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arXiv:1711.04119 · Entropy

Revealing Tripartite Quantum Discord with Tripartite Information Diagram

Wei-Ting Lee, Che-Ming Li

Published Nov 10, 2017• 2 citations
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Abstract

A new measure based on the tripartite information diagram is proposed for identifying quantum discord in tripartite systems. The proposed measure generalizes the mutual information underlying discord from bipartite to tripartite systems, and utilizes both one-particle and two-particle projective measurements to reveal the characteristics of the tripartite quantum discord. The feasibility of the proposed measure is demonstrated by evaluating the tripartite quantum discord for systems with states close to Greenberger–Horne–Zeilinger, W, and biseparable states. In addition, the connections between tripartite quantum discord and two other quantum correlations—namely genuine tripartite entanglement and genuine tripartite Einstein–Podolsky–Rosen steering—are briefly discussed. The present study considers the case of quantum discord in tripartite systems. However, the proposed framework can be readily extended to general N-partite systems.

DOI
10.3390/e19110602

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A new measure based on the tripartite information diagram is proposed for identifying quantum discord in tripartite systems. The proposed measure generalizes the mutual information underlying discord from bipartite to tripartite systems, and utilizes both one-particle and two-particle projective measurements to reveal the characteristics of the tripartite quantum discord. The feasibility of the proposed measure is demonstrated by evaluating the tripartite quantum discord for systems with states close to Greenberger–Horne–Zeilinger, W, and biseparable states. In addition, the connections between tripartite quantum discord and two other quantum correlations—namely genuine tripartite entanglement and genuine tripartite Einstein–Podolsky–Rosen steering—are briefly discussed. The present study considers the case of quantum discord in tripartite systems. However, the proposed framework can be readily extended to general N-partite systems.
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.
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.
A comparison is made between normal discord, super quantum discord, and entanglement for the tripartite Greenberger-Horne-Zeilinger (GHZ)-type squeezed states. For this aim, we start by studying the behavior of quantum discord for tripartite GHZ-type squeezed states. Also, we verify the monotonicity of the super quantum discord according to weaker quantum measurements. This was done by manipulating the amount of quantum information lost during the disruption of our quantum system and by controlling the measurement strength. Furthermore, we calculate the entanglement for our system.
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.
We consider a monogamy inequality of quantum discord in a pure tripartite state and show that it is equivalent to an inequality between quantum mutual information and entanglement of formation of two parties. Since this inequality does not hold for arbitrary bipartite states, quantum discord can generally be both monogamous and polygamous. We also carry out numerical calculations for some special states. The upper bounds of quantum discord and classical correlation are also discussed and we give physical analysis on the invalidness of a previous conjectured upper bound of quantum correlation. Our results provide new insights for further understanding of distributions of quantum correlations.
We investigate the possibility of multiple uses of a single copy of a three-qubit state for detecting genuine tripartite Einstein-Podolsky-Rosen (EPR) steering. A pure three-qubit state of either the Greenberger-Horne-Zeilinger (GHZ) type or WWW type is shared between two fixed observers in two wings and a sequence of multiple observers in the third wing who perform unsharp or nonprojective measurements. The measurement settings of each of the multiple observers in the third wing are independent and uncorrelated with the measurement settings and outcomes of the previous observers. In such setup, we investigate all possible types of ($2\ensuremath{\rightarrow}1$) and ($1\ensuremath{\rightarrow}2$) genuine tripartite EPR steering. For each case, we obtain an upper limit on the number of observers on the third wing who can demonstrate genuine EPR steering through the quantum violation of an appropriate tripartite steering inequality. We show that the GHZ state allows for a higher number of observers compared to the WWW state. Additionally, ($1\ensuremath{\rightarrow}2$) genuine steering is possible for a larger range of sharpness parameters compared to ($2\ensuremath{\rightarrow}1$) genuine steering cases.