spedalieri, S. L. Braunstein, N. J. Cerf, S. Lloyd: Optimality of Gaussian Discord, Phys. Rev. Lett. 113, 140405 (2014), available from arXiv:1309.2215v3 [3], 26 Nov 2014
16. ^ Huang, Yichen (21 March 2014). "Computing quantum discord is NP-complete". New Journal of Physics. 16 (3): 033027. arXiv:1305.5941. Bibcode:2014NJPh...16c3027H. doi:10.1088/1367-2630/16/3/033027.
17. ^ Chen, Qing; Zhang, Chengjie; Yu, Sixia; Yi, X. X.; Oh, C. H. (6 October 2011). "Quantum discord of two-qubit X states". Physical Review A. 84 (4): 042313. arXiv:1102.0181. doi:10.1103/PhysRevA.84.042313.
18. ^ Huang, Yichen (18 July 2013). "Quantum discord for two-qubit X states: Analytical formula with very small worst-case error". Physical Review A. 88 (1): 014302. arXiv:1306.0228. doi:10.1103/PhysRevA.88.014302.
19. ^ W. H. Zurek: Quantum discord and Maxwell’s demons", Physical Review A, vol. 67, 012320 (2003), abstract'
20. ^ D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, A. Winter: Operational interpretations of quantum discord, quant-ph, arXiv:1008.3205
21. ^ 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, Physical Review Letters, vol. 107, 070501 (2011) abstract (arXiv:1104.1596)
22. ^ Miranda Marquit: Quantum correlations – without entanglement, PhysOrg, August 24, 2011
23. ^ Animesh Datta, Anil Shaji, Carlton M. Caves: Quantum discord and the power of one qubit, arXiv:0709.0548v1 [quant-ph], 4 Sep 2007, p. 1
24. ^ 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, [4]'
25. ^ a b D. Girolami, T. Tufarelli, and G. Adesso, Characterizing Nonclassical Correlations via Local Quantum Uncertainty, Phys. Rev. Lett. 110, 240402 (2013) [5]
26. ^ a b D. Girolami et al., Quantum Discord Determines the Interferometric Power of Quantum States, Phys. Rev. Lett. 112, 210401 (2014) [6]
27. ^ S. Pirandola: Quantum discord as a resource for quantum cryptography, Sci. Rep. 4, 6956 (2014), available from [7]
28. ^ See [8] as well as [9] and citations therein
29. ^ 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 abstract, fulltext (arXiv)
30. ^ Animesh Datta: Quantum discord between relatively accelerated observers, arXiv:0905.3301v1 [quant-ph] 20 May 2009, [10]
31. ^ Dillenschneider, Raoul (16 December 2008). "Quantum discord and quantum phase transition in spin chains". Physical Review B. 78 (22): 224413. arXiv:0809.1723. doi:10.1103/PhysRevB.78.224413.
32. ^ Sarandy, M. S. (12 August 2009). "Classical correlation and quantum discord in critical systems". Physical Review A. 80 (2): 022108. arXiv:0905.1347. doi:10.1103/PhysRevA.80.022108.
33. ^ Werlang, T.; Trippe, C.; Ribeiro, G. A. P.; Rigolin, Gustavo (25 August 2010). "Quantum Correlations in Spin Chains at Finite Temperatures and Quantum Phase Transitions". Physical Review Letters. 105 (9): 095702. arXiv:1006.3332. doi:10.1103/PhysRevLett.105.095702. PMID 20868176.
34. ^ Huang, Yichen (11 February 2014). "Scaling of quantum discord in spin models". Physical Review B. 89 (5): 054410. arXiv:1307.6034. doi:10.1103/PhysRevB.89.054410.
35. ^ Jonathan Oppenheim, Michał Horodecki, Paweł Horodecki and Ryszard Horodecki:"Thermodynamical Approach to Quantifying Quantum Correlations" Physical Review Letters 89, 180402 (2002) [11]
36. ^ 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) [12]
37. ^ see for ex.: Animesh Datta, Sevag Gharibian: Signatures of non-classicality in mixed-state quantum computation, Physical Review A vol. 79, 042325 (2009) abstract, arXiv:0811.4003
38. ^ Matthias Lang, Anil Shaji, Carlton Caves: Entropic measures of nonclassical correlations, American Physical Society, APS March Meeting 2011, March 21–25, 2011, abstract #X29.007, arXiv:1105.4920
39. ^ 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, arXiv:1112.4318v2 (submitted on 19 December 2011, version of 21 December 2011)
40. ^ S. Lu, S. Fu: Geometric measure of quantum discord, Phys. Rev. A, vol. 82, no. 3, 034302 (2010)