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Journal of Physics A Mathematical and Theoretical

Robustness of quantum correlations against linear noise

Zhihua Guo, Huaixin Cao, Shixian Qu

Published Apr 4, 2016• 3 citations
Publisher

Abstract

Abstract Relative robustness of quantum correlations (RRoQC) of a bipartite state is firstly introduced relative to a classically correlated state. Robustness of quantum correlations (RoQC) of a bipartite state is then defined as the minimum of RRoQC of the state relative to all classically correlated ones. It is proved that as a function on quantum states, RoQC is nonnegative, lower semi-continuous and neither convex nor concave; especially, it is zero if and only if the state is classically correlated. Thus, RoQC not only quantifies the endurance of quantum correlations of a state against linear noise, but also can be used to distinguish between quantum and classically correlated states. Furthermore, the effects of local quantum channels on the robustness are explored and characterized.

DOI
10.1088/1751-8113/49/19/195301

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Abstract Relative robustness of quantum correlations (RRoQC) of a bipartite state is firstly introduced relative to a classically correlated state. Robustness of quantum correlations (RoQC) of a bipartite state is then defined as the minimum of RRoQC of the state relative to all classically correlated ones. It is proved that as a function on quantum states, RoQC is nonnegative, lower semi-continuous and neither convex nor concave; especially, it is zero if and only if the state is classically correlated. Thus, RoQC not only quantifies the endurance of quantum correlations of a state against linear noise, but also can be used to distinguish between quantum and classically correlated states. Furthermore, the effects of local quantum channels on the robustness are explored and characterized.
Quantum channels, in general, will disturb quantum states and will lead to decoherence (i.e., information leakage to the environment). For bipartite systems, local channels often cause more decoherence for global states than for local states due to correlations. Intuitively, the difference between these two kinds of decoherence (i.e., global and local ones) may be regarded as a quantifier characterizing certain aspects of correlations in the global bipartite states. Based on this idea, we introduce a quantifier of correlations (relative to a local channel) as the coherence difference and investigate its basic properties. We probe and quantify correlations relative to various channels including the unitary channels, the twirling channels, the projective measurements, and the weak measurements. In particular, we show that both product states and some natural classical-quantum states can be operationally characterized in terms of local channels: The product states are just those states with vanishing coherence difference relative to the twirling channel induced by the unitary group, whereas these natural classical-quantum states are just those states with vanishing coherence difference relative to the twirling channel induced by the subgroup of the unitary group that does not disturb the local states. We further illustrate the results by various examples.
Quantum correlations are fragile under noise, motivating quantitative measures that track not only their presence but also their degradation during physical evolution. We develop a robustness framework for the hierarchy of Bell nonlocality, EPR steering, and entanglement, establishing an ordering of the corresponding robustness measures and relating robustness to noise-induced transition times. We further formulate the instantaneous robustness decay along Lindblad dynamics as a directional derivative determined by both the evolution and the robustness functional. Applications to isotropic depolarizing noise and amplitude damping illustrate, respectively, the resulting resource transition times and the directional dependence of steering robustness. The framework provides an approach to characterizing the degradation and lifetime of bipartite quantum correlations under noisy dynamics.
The aim of this paper is to distinguish classical correlations from quantum ones. First, a new characterization of a classical correlated (CC) state is established. Corresponding results for the left and right classical correlations are also obtained. Second, a sufficient and necessary condition for a convex combination of two CC states to be CC is given. It is proved that the set CC of all CC states on is a perfect, nowhere dense and compact subset of the metric space of all states (density operators) on . Based on our new characterization of CC states, a quantity Q (ρ) is associated with a state ρ. It is shown that a state ρ is CC if and only if Q (ρ) = 0. In particular, we prove that the Werner state W λ in a two-qubit system with a single real parameter λ is CC if and only if λ = 0.25.
We consider how to quantify non-Gaussianity for the correlation of a bipartite quantum state by using various measures such as relative entropy and geometric distances. We first show that an intuitive approach, i.e., subtracting the correlation of a reference Gaussian state from that of a target non-Gaussian state, fails to yield a non-negative measure with monotonicity under local Gaussian channels. Our finding clearly manifests that quantum-state correlations generally have no Gaussian extremality. We therefore propose a different approach by introducing relevantly averaged states to address correlation. This enables us to define a non-Gaussianity measure based on, e.g., the trace-distance and the fidelity, fulfilling all requirements as a measure of non-Gaussian correlation. For the case of the fidelity-based measure, we also present readily computable lower bounds of non-Gaussian correlation.
One of the best signatures of nonclassicality in a quantum system is the existence of correlations that have no classical counterpart. Different methods for quantifying the quantum and classical parts of correlations are among the more actively studied topics of quantum-information theory over the past decade. Entanglement is the most prominent of these correlations, but in many cases unentangled states exhibit nonclassical behavior too. Thus distinguishing quantum correlations other than entanglement provides a better division between the quantum and classical worlds, especially when considering mixed states. Here different notions of classical and quantum correlations quantified by quantum discord and other related measures are reviewed. In the first half, the mathematical properties of the measures of quantum correlations are reviewed, related to each other, and the classical-quantum division that is common among them is discussed. In the second half, it is shown that the measures identify and quantify the deviation from classicality in various quantum-information-processing tasks, quantum thermodynamics, open-system dynamics, and many-body physics. It is shown that in many cases quantum correlations indicate an advantage of quantum methods over classical ones.
Abstract Local operation is an important tool to characterize the nonlocal aspects of multipartite quantum system. Exploiting the notion of resource theory of coherence, in this article, we establish a quantum correlation measure as the difference between the bipartite coherence and marginal state coherence. We study the Tsallis α-entropy (TαE) coherence based quantum correlation of bipartite state relative to different channels such as unitary channel, the twirling (unitary-induced) channel, projective measurements and weak measurements. It is shown that the quantum channel helps us to discriminate the product and classical-quantum states. We provide the operational interpretation of the correlation measure relative to the measurement in terms of classical uncertainty of channel. A closer connection between the correlation relative to the projective measurements and weak measurements is obtained in terms of measurement strength. As an illustration, we have studied the quantum correlations of well-known two-qubit states.