The effect of noninertial dynamics on static quantum resource theories is investigated. To this end, we first show the equivalence between noninertial effects and a completely positive, trace-preserving (CPTP) map. In this formulation, the Unruh effect is equivalent to a bosonic amplifier channel. The effect of this map on a generic quantum resource is investigated by studying the role of the CPTP map on the three core ingredients of a resource theory, namely, the free states, the free operations and the resource quantifiers. We show several general statements can be made about these three components of a resource theory in the presence of noninertial motion.
The effect of noninertial dynamics on static quantum resource theories is investigated. To this end, we first show the equivalence between noninertial effects and a completely positive, trace-preserving (CPTP) map. In this formulation, the Unruh effect is equivalent to a bosonic amplifier channel. The effect of this map on a generic quantum resource is investigated by studying the role of the CPTP map on the three core ingredients of a resource theory, namely, the free states, the free operations and the resource quantifiers. We show several general statements can be made about these three components of a resource theory in the presence of noninertial motion.
For an open quantum system to evolve under CPTP maps, assumptions are made on the initial correlations between the system and the environment. Hermitian-preserving trace-preserving (HPTP) maps are considered as the local dynamic maps beyond CPTP. In this paper, we provide a succinct answer to the question of what physical maps are in the HPTP realm by two approaches. The first is by taking one step out of the CPTP set, which provides us with Semi-Positivity (SP) TP maps. The second way is by examining the physicality of HPTP maps, which leads to Semi-Nonnegative (SN) TP maps. Physical interpretations and geometrical structures are studied for these maps. The non-CP SPTP maps Ψ correspond to the quantum non-Markovian process under the CP-divisibility definition (Ψ=Ξ∘Φ−1, where Ξ and Φ are CPTP). When removing the invertibility assumption on Φ, we land in the set of SNTP maps. A by-product of set relations is an answer to the following question -- what kind of dynamics the system will go through when the previous dynamic Φ is non-invertible. In this case, the only locally well-defined maps are in SN\SP, they live on the boundary of SN. Otherwise, the non-local information will be irreplaceable in the system's dynamic. With the understanding of physical maps beyond CPTP, we prove that the current quantum error correction scheme is still sufficient to correct quantum non-Markovian errors. In some special cases, lack of complete positivity could provide us with more error correction methods with less overhead.
We initiate the systematic study of resource theories of quantum channels, i.e. of the dynamics that quantum systems undergo by completely positive maps, in abstracto: Resources are in principle all maps from one quantum system to another, but some maps are deemed free. The free maps are supposed to satisfy certain axioms, among them closure under tensor products, under composition and freeness of the identity map (the latter two say that the free maps form a monoid). The free maps act on the resources simply by tensor product and composition. This generalizes the much-studied resource theories of quantum states, and abolishes the distinction between resources (states) and the free maps, which act on the former, leaving only maps, divided into resource-full and resource-free ones. We discuss the axiomatic framework of quantifying channel resources, and show two general methods of constructing resource monotones of channels. Furthermore, we show that under mild regularity conditions, each resource theory of quantum channels has a distinguished monotone, the robustness (and its smoothed version), generalizing the analogous concept in resource theories of states. We give an operational interpretation of the log-robustness as the amount of heat dissipation (randomness) required for resource erasure by random reversible free maps, valid in broad classes of resource theories of quantum channels. Technically, this is based on an abstract version of the recent convex-split lemma, extended from its original domain of quantum states to ordered vector spaces with sufficiently well-behaved base norms (which includes the case of quantum channels with diamond norm or variants thereof). Finally, we remark on several key issues concerning the asymptotic theory.
We initiate the systematic study of resource theories of quantum channels, i.e. of the dynamics that quantum systems undergo by completely positive maps, in abstracto: Resources are in principle all maps from one quantum system to another, but some maps are deemed free. The free maps are supposed to satisfy certain axioms, among them closure under tensor products, under composition and freeness of the identity map (the latter two say that the free maps form a monoid). The free maps act on the resources simply by tensor product and composition. This generalizes the much-studied resource theories of quantum states, and abolishes the distinction between resources (states) and the free maps, which act on the former, leaving only maps, divided into resource-full and resource-free ones. We discuss the axiomatic framework of quantifying channel resources, and show two general methods of constructing resource monotones of channels. Furthermore, we show that under mild regularity conditions, each resource theory of quantum channels has a distinguished monotone, the robustness (and its smoothed version), generalizing the analogous concept in resource theories of states. We give an operational interpretation of the log-robustness as the amount of heat dissipation (randomness) required for resource erasure by random reversible free maps, valid in broad classes of resource theories of quantum channels. Technically, this is based on an abstract version of the recent convex-split lemma, extended from its original domain of quantum states to ordered vector spaces with sufficiently well-behaved base norms (which includes the case of quantum channels with diamond norm or variants thereof). Finally, we remark on several key issues concerning the asymptotic theory.
We investigate resource nongenerating operations in both static and dynamical quantum resource theories. For the static scenarios, we derive a sufficient condition for state transformations under resource nongenerating operations. Then we construct a dynamical resource theory where resource nongenerating operations constitute the set of free operations, and we propose an axiomatic approach to quantify the dynamical resource. We further analyze the erasure of the dynamical resources. As applications, we establish bounds on the rate of state conversions under resource nongenerating operations in a generic convex resource theory and obtain capacity bounds for classical communication tasks assisted by dynamical coherence. Our results clarify the key roles of resource nongenerating operations in quantum information processing tasks.
Completely positive (CP) matrices are ubiquitous in modern science and technology with applications in optimization, graph theory, and quantum entanglement. Recently, Johnston \emph{et al.} [Linear Algebra and its Applications, 2022] have cast CP matrices into the framework of quantum resource theories, where CP states serve as free states and CP-preserving channels act as free operations. This work addresses several questions raised in their work. Specifically, we provide the necessary and sufficient conditions of CP-preserving channels in small dimensions, which are necessary in higher dimensions, and discuss the resource quantification via the trace distance of non-negativity. By constructing an explicit counterexample, we demonstrate that the trace-distance measure of non-negativity violates strong monotonicity. We also provide an alternative proof that every CPDNN channel $Φ:\MM_n\to \MM_2$ is CPCP. Additionally, we show that any unital CPDNN map $Φ:\MM_2\to \MM_n$ is also CPCP.
Chapter 9 introduces the framework of static quantum resource theories, which provide a structured approach for studying different types of quantum resources like entanglement and coherence. The chapter begins by laying out the structure of quantum resource theories, defining what constitutes a quantum resource and how it can be quantified, manipulated, and converted. The text discusses the role of free states and free operations in resource theories, as they form the basis for comparing resources. It introduces state-based resource theories, which focus on the resource content of quantum states, and affine resource theories, which are used to study various interconversions of quantum resources. Resource witnesses, a key concept, are explored as tools to detect the presence of a resource within a quantum state.