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arXiv:1809.02570 · Physical Review Letters

Robustness of Measurement, Discrimination Games, and Accessible Information

Paul Skrzypczyk, Noah Linden

Published Apr 10, 2019• 85 citations
arXivHTMLPDF

Abstract

We introduce a resource theory of measurement informativeness. This allows us to define an associated quantifier, which we call the robustness of measurement. It describes how much "noise" must be added to a measurement before it becomes completely uninformative. We show that this geometric quantifier has operational significance in terms of the advantage the measurement provides over guessing at random in a suitably chosen state discrimination game and that it is the single-shot generalization of the accessible information of a certain quantum-to-classical channel. Using this insight, we further show that the recently introduced robustness of asymmetry or coherence is the single-shot generalization of the accessible information of an ensemble. Finally, we discuss more generally the connection between robustness-based measures, discrimination problems, and information-theoretic quantities.

DOI
10.1103/physrevlett.122.140403

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We introduce a resource theory of measurement informativeness. This allows us to define an associated quantifier, which we call the robustness of measurement. It describes how much "noise" must be added to a measurement before it becomes completely uninformative. We show that this geometric quantifier has operational significance in terms of the advantage the measurement provides over guessing at random in a suitably chosen state discrimination game and that it is the single-shot generalization of the accessible information of a certain quantum-to-classical channel. Using this insight, we further show that the recently introduced robustness of asymmetry or coherence is the single-shot generalization of the accessible information of an ensemble. Finally, we discuss more generally the connection between robustness-based measures, discrimination problems, and information-theoretic quantities.
We consider the quantum resource theory of measurement informativeness and introduce a weight-based quantifier of informativeness. We show that this quantifier has operational significance from the perspective of quantum state exclusion, by showing that it precisely captures the advantage a measurement provides in minimising the error in this game. We furthermore introduce information theoretic quantities related to exclusion, in particular the notion of excludible information of a quantum channel, and show that for the case of quantum-to-classical channels it is determined precisely by the weight of informativeness. This establishes a three-way correspondence which sits in parallel to the recently discovered correspondence in quantum resource theories between robustness-based quantifiers, discrimination games, and accessible information. We conjecture that the new correspondence between a weight-based quantifier and an exclusion-based task found in this work is a generic correspondence that holds in the context of quantum resource theories.
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.
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.
We introduce operational tasks for measurement incompatibility in the form of multiobject quantum subchannel discrimination and exclusion games with prior information, where a player can simultaneously harness the resources contained within both a quantum state and a set of measurements. We show that any fully or partially resourceful pair of objects is useful for a suitably chosen multiobject subchannel discrimination and exclusion game with prior information. The advantage provided by a fully or partially resourceful object against all possible fully free objects in such a game can be quantified in a manner by the resource quantifiers of generalized robustness and the weight of a resource for discrimination and exclusion games, respectively. These results hold for arbitrary properties of quantum states as well as for arbitrary properties of sets of measurements closed under classical pre- and postprocessing and, consequently, include measurement incompatibility as a particular case. We furthermore show that these results are not exclusive to quantum theory, but can also be extended to the realm of general probabilistic theories.
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.
Recent developments surrounding resource theories have shown that any quantum state or measurement resource, with respect to a convex (and compact) set of resourceless objects, provides an advantage in a tailored subchannel or state discrimination task, respectively. Here we show that an analogous, more general result is also true in the case of dynamical quantum resources, i.e., channels and instruments. In the scenario we consider, the tasks associated to a resource are input-output games. The advantage a resource provides in these games is naturally quantified by a generalized robustness measure. We illustrate our approach by applying it to a broad collection of examples, including classical and measure-and-prepare channels, measurement and channel incompatibility, local operations assisted by classical communication, and steering, as well as discussing its applicability to other resources in, e.g., quantum thermodynamics. We finish by showing that our approach generalizes to higher-order dynamics where it can be used, for example, to witness causal properties of supermaps.