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arXiv:1402.4321 · New Journal of Physics

Measurement-induced nonlocality based on the trace norm

Ming‐Liang Hu, Heng Fan

Published Mar 3, 2015• 108 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/17/3/033004

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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.
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.
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.
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.
In quantum mechanics, general measurements often cause disturbance which may be exploited to quantify entanglement, nonlocality or quantumness. Imagine a bipartite state ρ ab shared by two parties a and b , and a von Neumann measurement performed locally on party a which does not disturb the local state ρ a ≔tr b ρ ab , but nevertheless may disturb the global state ρ ab . This disturbance is an indication of some kind of correlations or global effect in ρ ab which cannot be accounted for locally. We propose to use the maximum disturbance on ρ ab caused by locally non-disturbing measurements as a figure of merit quantifying the global effect (nonlocality), and investigate its fundamental properties. For general two-qubit states and some higher-dimensional symmetric states, we present analytic formulas for their global effects.
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 introduce a quantum Fisher information based measurement-induced nonlocality (QFI-MIN), which quantifies the maximal statistical distinguishability induced by locally invariant unitary dynamics. The proposed measure inherits desirable properties including positivity, local unitary invariance, monotonicity under local operations, and immunity to the local ancilla problem. Analytical expressions are obtained for pure states, arbitrary two-qubit states, and two-qubit X states, revealing a direct connection with entanglement for pure systems. We further establish clear operational interpretations of QFI-MIN in quantum parameter estimation, local channel discrimination, and correlation-assisted communication. Its behavior under amplitude damping, depolarizing, and generalized amplitude damping channels demonstrates robustness against environmental noise. The proposed framework provides a physically consistent and operationally meaningful quantifier of quantum correlations, linking nonlocality, quantum metrology, and quantum information processing