Measurement-induced nonlocality based on the trace norm

Ming-Liang Hu,Heng Fan

doi:10.1088/1367-2630/17/3/033004

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.

Highlights

Quantum physics differs in many aspects from our conventional intuition
We have introduced a well-defined measure of nonlocality by making using of the trace norm
It can remedy the undesirable property of the conventional measurement-induced nonlocality (MIN) which can be changed arbitrarily and reversibly by trivial local action on the subsystem

Summary

INTRODUCTION

Quantum physics differs in many aspects from our conventional intuition. One of such intriguing difference is the celebrated notion of nonlocality, which arises from the debate of the early 20th century among scientists. One line of the debate originated with Einstein, Podolsky, and Rosen, who proposed the thought experiment known as EPR paradox and the socalled “spooky-action-at-a-distance” [1]. Their predictions of quantum mechanics are, in sharp contrast to the conventional view, that physical processes should obey the principle of locality. We realized that nonlocality is a central concept of quantum mechanics, but may be used to improve the efficiency of many quantum information processing (QIP) tasks [9, 10] It is interrelated with other foundational theory of quantum mechanics such as uncertainty principle [11]. These delicate and intriguing features of nonlocality prompted a huge surge of people’s interest from the quantum physics community, with notable progresses be-

MIN QUANTIFIED BY HILBERT-SCHMIDT NORM

MIN BASED ON THE TRACE NORM

ANALYTICAL FORMULAS OF THE TRACE MIN

SUMMARY AND DISCUSSION