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Register a new userResource theories of multi-time processes: A window into quantum non-Markovianity
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We investigate the conditions under which an uncontrollable background processes may be harnessed by an agent to perform a task that would otherwise be impossible within their operational framework. This situation can be understood from the perspective of resource theory: rather than harnessing useful quantum states to perform tasks, we propose a resource theory of quantum processes across multiple points in time. Uncontrollable background processes fulfil the role of resources, and a new set of objects called superprocesses, corresponding to operationally implementable control of the system undergoing the process, constitute the transformations between them. After formally introducing a framework for deriving resource theories of multi-time processes, we present a hierarchy of examples induced by restricting quantum or classical communication within the superprocess - corresponding to a client-server scenario. The resulting nine resource theories have different notions of quantum or classical memory as the determinant of their utility. Furthermore, one of these theories has a strict correspondence between non-useful processes and those that are Markovian and, therefore, could be said to be a true quantum resource theory of non-Markovianity.
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We establish a convex resource theory of non-Markovianity under the constraint of small time intervals within the temporal evolution. We construct the free operations, free states and a generalized bona-fide measure of non-Markovianity. The framework satisfies the basic properties of a consistent resource theory. The proposed resource quantifier is lower bounded by the optimization free Rivas-Huelga-Plenio (RHP) measure of nonMarkovianity. We further define the robustness of non-Markovianity and show that it can directly be expressed as a function of the RHP measure of non-Markovianity. This enables a physical interpretation of the RHP measure.
The non-Markovianity of the stochastic process called the quantum semi-Markov (QSM) process is studied using a recently proposed quantification of memory based on the deviation from semigroup evolution and thus providing a unified description of divisible and indivisible channels. This is shown to bring out the property of QSM process to exhibit memory effects in the CP-divisible regime. An operational meaning to the non-Markovian nature of semi-Markov processes is also provided.
To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class $Omega^*$ of operations performed by three distant parties. A tripartite quantum state is a free state under $Omega^*$ if and only if it is a quantum Markov chain. We introduce a function of tripartite quantum states that we call the non-Markovianity of formation, and prove that it is a faithful measure of non-Markovianity, which is continuous and monotonically nonincreasing under a subclass $Omega$ of $Omega^*$. We consider a task in which the three parties generate a non-Markov state from scratch by operations in $Omega$, assisted with quantum communication from the third party to the others, which does not belong to $Omega$. We prove that the minimum cost of quantum communication required therein is asymptotically equal to the regularized non-Markovianity of formation. Based on this result, we provide a direct operational meaning to a measure of bipartite entanglement called the c-squashed entanglement.
Quantum resource theories (QRTs) offer a highly versatile and powerful framework for studying different phenomena in quantum physics. From quantum entanglement to quantum computation, resource theories can be used to quantify a desirable quantum effect, develop new protocols for its detection, and identify processes that optimize its use for a given application. Particularly, QRTs revolutionize the way we think about familiar properties of physical systems like entanglement, elevating them from just being interesting from a fundamental point of view to being useful in performing practical tasks. The basic methodology of a general QRT involves partitioning all quantum states into two groups, one consisting of free states and the other consisting of resource states. Accompanying the set of free states is a collection of free quantum operations arising from natural restrictions on physical systems, and that consists of all the physical processes allowed by the resource theory and which acts invariantly on the set of free states. The QRT then studies what information processing tasks become possible using the restricted operations. Despite the large degree of freedom in how one defines the free states and free operations, unexpected similarities emerge among different QRTs in terms of resource measures and resource convertibility. As a result, objects that appear quite distinct on the surface, such as entanglement and quantum reference frames, appear to have great similarity on a deeper structural level. In this article we review the general framework of a quantum resource theory, focusing on common structural features, operational tasks, and resource measures. To illustrate these concepts, an overview is provided on some of the more commonly studied QRTs in the literature.
The prevalent modus operandi within the framework of quantum resource theories has been to characterise and harness the resources within single objects, in what we can call emph{single-object} quantum resource theories. One can wonder however, whether the resources contained within multiple different types of objects, now in a emph{multi-object} quantum resource theory, can simultaneously be exploited for the benefit of an operational task. In this work, we introduce examples of such multi-object operational tasks in the form of subchannel discrimination and subchannel exclusion games, in which the player harnesses the resources contained within a state-measurement pair. We prove that for any state-measurement pair in which either of them is resourceful, there exist discrimination and exclusion games for which such a pair outperforms any possible free state-measurement pair. These results hold for arbitrary convex resources of states, and arbitrary convex resources of measurements for which classical post-processing is a free operation. Furthermore, we prove that the advantage in these multi-object operational tasks is determined, in a multiplicative manner, by the resource quantifiers of: emph{generalised robustness of resource} of both state and measurement for discrimination games and emph{weight of resource} of both state and measurement for exclusion games.
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