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arXiv:1906.04959 · Physical Review A

Simple bounds for one-shot pure-state distillation in general resource theories

Madhav Krishnan Vijayan, Eric Chitambar, Min-Hsiu Hsieh

Published Nov 4, 2020• 10 citations
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Abstract

We present bounds for distilling many copies of a pure state from an arbitrary initial state in a general quantum resource theory. Our bounds apply to operations that are able to generate no more than a $\ensuremath{\delta}$ amount of resource, where $\ensuremath{\delta}\ensuremath{\ge}0$ is a given parameter. To maximize applicability of our upper bound, we assume little structure on the set of free states under consideration besides a weak form of superadditivity of the function ${G}_{\mathrm{min}}(\ensuremath{\rho})$, which measures the overlap between $\ensuremath{\rho}$ and the set of free states. Our bounds are given in terms of this function and the robustness of resource. Known results in coherence and entanglement theory are reproduced in this more general framework.

DOI
10.1103/physreva.102.052403

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We present bounds for distilling many copies of a pure state from an arbitrary initial state in a general quantum resource theory. Our bounds apply to operations that are able to generate no more than a \ensuremathδ\ensuremath{\delta}\ensuremathδ amount of resource, where \ensuremathδ\ensuremath≥0\ensuremath{\delta}\ensuremath{\ge}0\ensuremathδ\ensuremath≥0 is a given parameter. To maximize applicability of our upper bound, we assume little structure on the set of free states under consideration besides a weak form of superadditivity of the function Gmin(\ensuremathρ){G}_{\mathrm{min}}(\ensuremath{\rho})Gmin​(\ensuremathρ), which measures the overlap between \ensuremathρ\ensuremath{\rho}\ensuremathρ and the set of free states. Our bounds are given in terms of this function and the robustness of resource. Known results in coherence and entanglement theory are reproduced in this more general framework.
We present a general framework of a resource theory based on the assumption of a) convexity and b) that the overlap of free states with maximally resourceful state is bounded. Using this structure, we derive bounds on the one-shot distillation rate for such a resource theory, thereby reproducing known bounds in coherence and entanglement. We use the free robustness and introduce a function GminG_minGm​in related to overlap between states to express our bounds. To deal with resource theories where the free robustness in not finite we introduce the notion of imperfect free operations which we call ϵ\epsilonϵ-resource generating operations and generalize the free robustness toϵ\epsilonϵ-free robustness. We construct an ϵ\epsilonϵ-resource generating map that achieves pure state transformations and derive the conditions for such a transformation in terms of the ϵ\epsilonϵ-free robustness.
A fundamental approach for the characterization and quantification of all kinds of resources is to study the conversion between different resource objects under certain constraints. Here we analyze, from a resource-nonspecific standpoint, the optimal efficiency of resource formation and distillation tasks with only a single copy of the given quantum state, thereby establishing a unified framework of one-shot quantum resource manipulation. We find general bounds on the optimal rates characterized by resource measures based on the smooth max- or min-relative entropies and hypothesis testing relative entropy, as well as the free robustness measure, providing them with general operational meanings in terms of optimal state conversion. Our results encompass a wide class of resource theories via the theory-dependent coefficients we introduce, and the discussions are solidified by important examples, such as entanglement, coherence, superposition, magic states, asymmetry, and thermal nonequilibrium.
Although it is well known that the amount of resources that can be asymptotically distilled from a quantum state or channel does not exceed the resource cost needed to produce it, the corresponding relation in the nonasymptotic regime hitherto has not been well understood.Here, we establish a quantitative relation between the one-shot distillable resource yield and dilution cost in terms of transformation errors involved in these processes.Notably, our bound is applicable to quantum state and channel manipulation with respect to any type of quantum resource and any class of free transformations thereof, encompassing broad types of settings, including entanglement, quantum thermodynamics, and quantum communication.We also show that our techniques provide strong converse bounds relating the distillable resource and the resource dilution cost in the asymptotic regime.Moreover, we introduce a class of channels that generalize the twirling maps encountered in many resource theories, and by directly connecting it with resource quantification, we compute analytically several smoothed resource measures and improve our one-shot yield-cost bound in relevant theories.We use these operational insights to exactly evaluate important measures for various resource states in the resource theory of magic states.
A fundamental problem in resource theory is to study the manipulation of the resource. Focusing on a general dynamical resource theory of quantum channels, here we consider tasks of one-shot resource distillation and dilution with a single copy of the resource. For any target of unitary channel or pure state preparation channel, we establish a universal strategy to determine upper and lower bounds on rates that convert between any given resource and the target. We show that the rates are related to resource measures based on the channel robustness and the channel hypothesis testing entropy, with regularization factors of the target resource measures. The strategy becomes optimal with converged bounds when the channel robustness is finite and measures of the target resource collapse to the same value. The single-shot result also applies to asymptotic parallel manipulation of channels to obtain asymptotic resource conversion rates. We give several examples of dynamical resources, including the purity, classical capacity, quantum capacity, non-uniformity, coherence, and entanglement of quantum channels. Our results are applicable to general dynamical resource theories with potential applications in quantum communication, fault-tolerant quantum computing, and quantum thermodynamics.
Pure states are an important resource in many quantum information processing protocols. However, even making a fixed pure state, say ∣0⟩|0\rangle∣0⟩, in the laboratory requires a considerable amount of effort. Often one ends up with a mixed state ρρρ whose classical description is nevertheless known. Hence it is important to develop protocols that extract a fixed pure state from a known mixed state. In this work, we study the problem of extracting a fixed pure state ∣0⟩A′∣0⟩B′|0\rangle^{A'} |0\rangle^{B'}∣0⟩A′∣0⟩B′ from a known pure state ρABρ^{AB}ρAB distributed between two parties AAA and BBB. Here, A′A'A′, B′B'B′ are subspaces of AAA, BBB and the total amount of purity extracted is log⁡∣A′∣+log⁡∣B′∣\log |A'| + \log |B'|log∣A′∣+log∣B′∣. The parties can borrow local pure ancilla, apply local unitary operations and send a message from AAA to BBB through a dephasing channel. If local pure ancilla is borrowed, it must be subtracted in order to properly account for the purity extracted. We obtain the most efficient achievable bounds on one shot distributed purity extraction, in terms of the rate of local ancilla borrowed by the protocol, while distilling pure qubits at the best known rate. Our protocols borrow little to no local pure ancilla. Our bounds improve upon the existing bounds for this problem in both one shot as well as asymptotic iid settings. In particular they subsume all the asymptotic iid results of Devetak and Krovi-Devetak. In addition, we derive upper bounds for the rate of distillation in the one shot setting, which nearly match our achievable bounds.
We develop a general framework for higher-order quantum resources that treats the resourcefulness of quantum states, channels, and supermaps (also called processes) in a unified manner. By generalizing well-established concepts from resource theories of states and channels, we introduce and study supermap resource monotones and supermap distillation and dilution. As free supermaps, we consider free networks (supermaps that can be implemented with free channels), compatible supermaps (sets of supermaps closed under arbitrary composition), and maximally free superchannels. Specializing to approximate channel-to-channel conversions, we show that under mild assumptions, the conversion error with respect to the diamond distance and the worst-case infidelity coincide. We then apply the general framework to resource theories of dynamical coherence, where we establish the existence of largest sets of compatible supermaps. By characterizing these sets with semidefinite constraints, we show that they are strictly larger than the corresponding free networks but do not contain all maximally free superchannels. We finish by investigating the operational differences of these sets in one-shot channel distillation and dilution, providing solutions for both tasks in the parallel setting and for dilution in the adaptive setting.