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arXiv:2408.06315 · Physical Review A

Dynamical resource theory of incompatibility preservability

Chung-Yun Hsieh, Benjamin Stratton, Chao-Hsien Wu, Huan-Yu Ku

Published Feb 13, 2025• 5 citations
arXivHTMLPDF

Abstract

The uncertainty principle is one of quantum theory's most foundational features. It underpins a quantum phenomenon called measurement incompatibility—two physical observables of a single quantum system may not always be measured simultaneously. Apart from being fundamentally important, measurement incompatibility is also a powerful resource in the broad quantum science and technologies, with wide applications to cryptography, communication, random number generation, and device-independent tasks. Since every physical system is unavoidably subject to noise, an important, yet still open, question is how to characterize the ability of noisy quantum dynamics to preserve measurement incompatibility. This work fills this gap by providing a resource theory of this ability, termed incompatibility preservability. We quantify incompatibility preservability by a robustness measure. Then, we introduce an operational task, entanglement-assisted filter game, to completely characterize both the robustness measure and the conversion of incompatibility preservability. Our results provide a general framework to describe how noisy dynamics affect the uncertainty principle's signature.

DOI
10.1103/physreva.111.022422

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The uncertainty principle is one of quantum theory's most foundational features. It underpins a quantum phenomenon called measurement incompatibility—two physical observables of a single quantum system may not always be measured simultaneously. Apart from being fundamentally important, measurement incompatibility is also a powerful resource in the broad quantum science and technologies, with wide applications to cryptography, communication, random number generation, and device-independent tasks. Since every physical system is unavoidably subject to noise, an important, yet still open, question is how to characterize the ability of noisy quantum dynamics to preserve measurement incompatibility. This work fills this gap by providing a resource theory of this ability, termed incompatibility preservability. We quantify incompatibility preservability by a robustness measure. Then, we introduce an operational task, entanglement-assisted filter game, to completely characterize both the robustness measure and the conversion of incompatibility preservability. Our results provide a general framework to describe how noisy dynamics affect the uncertainty principle's signature.
Measurement incompatibility is a fundamental quantum resource that enables advantages in many quantum information tasks, including cryptography and communication. However, unavoidable interactions between a system and its environment can degrade or even completely destroy measurement incompatibility; such a process is referred to as a measurement-incompatibility-annihilating channel. Recently, the capability of noisy quantum dynamics to preserve measurement incompatibility has been characterized within the resource theory of measurement incompatibility preservability. This motivates studying how to purify the preservation of measurement incompatibility and how to activate it from a measurement-incompatibility-annihilating channel. To this end, we first introduce our figures of merit as robustness-based resource monotones within this resource theory. We demonstrate that while pre-filtering operations can strengthen measurement incompatibility preservability, they cannot activate it from an incompatibility-annihilating channel, establishing a no-go theorem. Furthermore, we explicitly demonstrate that an incompatibility-annihilating channel can be stochastically activated via post-filtering operations. Our results provide a practical framework for exploiting measurement incompatibility in quantum information processing.
Abstract The existence of incompatible measurements, i.e. measurements which cannot be performed simultaneously on a single copy of a quantum state, constitutes an important distinction between quantum mechanics and classical theories. While incompatibility might at first glance seem like an obstacle, it turns to be a necessary ingredient to achieve the so-called quantum advantage in various operational tasks like random access codes or key distribution. To improve our understanding of how to quantify incompatibility of quantum measurements, we define and explore a family of incompatibility measures based on non-commutativity. We investigate some basic properties of these measures, we show that they satisfy some natural information-processing requirements and we fully characterize the pairs which achieve the highest incompatibility (in a fixed dimension). We also consider the behavior of our measures under different types of compositions. Finally, to link our new measures to existing results, we relate them to a robustness-based incompatibility measure and two operational scenarios: random access codes and entropic uncertainty relations.
Information is instrumental in our understanding of thermodynamics. Their interplay has been studied through completely degenerate Hamiltonians whereby the informational contributions to thermodynamic transformations can be isolated. In this setting, all states other than the maximally mixed state are considered to be in informational nonequilibrium. An important yet still open question is how to characterize the ability of quantum dynamics to preserve informational nonequilibrium. Here, the dynamical resource theory of informational nonequilibrium preservability is introduced to begin providing an answer to this question. A characterization of the allowed operations is given for qubit channels and the n-dimensional Weyl-covariant channels-a physically relevant subset of the general channels. An operational interpretation of a state discrimination game with Bell state measurements is given. Finally, an explicit link between a channel's classical capacity and its ability to preserve informational nonequilibrium is made.
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.
Quantum resources are certain features of the quantum world that provide advantages in certain information-theoretic, thermodynamic, or other useful operational tasks that are outside the realm of what classical theories can achieve. Quantum resource theories provide us with an elegant framework for studying these resources quantitatively and rigorously. While numerous state-based quantum resource theories have already been investigated, and to some extent, measurement-based resource theories have also been explored, instrument-based resource theories remain largely unexplored, with only a few notable exceptions. As quantum instruments are devices that provide both the classical outcomes of induced measurements and the post-measurement quantum states, they are quite important, especially for scenarios where multiple parties sequentially act on a quantum system. In this work, we study several instrument-based resource theories, namely (1) the resource theory of information preservability, (2) the resource theory of (strong) entanglement preservability, (3) the resource theory of (strong) incompatibility preservability, (4) the resource theory of traditional incompatibility, and (5) the resource theory of parallel incompatibility. Furthermore, we outline the hierarchies of these instrument-based resources and provide measures to quantify them. We then also established a relationship between our resource measure and the advantage in an information-theoretic task. In short, we provide a detailed framework for a wide variety of instrument-based quantum resource theories.
Measurement incompatibility describes two or more quantum measurements whose expected joint outcome on a given system cannot be defined. This purely nonclassical phenomenon provides a necessary ingredient in many quantum information tasks such as violating a Bell inequality or nonlocally steering part of an entangled state. In this Letter, we characterize incompatibility in terms of programmable measurement devices and the general notion of quantum programmability. This refers to the temporal freedom a user has in issuing programs to a quantum device. For devices with a classical control and classical output, measurement incompatibility emerges as the essential quantum resource embodied in their functioning. Based on the processing of programmable measurement devices, we construct a quantum resource theory of incompatibility. A complete set of convertibility conditions for programmable devices is derived based on quantum state discrimination with postmeasurement information.