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Operational Resource Theory of Quantum Channels

Yunchao Liu, Xiao Yuan

arXiv:1904.02680v3quant-ph

TL;DR

Quantum channels have often been treated as tools for manipulating state resources, rather than as resources that can themselves be converted. This paper develops a channel resource framework and rate bounds, then finds that channel coherence is asymptotically irreversible under maximally incoherent operations.

  • Problem

    Quantum channels have been studied for resource-theoretic properties, but not generally as operational resources that can themselves be manipulated and converted.

    Source: p. 1

  • Method

    The paper defines channel resource monotones from channel distance quantifiers, relates them to state-resource manipulation power, and studies asymptotic channel distillation and dilution.

    Source: pp. 1–2

  • Results

    The rate bounds applied to coherence under maximally incoherent operations show that its channel resource theory is asymptotically irreversible.

    Source: pp. 2, 5

  • Takeaways & Limitations

    Channel resource monotones bound state-resource manipulation power and asymptotic channel conversion rates in the settings studied.

    Source: p. 2

    The general case of other channel manipulation powers, potentially with different optimal resource channels, is left for future work.

    Source: p. 4

Abstract

from arXiv · show

Quantum resource theories have been widely studied to systematically characterize the non-classicality of quantum systems. Most resource theories focus on quantum states and study their interconversions. Although quantum channels are generally used as the tool for state manipulation, such a manipulation capability can be naturally regarded as a generalized quantum resource, leading to an open research direction in the resource theories of quantum channels. Various resource-theoretic properties of channels have been investigated, however, without treating channels themselves as operational resources that can also be manipulated and converted. In this Letter, we address this problem by first proposing a general resource framework for quantum channels and introducing resource monotones based on general distance quantifiers of channels. We study the interplay between channel and state resource theories by relating resource monotones of a quantum channel to its manipulation power of the state resource. Regarding channels as operational resources, we introduce asymptotic channel distillation and dilution, the most important tasks in an operational resource theory, and show how to bound the conversion rates with channel resource monotones. Finally, we apply our results to quantum coherence as an example and introduce the coherence of channels, which characterizes the coherence generation ability of channels. We consider asymptotic channel distillation and dilution with maximally incoherent operations and find the theory asymptotically irreversible, in contrast to the asymptotic reversibility of the coherence of states.

Properties of Channel Resource Monotones

The section establishes that the proposed channel resource quantifiers are monotones under free super-operations. It covers distance-based measures and the resource-generating and resource-boosting powers.

Properties of Channel Resource Monotones

  • The four proposed channel resource quantifiers satisfy the resource-monotone properties.The results include distance-based channel monotones and resource-generating and resource-boosting powers.Source: Properties of Channel Resource Monotones · p. 7
  • For any distance quantifier satisfying nonnegativity and data processing, R(N) and RΩ(N) are channel resource monotones.The monotonicity proof for RΩ uses the form of free super-operations and data processing.Source: Properties of Channel Resource Monotones · pp. 7–8
  • If the state resource measure satisfies nonnegativity and monotonicity, its resource-generating and resource-boosting powers are channel resource monotones.The proof uses the state measure's monotonicity under free operations.Source: Properties of Channel Resource Monotones · pp. 8–9

Interplay with State Resource Theories

The section relates channel resource monotones to the power of channels to generate state resources. It gives bounds under stated conditions and describes a case where a generating-power bound is tight.

Interplay with State Resource Theories

  • The section establishes general relationships between distance-based channel monotones and resource-generating power.It also presents a distinct lower bound on Rg and a resource-destroying-channel characterization.Source: Interplay with State Resource Theories · pp. 9–10
  • When the state distance quantifier obeys the triangle inequality, R(N) is bounded below by an optimized difference of distances to a state δ.The bound optimizes over input states and resource-nongenerating channels.Source: Interplay with State Resource Theories · p. 9
  • If a resource-destroying channel λ satisfies µ(ρ) = D(ρ, λ(ρ)), then the resource-generating power equals RΩ(N).The resource-destroying-channel quantifier also provides a closed-form expression, though it need not itself be a resource monotone.Source: Interplay with State Resource Theories · p. 10

Channel Distillation and Dilution

The section develops asymptotic channel distillation and dilution, relating their rates to channel resource measures. It distinguishes parallel and iterative protocols and gives rate bounds under specified assumptions.

Channel Distillation and Dilution

  • Parallel distillation and dilution protocols are contained in their iterative counterparts, giving Rdistill(N) ≤ ˜Rdistill(N) ≤ ˜Rdilute(N) ≤ Rdilute(N).The construction embeds parallel operations in the iterative protocol by choosing intermediate channels to be identities.Source: Channel Distillation and Dilution · p. 11
  • Any asymptotic channel resource measure bounds the distillation rate from above and the dilution rate from below.The proof states Rdistill(N) ≤ Rasymp(N) ≤ Rdilute(N).Source: Channel Distillation and Dilution · pp. 11–12
  • If the state resource measure is additive, the channel resource-generating and resource-boosting powers are superadditive under channel tensor products.For example, Rg(N ⊗ M) ≥ Rg(N) + Rg(M), and likewise for Rb.Source: Channel Distillation and Dilution · p. 12
  • When Rg satisfies (R3) and (R5), the parallel distillation rate is bounded by Rg, with equality if Rg also satisfies (R4’).The lower bound is Rdistill(N) ≥ Rg(N); the result supplies the corresponding upper bound under the stated conditions.Source: Channel Distillation and Dilution · p. 13
  • The iterative distillation rate is at most Rb, with equality when the state resource theory is asymptotically reversible and µ = µd.The achieving protocol repeatedly distills and dilutes resource states while using the channel.Source: Channel Distillation and Dilution · pp. 13–14
  • Under the stated channel-resource assumption, parallel and iterative dilution rates are equal and are at least Rb when the specified regularity conditions hold.The optimal resource channel is assumed to be the constant channel outputting the maximal unit resource state.Source: Channel Distillation and Dilution · pp. 12, 14

Applications in the Resource Theory of Coherence

The coherence application defines channel coherence measures and applies the general rate bounds to maximally incoherent operations. It finds unequal parallel and iterative distillation rates in general and characterizes the asymptotic dilution rate.

Applications in the Resource Theory of Coherence

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