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arXiv:1810.03882 · Physical Review A

Epsilon-smooth measure of coherence

Zhengjun Xi, Shanshan Yuwen

Published Jan 7, 2019• 16 citations
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Abstract

In this paper, by minimizing the coherence quantifiers over all states in an $\ensuremath{\epsilon}$ ball around a given state, we define a generalized smooth quantifier, called the $\ensuremath{\epsilon}$-smooth measure of coherence. We use it to estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in an actually prepared state, and it can been interpreted as the minimal coherence guaranteed to present in an $\ensuremath{\epsilon}$ ball around given quantum state. We find that the $\ensuremath{\epsilon}$-smooth measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the $\ensuremath{\epsilon}$-smooth measure of coherence is continuous even if the original coherence quantifier is not. We also study the $\ensuremath{\epsilon}$-smooth measure of distance-based coherence quantifiers, and some interesting properties are given. Moreover, we discuss the dual form of the $\ensuremath{\epsilon}$-smooth measure of coherence by maximizing over all states in an $\ensuremath{\epsilon}$ ball around the given state and show that the dual $\ensuremath{\epsilon}$-smooth measure of coherence provides an upper bound of one-shot coherence distillation.

DOI
10.1103/physreva.99.012308

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In this paper, by minimizing the coherence quantifiers over all states in an \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ ball around a given state, we define a generalized smooth quantifier, called the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-smooth measure of coherence. We use it to estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in an actually prepared state, and it can been interpreted as the minimal coherence guaranteed to present in an \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ ball around given quantum state. We find that the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-smooth measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-smooth measure of coherence is continuous even if the original coherence quantifier is not. We also study the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-smooth measure of distance-based coherence quantifiers, and some interesting properties are given. Moreover, we discuss the dual form of the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-smooth measure of coherence by maximizing over all states in an \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ ball around the given state and show that the dual \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-smooth measure of coherence provides an upper bound of one-shot coherence distillation.
To estimate the difference between the expected state and the actually prepared state and quantify quantum coherence contained in actually prepared state, we discuss the smooth version of coherence quantifiers which depend on a precision parameter ϵ\epsilonϵ. In this paper, we call it the ϵ\epsilonϵ-measure of coherence, it can be interpreted as the minimal coherence guaranteed to present in an ϵ\epsilonϵ ball around given quantum state. We find that the ϵ\epsilonϵ-measure of any coherence monotone is still a coherence monotone, but it does not satisfy monotonicity on average under incoherent operations. We show the ϵ\epsilonϵ-measure of coherence is continuous even if the original coherence quantifier is not. In particular, we study some properties of the ϵ\epsilonϵ-measure of distance-based coherence quantifier.
The quantification of state-based quantum ``resources'' such as entanglement, coherence, and nonstabilizer states lies at the heart of quantum science and technology, providing potential advantages over classical methods. In a realistic scenario, due to the imperfections and uncertainties in physical devices, we are unable to perfectly prepare or detect the true quantum states. Consequently, it is necessary to study the quantification of quantum resources under such circumstances. In this work, by focusing on the state-based quantum resource theory, we introduce a family of resource measures called $\ensuremath{\epsilon}$ measure that relies on a precision parameter to address this issue. This family of resource measures inherits the fundamental properties of the original resource measure, such as weak monotonicity, convexity, monogamy, and so forth. Furthermore, the $\ensuremath{\epsilon}$ measure remains continuous irrespective of whether the original measure is continuous or not. We also investigate the $\ensuremath{\epsilon}$ measure of distance-based resource quantifiers, and some interesting properties are presented. As part of the applications, we derived several formulas for the $\ensuremath{\epsilon}$ measure of coherence, nonstabilizerness, asymmetry, nonuniformity, and imaginarity. Additionally, we offered an upper bound for the $\ensuremath{\epsilon}$ measure of the resource rank. Finally, we outline how this work can be extended to channel-based resources among others.
Abstract In this work we investigate how to quantify the coherence of quantum measurements. First, we establish a resource theoretical framework to address the coherence of measurement and show that any statistical distance can be adopted to define a coherence monotone of measurement. For instance, the relative entropy fulfills all the required properties as a proper monotone. We specifically introduce a coherence monotone of measurement in terms of off-diagonal elements of positive-operator-valued measure components. This quantification provides a lower bound on the robustness of measurement-coherence that has an operational meaning as the maximal advantage over all incoherent measurements in state discrimination tasks. Finally, we propose an experimental scheme to assess our quantification of measurement-coherence and demonstrate it by performing an experiment using a single qubit on IBM Q processor.
We associate to every entanglement measure a family of measures which depend on a precision parameter, and which we call epsilon-measures of entanglement. Their definition aims at addressing a realistic scenario in which we need to estimate the amount of entanglement in a state that is only partially known. We show that many properties of the original measure are inherited by the family, in particular weak monotonicity under transformations applied by means of Local Operations and Classical Communication (LOCC). On the other hand, they may increase on average under stochastic LOCC. Remarkably, the epsilon-version of a convex entanglement measure is continuous even if the original entanglement measure is not, so that the epsilon-version of an entanglement measure may be actually considered a smoothed version of it.
Quantifying coherence is an essential endeavor for both quantum foundations and quantum technologies. Here, the robustness of coherence is defined and proven to be a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task.
The notion of measurement is of both foundational and instrumental significance in quantum mechanics, and coherence destroyed by measurements (decoherence) lies at the very heart of quantum to classical transition. Qualitative aspects of this spirit have been widely recognized and analyzed ever since the inception of quantum theory. However, axiomatic and quantitative investigations of coherence are attracting great interest only recently with several figures of merit for coherence introduced [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. While these resource theoretic approaches have many appealing and intuitive features, they rely crucially on various notions of incoherent operations which are sophisticated, subtle, and not uniquely defined, as have been critically assessed [Chitambar and Gour, Phys. Rev. Lett. 117, 030401 (2016)]. In this paper, we elaborate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate, and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of a measurement (with respect to a state). Consequently, coherence measures may be set into correspondence with measures of quantum uncertainty. In particular, we take average quantum Fisher information as a measure of quantum uncertainty, and introduce the corresponding measure of coherence, which is demonstrated to exhibit desirable properties. Implications for interpreting quantum purity as maximal coherence, and quantum discord as minimal coherence, are illustrated.