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] 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] 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] 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. [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

<span class="mw-page-title-main">Quantum teleportation</span> Physical phenomenon

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light.

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

Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

<span class="mw-page-title-main">Quantum decoherence</span> Loss of quantum coherence

Quantum decoherence is the loss of quantum coherence, the process in which a system's behaviour changes from that which can be explained by quantum mechanics to that which can be explained by classical mechanics. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.

In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the principle of locality. These are models, usually deterministic, that attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying, but inaccessible variables, with the additional requirement that distant events be statistically independent.

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 , the conditional entropy is written , or , depending on the notation being used for the von Neumann entropy. The quantum conditional entropy was defined in terms of a conditional density operator by Nicolas Cerf and Chris Adami, who showed that quantum conditional entropies can be negative, something that is forbidden in classical physics. The negativity of quantum conditional entropy is a sufficient criterion for quantum non-separability.

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.

The Peres–Horodecki criterion is a necessary condition, for the joint density matrix of two quantum mechanical systems and , to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2×2 and 2×3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. The theorem was discovered in 1996 by Asher Peres and the Horodecki family

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.

<span class="mw-page-title-main">LOCC</span> Method in quantum computation and communication

LOCC, or local operations and classical communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed conditioned on the information received.

Squashed entanglement, also called CMI entanglement, is an information theoretic measure of quantum entanglement for a bipartite quantum system. If is the density matrix of a system composed of two subsystems and , then the CMI entanglement of system is defined by

In physics, the no-broadcasting theorem is a result of quantum information theory. In the case of pure quantum states, it is a corollary of the no-cloning theorem. The no-cloning theorem for pure states says that it is impossible to create two copies of an unknown state given a single copy of the state. Since quantum states cannot be copied in general, they cannot be broadcast. Here, the word "broadcast" is used in the sense of conveying the state to two or more recipients. For multiple recipients to each receive the state, there must be, in some sense, a way of duplicating the state. The no-broadcast theorem generalizes the no-cloning theorem for mixed states.

In the case of systems composed of subsystems, the classification of quantum-entangledstates is richer than in the bipartite case. Indeed, in multipartite entanglement apart from fully separable states and fully entangled states, there also exists the notion of partially separable states.

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 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
  4. 1 2 3 4 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
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  6. ↑ For a succinct overview see for ex arXiv : 0809.1723
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  29. ↑ See as well as and citations therein
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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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