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arXiv:1112.0736 · Physical Review A

Measurement-induced nonlocality based on the relative entropy

Zhengjun Xi, Xiaoguang Wang, Yongming Li

Published Apr 25, 2012• 54 citations
arXivHTMLPDF

Abstract

We quantify the measurement-induced nonlocality [Luo and Fu, Phys. Rev. Lett. 106, 120401 (2011)] from the perspective of the relative entropy. This quantification leads to an operational interpretation for the measurement-induced nonlocality, namely, it is the maximal entropy increase after the locally invariant measurements. The relative entropy of nonlocality is upper bounded by the entropy of the measured subsystem. We establish a relationship between the relative entropy of nonlocality and the geometric nonlocality based on the Hilbert-Schmidt norm, and show that it is equal to the maximal distillable entanglement. Several trade-off relations are obtained for tripartite pure states. We also give explicit expressions for the relative entropy of nonlocality for Bell-diagonal states.

DOI
10.1103/physreva.85.042325

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We quantify the measurement-induced nonlocality [Luo and Fu, Phys. Rev. Lett. 106, 120401 (2011)] from the perspective of the relative entropy. This quantification leads to an operational interpretation for the measurement-induced nonlocality, namely, it is the maximal entropy increase after the locally invariant measurements. The relative entropy of nonlocality is upper bounded by the entropy of the measured subsystem. We establish a relationship between the relative entropy of nonlocality and the geometric nonlocality based on the Hilbert-Schmidt norm, and show that it is equal to the maximal distillable entanglement. Several trade-off relations are obtained for tripartite pure states. We also give explicit expressions for the relative entropy of nonlocality for Bell-diagonal states.
We interpret the maximum global effect caused by locally invariant measurements as measurement-induced nonlocality, which is in some sense dual to the geometric measure of quantum discord [Dakic, Vedral, and Brukner, Phys. Rev. Lett. 105, 190502 (2010)]. We quantify measurement-induced nonlocality from a geometric perspective in terms of measurements, and obtain analytical formulas for any dimensional pure states and 2 × n dimensional mixed states. We further derive a tight upper bound to measurement-induced nonlocality in general case. The physical significance of measurement-induced nonlocality is discussed in the context of correlations, entanglement, quantumness, and cryptographic communication.
Nonlocality is one unique property of quantum mechanics that differs from the classical world. One of its quantifications can be properly described as the maximum global effect caused by locally invariant measurements, known as measurement-induced nonlocality (MIN) (2011 Phys. Rev. Lett. 106 120401). Here, we propose quantifying the MIN by the trace norm. We show explicitly that this measure is monotonically decreasing under the action of a completely positive trace-preserving map, which is the general local quantum operation, on the unmeasured party for the bipartite state. This property avoids the undesirable characteristic appearing in the known measure of MIN defined by the Hilbert–Schmidt norm which may be increased or decreased by trivial local reversible operations on the unmeasured party. We obtain analytical formulas of the trace-norm MIN for any -dimensional pure state, two-qubit state, and certain high-dimensional states. As with other quantum correlation measures, the newly defined MIN can be directly applied to various models for physical interpretations.
Measurement-induced nonlocality (MIN), which describes the maximum global effect caused by locally invariant measurements, was introduced by Luo and Fu ( Phys. Rev. Lett. , 106 (2011) 120401). In this paper, a new measure of MIN based on Wigner-Yanase skew information is proposed. It is shown that this measure not only has good computability but also eliminates the noncontractivity problem appearing in the original measure of MIN defined by the Hilbert-Schmidt norm. The analytical formulas of MIN based on Wigner-Yanase skew information for any pure states, ( 2 × n ) -dimensional mixed states, and some higher-dimensional symmetric states are presented. Furthermore, the tight upper bound to MIN based on Wigner-Yanase skew information in the general case is also derived.
Abstract Measurement-induced nonlocality (MIN), a quantum correlation measure for bipartite systems, is an indicator of maximal global effects due to locally invariant von Neumann projective measurements. It is originally defined as the maximal square of the Hilbert–Schmidt norm of the difference between pre- and post-measurement states. In this article, we propose a new form of MIN based on affinity. This quantity satisfies all the criteria of a bona fide measure of quantum correlation measures. This quantity is evaluated for both arbitrary pure and 2 × n dimensional (qubit–qudit) mixed states. The operational meaning of the proposed quantity is interpreted in terms of the interferometric power of the quantum state. We apply these results on two-qubit mixed states, such as the Werner, isotropic and Bell diagonal states.
Abstract We introduce a geometric framework for studying Bell nonlocality in Hilbert space, where, for a given quantum state, nonlocality is quantified by the distance between the state and the set of local states. This approach applies to any Bell inequality and any measurement scenario. Whenever the local set is characterized, the proposed nonlocality measure can be computed explicitly. As a general result, we prove that for any scenario in arbitrary dimension the closest local state to a Werner state is itself a Werner state, and analogously, the closest local state to an isotropic state is again isotropic. In the two-qubit case, we further show that the closest local state to a Bell-diagonal state is Bell-diagonal as well. These structural results are independent of the specific Bell inequality considered, thus revealing intrinsic geometric features of these families of states and providing significant simplifications for computing the proposed measures. For the Clauser–Horne–Shimony–Holt inequality in two-qubit systems and the Collins-Gisin-Linden-Massar-Popescu inequality for two qudits of arbitrary finite dimension, we derive explicit geometric measures of nonlocality for Bell-diagonal, Werner, and isotropic states using various distance metrics, including the trace, Hellinger, Hilbert–Schmidt distances, and relative entropy. Furthermore, we prove in all generality that for all scenarios in which the local set is not fully characterized, the geometric measures provide rigorous lower bounds on nonlocality.