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arXiv:1110.1664 · Physical Review A

Unification of different views of decoherence and discord

Patrick J. Coles

Published Apr 3, 2012• 57 citations
arXivHTMLPDF

Abstract

Macroscopic behavior such as the lack of interference patterns has been attributed to ``decoherence,'' a word with several possible definitions such as (1) the loss of off-diagonal density matrix elements, (2) the flow of information to the environment, (3) the loss of complementary information, and (4) the loss of the ability to create entanglement in a measurement. In this article, we attempt to unify these distinct definitions by providing general quantitative connections between them, valid for all finite-dimensional quantum systems or quantum processes. The most important application of our results is to the understanding of quantum discord, a measure of the nonclassicality of the correlations between quantum systems. We show that some popular measures of discord measure the information missing from the purifying system and hence quantify security, which can be stated operationally in terms of distillable secure bits. The results also give some strategies for constructing discord measures.

DOI
10.1103/physreva.85.042103

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Macroscopic behavior such as the lack of interference patterns has been attributed to ``decoherence,'' a word with several possible definitions such as (1) the loss of off-diagonal density matrix elements, (2) the flow of information to the environment, (3) the loss of complementary information, and (4) the loss of the ability to create entanglement in a measurement. In this article, we attempt to unify these distinct definitions by providing general quantitative connections between them, valid for all finite-dimensional quantum systems or quantum processes. The most important application of our results is to the understanding of quantum discord, a measure of the nonclassicality of the correlations between quantum systems. We show that some popular measures of discord measure the information missing from the purifying system and hence quantify security, which can be stated operationally in terms of distillable secure bits. The results also give some strategies for constructing discord measures.
Quantum discord, a measure of quantum correlation beyond entanglement, is initially defined as the discord between two classically equivalent while quantum discordant definitions of mutual information. In this paper, we report some new interpretations of discord which rely on the differences between measurement-induced effects on the local measured system and the whole system. Specifically, with proper quantitative definitions introduced in Buscemi et al (2008 Phys. Rev. Lett. 100 210504), we find that quantum discord can be interpreted as the differences of measurement-induced disturbance or information gain on the local measured system and on the whole system. Combined with previous similar results based on measurement-induced entanglement and decoherence, our results provide a unified view on quantum discord.
Quantum discord is the quantitative difference between two alternative expressions for bipartite mutual information, given respectively in terms of two distinct definitions for the conditional entropy. By constructing a stochastic model of shared states, classical discord can be similarly defined, quantifying the presence of some stochasticity in the measurement process. Therefore, discord can generally be understood as a quantification of the system's state disturbance due to local measurements, be it quantum or classical. We establish an operational meaning of classical discord in the context of state merging with noisy measurement and thereby show the quantum-classical separation in terms of a negative conditional entropy.
Measurements of quantum systems disturb their states. To quantify this nonclassical characteristic, Zurek and Ollivier [Phys. Rev. Lett. 88, 017901 (2001)] introduced the quantum discord, a quantum correlation that can be nonzero even when entanglement in the system is zero. Discord has aroused great interest as a resource that is more robust against the effects of decoherence and offers the exponential speed-up of certain computational algorithms. Here, we study general two-level bipartite systems and give general results on the relationship between discord, entanglement, and linear entropy. We also identify the states for which discord takes a maximal value for a given entropy or entanglement, thus placing strong bounds on entanglement-discord and entropy-discord relations. We find out that although discord and entanglement are identical for pure states, they differ when generalized to mixed states as a result of the difference in the method of generalization.
Quantum discord expresses a fundamental nonclassicality of correlations that is more general than entanglement, but that, in its standard definition, is not easily evaluated. We derive a hierarchy of computationally efficient lower bounds to the standard quantum discord. Every nontrivial element of the hierarchy constitutes by itself a valid discordlike measure, based on a fundamental feature of quantum correlations: their lack of shareability. Our approach emphasizes how the difference between entanglement and discord depends on whether shareability is intended as a static property or as a dynamical process.
Recent measures of nonclassical correlations are motivated by different notions of classicality and operational means. Quantum discord has received a great deal of attention in studies involving quantum computation, metrology, dynamics, many-body physics, and thermodynamics. In this article I show how quantum discord is different from quantum entanglement from a pedagogical point of view. I begin with a pedagogical introduction to quantum entanglement and quantum discord, followed by a historical review of quantum discord. Next, I give a novel definition of quantum discord in terms of any classically extractable information, an approach that is fitting for the current avenues of research. Lastly, I put forth several arguments for why discord is an interesting quantity to study and why it is of interest to so many researchers in the community.
Quantum discord is a more general measure of quantum correlations than entanglement and has been proposed as a resource in certain quantum information processing tasks. The computation of discord is mostly confined to two-qubit systems for which an analytical calculational scheme is available. The utilization of quantum correlations in quantum information-based applications is limited by the problem of decoherence, i.e., the loss of coherence due to the inevitable interaction of a quantum system with its environment. The dynamics of quantum correlations due to decoherence may be studied in the Kraus operator formalism for different types of quantum channels representing system-environment interactions. In this review, we describe the salient features of the dynamics of classical and quantum correlations in a two-qubit system under Markovian (memoryless) time evolution. The two-qubit state considered is described by the reduced density matrix obtained from the ground state of a spin model. The models considered include the transverse-field XY model in one dimension, a special case of which is the transverse-field Ising model, and the XXZ spin chain. The quantum channels studied include the amplitude damping, bit-flip, bit-phase-flip and phase-flip channels. The Kraus operator formalism is briefly introduced and the origins of different types of dynamics discussed. One can identify appropriate quantities associated with the dynamics of quantum correlations which provide signatures of quantum phase transitions in the spin models. Experimental observations of the different types of dynamics are also mentioned.