Nan Li

Decorrelating capabilities of operations with application to decoherence

Decoherence, interpreted broadly, is essentially the leakage of system information into the environment and is often accompanied by dissipation. The basic questions arise: how to quantify decoherence induced by an operation and how to quantitatively compare decoherence induced by different operations. In this paper, based on a joint ancilla-system-environment tripartite purification for the initial system state and the operation, and by exploiting the intrinsic relations between the loss of correlations in the ancilla-system and the correlations established in the system-environment, we characterize and quantify decoherence from a decorrelating perspective. For this purpose, we first address the issue of separating and quantifying the classical and quantum parts of decorrelation. By use of the canonical isomorphism between operations and bipartite states, we propose two intrinsic decorrelation measures: One is the classical decorrelation based on the loss of classical correlations, and the other is the quantum decorrelation based on the loss of quantum correlations. With the help of quantum decorrelation, we introduce an intuitive measure of (quantum) decoherence. We further employ these informational quantities to analyze some widely used channels such as the complete decoherent channel, the depolarizing channel, the bit-flip channel, the transpose depolarizing channel, the amplitude damping channel, and the phasemorexa0» damping channel. Our analysis illustrates the intriguing interplay between classical and quantum decorrelations and sheds some light on the informational nature of decoherence.«xa0less

Relative entropy between quantum ensembles

Relative entropy between two quantum states, which quantifies to what extent the quantum states can be distinguished via whatever methods allowed by quantum mechanics, is a central and fundamental quantity in quantum information theory. However, in both theoretical analysis (such as selective measurements) and practical situations (such as random experiments), one is often encountered with quantum ensembles, which are families of quantum states with certain prior probability distributions. How can we quantify the quantumness and distinguishability of quantum ensembles? In this paper, by use of a probabilistic coupling technique, we propose a notion of relative entropy between quantum ensembles, which is a natural generalization of the relative entropy between quantum states. This generalization enjoys most of the basic and important properties of the original relative entropy. As an application, we use the notion of relative entropy between quantum ensembles to define a measure for quantumness of quantum ensembles. This quantity may be useful in quantum cryptography since in certain circumstances it is desirable to encode messages in quantum ensembles which are the most quantum, thus the most sensitive to eavesdropping. By use of this measure of quantumness, we demonstrate that a set consisting of two pure states is the most quantum when the states are 45° apart.

Quantumness of bipartite states in terms of conditional entropies

Quantum discord, as defined by Olliver and Zurek (2002 Phys. Rev. Lett. 88 017901) as the difference of two natural quantum extensions of the classical mutual information, plays an interesting role in characterizing quantumness of correlations. Inspired by this idea, we will study quantumness of bipartite states arising from different quantum analogs of the classical conditional entropy. Our approach is intrinsic, in contrast to the Olliver–Zurek method that involves extrinsic local measurements. For this purpose, we introduce two alternative variants of quantum conditional entropies via conditional density operators, which in turn are intuitive quantum extensions of equivalent classical expressions for the conditional probability. The significance of these quantum conditional entropies in characterizing quantumness of bipartite states is illustrated through several examples.

How quantum is a quantum ensemble

In the Hilbert space operator formalism of quantum mechanics, a single quantum state, which is represented by a density operator, can be regarded as classical in the sense that it can always be diagonalized. However, a quantum ensemble, which is represented by a family of quantum states together with a probability distribution specifying the probability of the occurrence of each state, cannot be diagonalized simultaneously in generic cases, and possesses intrinsic quantum features as long as the involved quantum states are not commutative. The natural question arises as how to quantify its quantumness. By virtue of a canonical correspondence between quantum ensembles and classical-quantum bipartite states, we propose an intuitive entropic quantity which captures certain quantum features of quantum ensembles, and compare it with that defined as the gap between the Holevo quantity and the accessible information. Implications for quantum cryptography and relations to quantum channel capacities are indicated. Some illustrative examples are worked out.

Weak superadditivity of skew information

We revisit the superadditivity conjecture for skew information which has recently been disproved by Hansen (2007 J. Stat. Phys. 126 643). We establish two weak forms of superadditivity which are conjectured to be optimal. Our results show that in a certain sense the superadditivity is true with 50% off.

Quantumness-generating capability of quantum dynamics

We study quantumness-generating capability of quantum dynamics, where quantumness refers to the noncommutativity between the initial state and the evolving state. In terms of the commutator of the square roots of the initial state and the evolving state, we define a measure to quantify the quantumness-generating capability of quantum dynamics with respect to initial states. Quantumness-generating capability is absent in classical dynamics and hence is a fundamental characteristic of quantum dynamics. For qubit systems, we present an analytical form for this measure, by virtue of which we analyze several prototypical dynamics such as unitary dynamics, phase damping dynamics, amplitude damping dynamics, and random unitary dynamics (Pauli channels). Necessary and sufficient conditions for the monotonicity of quantumness-generating capability are also identified. Finally, we compare these conditions for the monotonicity of quantumness-generating capability with those for various Markovianities and illustrate that quantumness-generating capability and quantum Markovianity are closely related, although they capture different aspects of quantum dynamics.

Fisher Concord: Efficiency of Quantum Measurement

Abstract By comparing measurement-induced classical Fisher information of parameterized quantum states with quantum Fisher information,we study the notion of Fisher concord (as abbreviation of the concord between the classical and the quantum Fisher information), which is an information-theoretic measure of quantum states and quantum measurements based on both classical and quantum Fisher information. Fisher concord is defined by multiplying the inverse square root of quantum Fisher information matrix to measurement-induced classical Fisher information matrix on both sides, and quantifies the relative accessibility of parameter information from quantum measurements (alternatively, the efficiency of quantum measurements in extracting parameter information). It reduces to the ratio of the classical Fisher information to quantum Fisher information in any single parameter scenario. In general, Fisher concord is a symmetric matrix which depends on both quantum states and quantum measurements. Some basic properties of Fisher concord are elucidated. The significance of Fisher concord in quantifying the interplay between classicality and quantumness in parameter estimation and in characterizing the ef- ficiency of quantum measurements are illustrated through several examples, and some information conservation relations in terms of Fisher concord are exhibited.