Quantum discord

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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 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 [1] [2] and, independently by Leah 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). Entropy-mutual-information-relative-entropy-relation-diagram.svg
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:

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;

The difference between the two expressions defines the basis-dependent quantum discord

which is asymmetrical in the sense that can differ from . [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]

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] Quantum discord must be non-negative and 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] 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] Non-linear classicality witnesses have been implemented with Bell-state measurements in photonic systems. [23]

Quantum discord has been seen as a possible basis for the performance in terms of quantum computation ascribed to certain mixed-state quantum systems, [24] 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. [25] 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. [26] [27] A recent work [28] 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. [29] 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. [30] 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. [31]

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. [32] [33] [34] [35]

Alternative measures

An operational measure, in terms of distillation of local pure states, is the 'quantum deficit'. [36] The one-way and zero-way versions were shown to be equal to the relative entropy of quantumness. [37]

Other measures of nonclassical correlations include the measurement induced disturbance (MID) measure and the localized noneffective unitary (LNU) distance [38] and various entropy-based measures. [39]

There exists a geometric indicator of discord based on Hilbert-Schmidt distance, [5] which obeys a factorization law, [40] can be put in relation to von Neumann measurements, [41] but is not in general a faithful measure.

Faithful, computable and operational measures of discord-type correlations are the local quantum uncertainty [26] and the interferometric power. [27]

Related Research Articles

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<span class="mw-page-title-main">Quantum entanglement</span> Correlation between quantum systems

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In quantum mechanics, einselections, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descriptions of reality from quantum descriptions. In this approach, classicality is described as an emergent property induced in open quantum systems by their environments. Due to the interaction with the environment, the vast majority of states in the Hilbert space of a quantum open system become highly unstable due to entangling interaction with the environment, which in effect monitors selected observables of the system. After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into an uncertain state which can be expressed as a mixture of simple pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of pure superpositions of pointer states. These 'pointer states' are stable despite environmental interaction. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement and superposition.

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In quantum mechanics, separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are multipartite quantum states that can be written as a tensor product of states in each space. The physical intuition behind these definitions is that product states have no correlation between the different degrees of freedom, while separable states might have correlations, but all such correlations can be explained as due to a classical random variable, as opposed as being due to entanglement.

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In quantum information science, the concurrence is a state invariant involving qubits.

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 entanglement given by the conditional quantum entropy, with the Von Neumann entropy, . It thus provides an operational meaning to this quantity.

In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute. It is a measure deriving from the PPT criterion for separability. It has shown to be an entanglement monotone and hence a proper measure of entanglement.

In quantum information and quantum computation, an entanglement monotone or entanglement measure is a function that quantifies the amount of entanglement present in a quantum state. Any entanglement monotone is a nonnegative function whose value does not increase under local operations and classical communication.

In quantum physics, the "monogamy" of quantum entanglement refers to the fundamental property that it cannot be freely shared between arbitrarily many parties.

The entanglement of formation is a quantity that measures the entanglement of a bipartite quantum state.

Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory.

References

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  2. 1 2 3 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
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  6. ↑ For a succinct overview see for ex arXiv : 0809.1723
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  39. ↑ 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
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