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arXiv:1901.08566 · Quantum

Operational relevance of resource theories of quantum measurements

Michał Oszmaniec, Tanmoy Biswas

Published Apr 26, 2019• 88 citations
arXivHTMLPDF

Abstract

For any resource theory it is essential to identify tasks for which resource objects offer advantage over free objects. We show that this identification can always be accomplished for resource theories of quantum measurements in which free objects form a convex subset of measurements on a given Hilbert space. To this aim we prove that every resourceful measurement offers advantage for some quantum state discrimination task. Moreover, we give an operational interpretation of robustness, which quantifies the minimal amount of noise that must be added to a measurement to make it free. Specifically, we show that this geometric quantity is related to the maximal relative advantage that a resourceful measurement offers in a class of minimal-error state discrimination (MESD) problems. Finally, we apply our results to two classes of free measurements: incoherent measurements (measurements that are diagonal in the fixed basis) and separable measurements (measurements whose effects are separable operators). For both of these scenarios we find, in the asymptotic setting in which the dimension or the number of particles increase to infinity, the maximal relative advantage that resourceful measurements offer for state discrimination tasks.

DOI
10.22331/q-2019-04-26-133

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For any resource theory it is essential to identify tasks for which resource objects offer advantage over free objects. We show that this identification can always be accomplished for resource theories of quantum measurements in which free objects form a convex subset of measurements on a given Hilbert space. To this aim we prove that every resourceful measurement offers advantage for some quantum state discrimination task. Moreover, we give an operational interpretation of robustness, which quantifies the minimal amount of noise that must be added to a measurement to make it free. Specifically, we show that this geometric quantity is related to the maximal relative advantage that a resourceful measurement offers in a class of minimal-error state discrimination (MESD) problems. Finally, we apply our results to two classes of free measurements: incoherent measurements (measurements that are diagonal in the fixed basis) and separable measurements (measurements whose effects are separable operators). For both of these scenarios we find, in the asymptotic setting in which the dimension or the number of particles increase to infinity, the maximal relative advantage that resourceful measurements offer for state discrimination tasks.
A central problem in quantum resource theory is to give operational meaning to quantum resources that can provide clear advantages in certain physical tasks compared to the convex set of resource-free states. We propose to extend this basic principle by defining the relative superiority of resources over a specific convex set of resource states, also provide a relative advantage in physical tasks based on this extended principle. This allows the generalized robustness measure to quantify the relative maximal advantage due to a given resource state over a specific convex set of resource states in the subchannel discrimination, thereby showing that the operational interpretation of resource measures also holds in a relative perspective. In addition, we offer a new framework for defining the deficiency of a given state in physical tasks compared to the set of maximum resource states. The geometric measure we provide satisfies the conditions of the framework for quantum coherence and entanglement, and it accurately quantifies the minimal disadvantage due to a given state compared to maximumresourcestates inthe subchannel discrimination in certain situations. These two extensions and new interpretations expand the scope of quantum resource theories and provide a more comprehensive operational interpretation.
One of the central problems in the study of quantum resource theories is to provide a given resource with an operational meaning, characterizing physical tasks in which the resource can give an explicit advantage over all resourceless states. We show that this can always be accomplished for all convex resource theories. We establish in particular that any resource state enables an advantage in a channel discrimination task, allowing for a strictly greater success probability than any state without the given resource. Furthermore, we find that the generalized robustness measure serves as an exact quantifier for the maximal advantage enabled by the given resource state in a class of subchannel discrimination problems, providing a universal operational interpretation to this fundamental resource quantifier. We also consider a wider range of subchannel discrimination tasks and show that the generalized robustness still serves as the operational advantage quantifier for several well-known theories such as entanglement, coherence, and magic.
We introduce the witness robustness of quantum measurements, a resource quantifier whose admissible noise consists of tuples of free-state witnesses rather than physical measurements. We establish its operational interpretation: it quantifies the maximal advantage that a measurement can provide over free measurements in discriminating an ensemble composed entirely of free states. Unlike the standard and generalized robustnesses, the witness robustness is not faithful in general, reflecting the fact that a resourceful measurement need not be useful when only free states can be prepared. We identify conditions under which faithfulness is recovered and show that, in resource theories admitting a resource-destroying map, the witness robustness vanishes for every measurement. We also establish fundamental properties, including convexity and monotonicity. Finally, we derive analytical results for projective measurements in single-qubit magic and for binary pure-state projective measurements in the two-qubit PPT entanglement theory.
Identifying what quantum-mechanical properties are useful to untap a superior performance in quantum technologies is a pivotal question. Quantum resource theories provide a unified framework to analyze and understand such properties, as successfully demonstrated for entanglement and coherence. While these are examples of convex resources, for which quantum advantages can always be identified, many physical resources are described by a nonconvex set of free states and their interpretation has so far remained elusive. Here we address the fundamental question of the usefulness of quantum resources without convexity assumption, by providing two operational interpretations of the generalized robustness measure in general resource theories. First, we characterize the generalized robustness in terms of a nonlinear resource witness and reveal that any state is more advantageous than a free one in some multicopy channel discrimination task. Next, we consider a scenario where a theory is characterized by multiple constraints and show that the generalized robustness coincides with the worst-case advantage in a single-copy channel discrimination setting. Based on these characterizations, we conclude that every quantum resource state shows a qualitative and quantitative advantage in discrimination problems in a general resource theory even without any specification on the structure of the free states.
Some quantum measurements cannot be performed simultaneously; i.e., they are incompatible. Here we show that every set of incompatible measurements provides an advantage over compatible ones in a suitably chosen quantum state discrimination task. This is proven by showing that the robustness of incompatibility, a quantifier of how much noise a set of measurements tolerates before becoming compatible, has an operational interpretation as the advantage in an optimally chosen discrimination task. We also show that if we take a resource-theory perspective of measurement incompatibility, then the guessing probability in discrimination tasks of this type forms a complete set of monotones that completely characterize the partial order in the resource theory. Finally, we make use of previously known relations between measurement incompatibility and Einstein-Podolsky-Rosen steering to also relate the latter with quantum state discrimination.
Abstract Which quantum phenomena are advantageous for information processing tasks? By classifying quantum states and operations as resourceful versus non-resourceful, or free, the mathematical formalism of quantum resource theories helps to address such questions. For the task of discriminating channels applied to a probe state, it has been shown that under certain conditions—namely, the set of free states being convex in any dimension or possibly non-convex but finite dimensional—every resourceful probe state can provide an advantage, quantified by a resource monotone known as generalized robustness. In this work, bypassing the limitations of previous studies, we define the generalized robustness for an arbitrary resource theory and show that it admits two operational interpretations. Firstly, it provides an upper bound on the maximal advantage in channel discrimination tasks implemented on multiple copies of the probe states. Secondly, in many physically relevant theories, it quantifies the advantage in single-copy channel discrimination tasks in a worst case scenario. We further present a general construction of multi-copy resource witnesses and provide practical methods to bound the generalized robustness in experiments. Finally, we apply our results to a key case study in continuous variable quantum information: the resource theory of non-Gaussianity. This theory is naturally defined by a set of free states (Gaussian states) that is non-convex and infinite-dimensional. Our work then shows conclusively that all non-Gaussian states can provide an advantage in some channel discrimination task, even those that are simply mixtures of Gaussian states and are typically disregarded for other quantum information tasks. To illustrate our findings, we provide exact formulas for the robustness of non-Gaussianity of Fock states, along with an analysis of the robustness for a family of non-Gaussian states within the convex hull of Gaussian states.