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arXiv:1110.2530 · Physical Review A

Quantumness of correlations revealed in local measurements exceeds entanglement

Marco Piani, Gerardo Adesso

Published Apr 3, 2012• 54 citations
arXivHTMLPDF

Abstract

We analyze a family of measures of general quantum correlations for composite systems, defined in terms of the bipartite entanglement necessarily created between systems and apparatuses during local measurements. For every entanglement monotone $E$, this operational correspondence provides a different measure ${Q}_{E}$ of quantum correlations. Examples of such measures are the relative entropy of quantumness, the quantum deficit, and the negativity of quantumness. In general, we prove that any so-defined quantum correlation measure is always greater than (or equal to) the corresponding entanglement between the subsystems, ${Q}_{E}\ensuremath{\ge}E$, for arbitrary states of composite quantum systems. We analyze qualitatively and quantitatively the flow of correlations in iterated measurements, showing that general quantum correlations and entanglement can never decrease along von Neumann chains, and that genuine multipartite entanglement in the initial state of the observed system always gives rise to genuine multipartite entanglement among all subsystems and all measurement apparatuses at any level in the chain. Our results provide a comprehensive framework to understand and quantify general quantum correlations in multipartite states.

DOI
10.1103/physreva.85.040301

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We analyze a family of measures of general quantum correlations for composite systems, defined in terms of the bipartite entanglement necessarily created between systems and apparatuses during local measurements. For every entanglement monotone EEE, this operational correspondence provides a different measure QE{Q}_{E}QE​ of quantum correlations. Examples of such measures are the relative entropy of quantumness, the quantum deficit, and the negativity of quantumness. In general, we prove that any so-defined quantum correlation measure is always greater than (or equal to) the corresponding entanglement between the subsystems, QE\ensuremath≥E{Q}_{E}\ensuremath{\ge}EQE​\ensuremath≥E, for arbitrary states of composite quantum systems. We analyze qualitatively and quantitatively the flow of correlations in iterated measurements, showing that general quantum correlations and entanglement can never decrease along von Neumann chains, and that genuine multipartite entanglement in the initial state of the observed system always gives rise to genuine multipartite entanglement among all subsystems and all measurement apparatuses at any level in the chain. Our results provide a comprehensive framework to understand and quantify general quantum correlations in multipartite states.
We analyze the general nonclassicality of correlations of a composite quantum system as measured by the negativity of quantumness. The latter corresponds to the minimum entanglement, as quantified by the negativity that is created between the system and an apparatus that is performing local measurements on a selection of subsystems. The negativity of quantumness thus quantifies the degree of nonclassicality on the measured subsystems. We demonstrate a number of possible different interpretations for this measure and for the concept of quantumness of correlations in general. In particular, for general bipartite states in which the measured subsystem is a qubit, the negativity of quantumness acquires a geometric interpretation as the minimum trace distance from the set of classically correlated states. This can be further reinterpreted as minimum disturbance, with respect to trace norm, due to a local measurement or a nontrivial local unitary operation. We calculate the negativity of quantumness in closed form for Werner and isotropic states and for all two-qubit states for which the reduced state of the system that is locally measured is maximally mixed; this includes all Bell diagonal states. We discuss the operational significance and potential role of the negativity of quantumness in quantum information processing.
We show that a von Neumann measurement on a part of a composite quantum system unavoidably creates distillable entanglement between the measurement apparatus and the system if the state has nonzero quantum discord. The minimal distillable entanglement is equal to the one-way information deficit. The quantum discord is shown to be equal to the minimal partial distillable entanglement that is the part of entanglement which is lost, when we ignore the subsystem which is not measured. We then show that any entanglement measure corresponds to some measure of quantum correlations. This powerful correspondence also yields necessary properties for quantum correlations. We generalize the results to multipartite measurements on a part of the system and on the total system.
ABSTRACT Characterizing the entanglement in multipartite quantum systems remains a central challenge in the theory of quantum entanglement. We establish a comprehensive framework for constructing ‐entanglement measures by leveraging bipartite entanglement measures and general ‐entanglement measures () with arithmetic and geometric averaging schemes. Numerical examples demonstrate that these measures outperform the existing ones, such as ‐ME concurrence, geometric mean (k‐GM) concurrence, and the generalized ‐entanglement measure by inducing distinct entanglement orderings and offering superior resolution. Furthermore, we reveal the operational relationship between the ‐entanglement and the local quantum coherence, verifying that the persistence of entanglement is intrinsically constrained by the system's capacity to maintain the local coherence. Finally, to overcome the inherent limitation of the conventional three‐tangle in characterizing the genuine multipartite entanglement of the GHZ‐W mixed states, we design a novel GME measure. Numerical results confirm that this measure effectively eliminates the detection blind spots where vanishes, providing a complete and practical mathematical tool for the quantitative analysis of genuine multipartite quantum correlations.
Having the quantum correlations in a general mixed or pure bipartite state in mind, the part of information accessible by simultaneous measurement on both subsystems is shown never to exceed the part accessible by measurement on one subsystem, which, in turn is proved not to exceed the von Neumann mutual information. A particular pair of (opposite-subsystem) observables is shown to be responsible both for the amount of quasiclassical correlations and for that of the purely quantum entanglement in the pure-state case: the former via simultaneous subsystem measurements, and the latter through the entropy of coherence or of incompatibility, which is defined for the general case. The observables at issue are so-called twin observables. A general definition of the latter is given in terms of their detailed properties.
In this paper, we investigate the characterization of multipartite entanglement based on the unified (q,s)(q,s)(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure Eq,sA∣B(ρ)E_{q,s}^{A|B}(ρ)Eq,sA∣B​(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) for quantifying kkk-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters qqq and sss are in certain ranges, Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) with respect to the parameters qqq and sss for fixed quantum states. In addition, for fixed qqq and sss, we also give their order relations for any two pure states.
We study the quantumness of correlations for ensembles of bi- and multi-partite systems and relate it to the task of quantum data hiding. Quantumness is here intended in the sense of minimum average disturbance under local measurements. We consider a very general framework, but focus on local complete von Neumann measurements as cause of the disturbance, and, later on, on the trace-distance as quantifier of the disturbance. We discuss connections with entanglement and previously defined notions of quantumness of correlations. We prove that a large class of quantifiers of the quantumness of correlations are entanglement monotones for pure bipartite states. In particular, we define an entanglement of disturbance for pure states, for which we give an analytical expression. Such a measure coincides with negativity and concurrence for the case of two qubits. We compute general bounds on disturbance for both single states and ensembles, and consider several examples, including the uniform Haar ensemble of pure states, and pairs of qubit states. Finally, we show that the notion of ensemble quantumness of correlations is most relevant in quantum data hiding. Indeed, while it is known that entanglement is not necessary for a good quantum data hiding scheme, we prove that ensemble quantumness of correlations is.