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arXiv:1105.2548 · Physical Review A

Global quantum discord in multipartite systems

C. C. Rulli, M. S. Sarandy

Published Oct 11, 2011• 311 citations
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Abstract

We propose a global measure for quantum correlations in multipartite systems, which is obtained by suitably recasting the quantum discord in terms of relative entropy and local von Neumann measurements. The measure is symmetric with respect to subsystem exchange and is shown to be nonnegative for an arbitrary state. As an illustration, we consider tripartite correlations in the Werner-GHZ (Greenberger-Horne-Zeilinger) state and multipartite correlations at quantum criticality. In particular, in contrast with the pairwise quantum discord, we show that the global quantum discord is able to characterize the infinite-order quantum phase transition in the Ashkin-Teller spin chain.

DOI
10.1103/physreva.84.042109

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We propose a global measure for quantum correlations in multipartite systems, which is obtained by suitably recasting the quantum discord in terms of relative entropy and local von Neumann measurements. The measure is symmetric with respect to subsystem exchange and is shown to be nonnegative for an arbitrary state. As an illustration, we consider tripartite correlations in the Werner-GHZ (Greenberger-Horne-Zeilinger) state and multipartite correlations at quantum criticality. In particular, in contrast with the pairwise quantum discord, we show that the global quantum discord is able to characterize the infinite-order quantum phase transition in the Ashkin-Teller spin chain.
Quantum discord is a measure of nonclassical correlations in quantum systems. While the bipartite version of quantum discord is experimentally well-studied, the multipartite version has never been convincingly measured. In this study, we experimentally investigate tripartite quantum discord using an NMR quantum information processor. Building on a theoretical framework for conditional projective measurements and quantum conditional mutual information, we quantify the tripartite quantum discord and its contributions in different three-qubit states such as the Greenberger-Horne-Zeilinger (GHZ) and Werner (W) states as well as classical mixtures of biseparable Bell states, and classical mixtures of product states. The experiments employed full quantum state tomography and temporal averaging to prepare mixed states, achieving fidelities exceeding 95%. Our results confirm that quantum discord persists even in the absence of entanglement, highlighting its utility as a broader indicator of quantum correlations. Furthermore, we validate the nonconvexity of discord, confirming that classical mixtures of zero-discord states can exhibit nonzero discord. This experimentally confirms that quantum discord does not fit into the framework of resource theory. This work establishes a robust methodology for measuring quantum discord, illuminating the structure and distribution of quantum correlations in multipartite systems.
The quantum discord is used as measure of quantum correlations for two families of multipartite coherent states. The first family interpolates between generalized GHZ states and generalized Werner states. The second one is an interpolation between generalized GHZ and the ground state of the multipartite quantum system. Two inequivalent ways to split the system in a pair of qubits are introduced. The explicit expressions of quantum quantum discord in multipartite coherent states are derived. Its evaluation uses the Koashi–Winter relation in optimizing the conditional entropy. The temporal evolution of quantum correlations (quantum discord and entanglement) is also discussed.
A generalization of quantum discord to multipartite systems is proposed. A key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems. It is by construction zero only for systems with classically correlated subsystems and is a non-negative quantity, giving a measure of the total nonclassical correlations in the multipartite system with respect to a fixed measurement ordering. For the tripartite case, we show that the discord can be decomposed into contributions resulting from changes induced by nonclassical correlation breaking measurements in the conditional mutual information and tripartite mutual information. The former gives a measure of the bipartite nonclassical correlations and is a non-negative quantity, while the latter is related to the monogamy of the nonclassical correlations.
Within the unified framework of exploiting the relative entropy as a distance measure of quantum correlations, we make explicit the hierarchical structure of quantum coherence, quantum discord, and quantum entanglement in multipartite systems. On this basis, we define a basis-independent measure of quantum coherence and prove that it is exactly equivalent to quantum discord. Furthermore, since the original relative entropy of coherence is a basis-dependent quantity, we investigate the local and nonlocal unitary creation of quantum coherence, focusing on the two-qubit unitary gates. Intriguingly, our results demonstrate that nonlocal unitary gates do not necessarily outperform the local unitary gates. Finally, the additivity relationship of quantum coherence in tripartite systems is discussed in detail, where the strong subadditivity of von Neumann entropy plays an essential role.
Quantum discord is a measure of quantum correlations beyond the entanglement-separability paradigm. It is conceptualized by using the von Neumann entropy as a measure of disorder. We introduce a class of quantum correlation measures as differences between total and classical correlations, in a shared quantum state, in terms of the sandwiched relative Rényi and Tsallis entropies. We compare our results with those obtained by using the traditional relative entropies. We find that the measures satisfy all the plausible axioms for quantum correlations. We evaluate the measures for shared pure as well as paradigmatic classes of mixed states. We show that the measures can faithfully detect the quantum critical point in the transverse quantum Ising model and find that they can be used to remove an unquieting feature of nearest-neighbor quantum discord in this respect. Furthermore, the measures provide better finite-size scaling exponents of the quantum critical point than the ones for other known order parameters, including entanglement and information-theoretic measures of quantum correlations.
We introduce a monogamy inequality for quantum correlations, which implies that the sum of pairwise quantum correlations is upper limited by the amount of multipartite quantum correlations as measured by the global quantum discord. This monogamy bound holds either for pure or mixed quantum states provided that bipartite quantum discord does not increase under discard of subsystems. We illustrate the monogamy behavior for multipartite pure states with Schmidt decomposition as well as for WWW-GHZ mixed states. As a by-product, we apply the monogamy bound to investigate residual multipartite correlations.