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arXiv:1402.2870 · New Journal of Physics

Discriminating strength: a bona fide measure of non-classical correlations

A Farace, Antonella De Pasquale, L Rigovacca, Vittorio Giovannetti

Published Jul 4, 2014• 39 citations
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Abstract

A new measure of non-classical correlations is introduced and characterized. It tests the ability of using a state ρ of a composite system AB as a probe for a quantum illumination task (e.g. see Lloyd 2008 Science 321 1463 ), in which one is asked to remotely discriminate between the two following scenarios: (i) either nothing happens to the probe, or (ii) the subsystem A is transformed via a local unitary whose properties are partially unspecified when producing ρ . This new measure can be seen as the discrete version of the recently introduced interferometric power measure (Girolami et al 2013 e-print arXiv: 1309.1472 ) and, at least for the case in which A is a qubit, it is shown to coincide (up to an irrelevant scaling factor) with the local quantum uncertainty measure of Girolami, Tufarelli and Adesso (2013 Phys. Rev. Lett. 110 240402 ). Analytical expressions are derived which allow us to formally prove that, within the set of separable configurations, the maximum value of our non-classicality measure is achieved over the set of quantum-classical states (i.e. states ρ which admit a statistical unravelling where each element of the associated ensemble is distinguishable via local measures on B ).

DOI
10.1088/1367-2630/16/7/073010

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A new measure of non-classical correlations is introduced and characterized. It tests the ability of using a state ρ of a composite system AB as a probe for a quantum illumination task (e.g. see Lloyd 2008 Science 321 1463 ), in which one is asked to remotely discriminate between the two following scenarios: (i) either nothing happens to the probe, or (ii) the subsystem A is transformed via a local unitary whose properties are partially unspecified when producing ρ . This new measure can be seen as the discrete version of the recently introduced interferometric power measure (Girolami et al 2013 e-print arXiv: 1309.1472 ) and, at least for the case in which A is a qubit, it is shown to coincide (up to an irrelevant scaling factor) with the local quantum uncertainty measure of Girolami, Tufarelli and Adesso (2013 Phys. Rev. Lett. 110 240402 ). Analytical expressions are derived which allow us to formally prove that, within the set of separable configurations, the maximum value of our non-classicality measure is achieved over the set of quantum-classical states (i.e. states ρ which admit a statistical unravelling where each element of the associated ensemble is distinguishable via local measures on B ).
We present a quantifier of nonclassical correlations for bipartite, multimode Gaussian states. It is derived from the Discriminating Strength measure, introduced for finite dimensional systems in Farace et al., [New J. Phys. 16, 073010 (2014)]. As the latter the new measure exploits the quantum Chernoff bound to gauge the susceptibility of the composite system with respect to local perturbations induced by unitary gates extracted from a suitable set of allowed transformations (the latter being identified by posing some general requirements). Closed expressions are provided for the case of two-mode Gaussian states obtained by squeezing or by linearly mixing via a beam splitter a factorized two-mode thermal state. For these density matrices, we study how nonclassical correlations are related with the entanglement present in the system and with its total photon number.
Local noise can produce quantum correlations on an initially classically correlated state, provided that it is not represented by a unital or semiclassical channel [Phys. Rev. Lett. 107, 170502 (2011)]. We find the power of any given local channel for producing quantum correlations on an initially classically correlated state. We introduce a computable measure for quantifying the quantum correlations in quantum-classical states, which is based on the noncommutativity of ensemble states in one party of the composite system. Using this measure we show that the amount of quantum correlations produced is proportional to the classical correlations in the initial state. The power of an arbitrary channel for producing quantum correlations is found by averaging over all possible initial states. Finally, we compare our measure with the geometrical measure of quantumness for a subclass of quantum-classical sates, for which we have been able to find a closed analytical expression.
The threshold between classical and nonclassical two-qubit states is drawn at the place when these states can no longer be described by classical correlations, i.e., quantum discord or entanglement appear. However, to check if the correlations are classical (in terms of quantum discord and entanglement) it is sufficient to witness the lack of quantum discord because its zero value implies the lack of entanglement. We explain how the indicator of quantum discord introduced by Girolami and Adesso [Phys. Rev. Lett. 108, 150403 (2012)] can be practically measured in linear-optical systems using standard beam splitters and photon detectors. We study the efficiency of the setup assuming both ideal and real components and show that the efficiency of the proposed implementation is better than the full two-photon quantum tomography. Thus, we demonstrate that a class of experiments previously available on NMR platform can be implemented in optical systems.
We introduce an alternative method based on local quantum uncertainty to measure non-Markovianity for an open quantum system. The local quantum uncertainty proposed by Girolami etalet aletal. [Phys. Rev. Lett. 110, 240402 (2013)] is defined by the minimum skew information from a local measurement perspective. The measure effectively captures the characteristics of non-Markovianity and has a clearly physical interpretation: in correlated bipartite systems, the non-Markovian (Markovian) process of a considered open subsystem corresponds to the increase (decrease) of quantum uncertainty of local observables of another measured free subsystem. Then the local quantum uncertainty is further discussed by carefully handling the square root of the density operator for the bipartite $2\ensuremath{\bigotimes}d$ systems. Specifically, for a two-qubit system we give an explicit expression of the local quantum uncertainty. Furthermore, by applying the measure to three typical non-Markovian noise channels, we find that the conditions of detecting the non-Markovianity are consistent with the results of existing measures for non-Markovianity, i.e., information backflow and divisibility.
Abstract The discriminating strength D S ( ρ AB ) induced by local Gaussian unitary operators for any ( n + m )-mode Gaussian state ρ AB is introduced in [Phys. Rev. A 83 (2011) 042325]. In this paper, we further discuss the quantity by restricting to Hilbert-Schmidt norm. The analytic formulas of D S for two-mode squeezed thermal states and mixed thermal states are given. Then, the relationship between D S ( ρ AB ) and D S ( ( I ⊗ Φ ) ( ρ AB ) ) for some special Gaussian channels Φ is discussed. In addition, D S is compared with Gaussian entanglement for symmetric squeezed thermal states .
The interferometric power of a bipartite quantum state quantifies the precision, measured by quantum Fisher information, that such a state enables for the estimation of a parameter embedded in a unitary dynamics applied to one subsystem only, in the worst-case scenario where a full knowledge of the generator of the dynamics is not available a priori. For finite-dimensional systems, this quantity was proven to be a faithful measure of quantum correlations beyond entanglement. Here we extend the notion of interferometric power to the technologically relevant setting of optical interferometry with continuous-variable probes. By restricting to Gaussian local dynamics, we obtain a closed formula for the interferometric power of all two-mode Gaussian states. We identify separable and entangled Gaussian states which maximize the interferometric power at fixed mean photon number of the probes and discuss the associated metrological scaling. At fixed entanglement of the probes, highly thermalized states can guarantee considerably larger precision than pure two-mode squeezed states.