Tight constraints on probabilistic convertibility of quantum states

We develop two general approaches to characterising the manipulation of quantum states by means of probabilistic protocols constrained by the limitations of some quantum resource theory. First, we give a general necessary condition for the existence of a physical transformation between quantum states, obtained using a recently introduced resource monotone based on the Hilbert projective metric. In all affine quantum resource theories (e.g. coherence, asymmetry, imaginarity) as well as in entanglement distillation, we show that the monotone provides a necessary and sufficient condition for one-shot resource convertibility under resource-non-generating operations, and hence no better restrictions on all probabilistic protocols are possible. We use the monotone to establish improved bounds on the performance of both one-shot and many-copy probabilistic resource distillation protocols. Complementing this approach, we introduce a general method for bounding achievable probabilities in resource transformations under resource-non-generating maps through a family of convex optimisation problems. We show it to tightly characterise single-shot probabilistic distillation in broad types of resource theories, allowing an exact analysis of the trade-offs between the probabilities and errors in distilling maximally resourceful states. We demonstrate the usefulness of both of our approaches in the study of quantum entanglement distillation.Comment: 46 pages, 3 figures. Technical companion paper to Phys. Rev. Lett. 128, 110505 (2022) [arXiv:2109.04481]; contains mostly content that was split off from arXiv:2109.04481v1, plus a lot of clarifications, extensions, and additional examples. v3: Accepted versio

Reversibility of quantum resources through probabilistic protocols

Among the most fundamental questions in the manipulation of quantum resources such as entanglement is the possibility of reversibly transforming all resource states. The most important consequence of this would be the identification of a unique entropic resource measure that exactly quantifies the limits of achievable transformation rates. Remarkably, previous results claimed that such asymptotic reversibility holds true in very general settings; however, recently those findings have been found to be incomplete, casting doubt on the conjecture. Here we show that it is indeed possible to reversibly interconvert all states in general quantum resource theories, as long as one allows protocols that may only succeed probabilistically. Although such transformations have some chance of failure, we show that their success probability can be ensured to be bounded away from zero, even in the asymptotic limit of infinitely many manipulated copies. As in previously conjectured approaches, the achievability here is realised through operations that are asymptotically resource non-generating. Our methods are based on connecting the transformation rates under probabilistic protocols with strong converse rates for deterministic transformations. We strengthen this connection into an exact equivalence in the case of entanglement distillation.Comment: 6+10 page

Computable lower bounds on the entanglement cost of quantum channels

A class of lower bounds for the entanglement cost of any quantum state was recently introduced in [arXiv:2111.02438] in the form of entanglement monotones known as the tempered robustness and tempered negativity. Here we extend their definitions to point-to-point quantum channels, establishing a lower bound for the asymptotic entanglement cost of any channel, whether finite or infinite dimensional. This leads, in particular, to a bound that is computable as a semidefinite program and that can outperform previously known lower bounds, including ones based on quantum relative entropy. In the course of our proof we establish a useful link between the robustness of entanglement of quantum states and quantum channels, which requires several technical developments such as showing the lower semicontinuity of the robustness of entanglement of a channel in the weak*-operator topology on bounded linear maps between spaces of trace class operators.Comment: 24 pages. Technical companion paper to [arXiv:2111.02438], now published as [Nat. Phys. 19, 184-189 (2023)]. In v2, which is close to the published version, we improved the presentation and corrected a few typo

Generating entanglement between two-dimensional cavities in uniform acceleration

Moving cavities promise to be a suitable system for relativistic quantum information processing. It has been shown that an inertial and a uniformly accelerated one-dimensional cavity can become entangled by letting an atom emit an excitation while it passes through the cavities, but the acceleration degrades the ability to generate entanglement. We show that in the two-dimensional case the entanglement generated is affected not only by the cavity's acceleration but also by its transverse dimension which plays the role of an effective mass

No second law of entanglement manipulation after all

We prove that the theory of entanglement manipulation is asymptotically irreversible under all non-entangling operations, showing from first principles that reversible entanglement transformations require the generation of entanglement in the process. Entanglement is thus shown to be the first example of a quantum resource that does not become reversible under the maximal set of free operations, that is, under all resource non-generating maps. Our result stands in stark contrast with the reversibility of quantum and classical thermodynamics, and implies that no direct counterpart to the second law of thermodynamics can be established for entanglement -- in other words, there exists no unique measure of entanglement governing all axiomatically possible state-to-state transformations. This completes the solution of a long-standing open problem [Problem 20 in arXiv:quant-ph/0504166]. We strengthen the result further to show that reversible entanglement manipulation requires the creation of exponentially large amounts of entanglement according to monotones such as the negativity. Our findings can also be extended to the setting of point-to-point quantum communication, where we show that there exist channels whose parallel simulation entanglement cost exceeds their quantum capacity, even under the most general quantum processes that preserve entanglement-breaking channels. The main technical tool we introduce is the tempered logarithmic negativity, a single-letter lower bound on the entanglement cost that can be efficiently computed via a semi-definite program.Comment: 16+30 pages, 3 figures. v2: minor clarification

Distillable entanglement under dually non-entangling operations

Computing the exact rate at which entanglement can be distilled from noisy quantum states is one of the longest-standing questions in quantum information. We give an exact solution for entanglement distillation under the set of dually non-entangling (DNE) operations -- a relaxation of the typically considered local operations and classical communication, comprising all channels which preserve the sets of separable states and measurements. We show that the DNE distillable entanglement coincides with a modified version of the regularised relative entropy of entanglement in which the arguments are measured with a separable measurement. Ours is only the second known regularised formula for the distillable entanglement under any class of free operations in entanglement theory, after that given by Devetak and Winter for one-way LOCCs. An immediate consequence of our finding is that, under DNE, entanglement can be distilled from any entangled state. As our second main result, we construct a general upper bound on the DNE distillable entanglement, using which we prove that the separably measured relative entropy of entanglement can be strictly smaller than the regularisation of the standard relative entropy of entanglement. This solves an open problem in [Li/Winter, CMP 326, 63 (2014)].Comment: 7+26 page

One-shot manipulation of dynamical quantum resources

We develop a unified framework to characterize one-shot transformations of dynamical quantum resources in terms of resource quantifiers, establishing universal conditions for exact and approximate transformations in general resource theories. Our framework encompasses all dynamical resources represented as quantum channels, including those with a specific structure -- such as boxes, assemblages, and measurements -- thus immediately applying in a vast range of physical settings. For the particularly important manipulation tasks of distillation and dilution, we show that our conditions become necessary and sufficient for broad classes of important theories, enabling an exact characterization of these tasks and establishing a precise connection between operational problems and resource monotones based on entropic divergences. We exemplify our results by considering explicit applications to: quantum communication, where we obtain exact expressions for one-shot quantum capacity and simulation cost assisted by no-signalling, separability-preserving, and positive partial transpose-preserving codes; as well as to nonlocality, contextuality, and measurement incompatibility, where we present operational applications of a number of relevant resource measures.Comment: 5+10 pages. Made some changes in presentation and added minor clarifications. Accepted in Physical Review Letter

Fundamental limitations on distillation of quantum channel resources

Quantum channels underlie the dynamics of quantum systems, but in many practical settings it is the channels themselves that require processing. We establish universal limitations on the processing of both quantum states and channels, expressed in the form of no-go theorems and quantitative bounds for the manipulation of general quantum channel resources under the most general transformation protocols. Focusing on the class of distillation tasks -- which can be understood either as the purification of noisy channels into unitary ones, or the extraction of state-based resources from channels -- we develop fundamental restrictions on the error incurred in such transformations and comprehensive lower bounds for the overhead of any distillation protocol. In the asymptotic setting, our results yield broadly applicable bounds for rates of distillation. We demonstrate our results through applications to fault-tolerant quantum computation, where we obtain state-of-the-art lower bounds for the overhead cost of magic state distillation, as well as to quantum communication, where we recover a number of strong converse bounds for quantum channel capacity.Comment: 15+25 pages, 4 figures. v3: close to published version (changes in presentation, title modified; main results unaffected). See also related work by Fang and Liu at arXiv:2010.1182

Benchmarking one-shot distillation in general quantum resource theories

We study the one-shot distillation of general quantum resources, providing a unified quantitative description of the maximal fidelity achievable in this task, and revealing similarities shared by broad classes of resources. We establish fundamental quantitative and qualitative limitations on resource distillation applicable to all convex resource theories. We show that every convex quantum resource theory admits a meaningful notion of a pure maximally resourceful state which maximizes several monotones of operational relevance and finds use in distillation. We endow the generalized robustness measure with an operational meaning as an exact quantifier of performance in distilling such maximal states in many classes of resources including bi- and multipartite entanglement, multi-level coherence, as well as the whole family of affine resource theories, which encompasses important examples such as asymmetry, coherence, and thermodynamics.Comment: 8+5 pages, 1 figure. v3: fixed (inconsequential) error in Lemma 1