We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorisation relation, e.g. the resource theories of entanglement and coherence (for pure state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a non-trivial trade-off between the rate rnat which n copies of a resource state ρ can be transformed into nrncopies of another resource state σ, and the error level εnof the interconversion process, as a function of n. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion rnapproaches its optimum in the asymptotic limit of unbounded resources (n → ∞), while the error εnvanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behaviour which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite n regime.
We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorisation relation, e.g. the resource theories of entanglement and coherence (for pure state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a non-trivial trade-off between the rate rnat which n copies of a resource state ρ can be transformed into nrncopies of another resource state σ, and the error level εnof the interconversion process, as a function of n. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion rnapproaches its optimum in the asymptotic limit of unbounded resources (n → ∞), while the error εnvanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behaviour which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite n regime.
We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorization relation, e.g., the resource theories of entanglement and coherence (for pure-state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a nontrivial trade-off between the rate rn at which n copies of a resource state $\ensuremath{\rho}$ can be transformed into nrn copies of another resource state $\ensuremath{\sigma}$, and the error level ${\ensuremath{\epsilon}}_{n}$ of the interconversion process, as a function of n. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion rn approaches its optimum in the asymptotic limit of unbounded resources ($n\ensuremath{\rightarrow}\ensuremath{\infty}$), while the error ${\ensuremath{\epsilon}}_{n}$ vanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behavior which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite n regime.
We identify and explore the intriguing property of resource resonance arising within resource theories of entanglement, coherence, and thermodynamics. While the theories considered are reversible asymptotically, the same is generally not true in realistic scenarios where the available resources are bounded. The finite-size effects responsible for this irreversibility could potentially prohibit small quantum information processors or thermal machines from achieving their full potential. Nevertheless, we show here that by carefully engineering the resource interconversion process any such losses can be greatly suppressed. Our results are predicted by higher order expansions of the trade-off between the rate of resource interconversion and the achieved fidelity, and are verified by exact numerical optimizations of the appropriate underlying approximate majorization conditions.
Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.
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.
The moderate deviation regime is concerned with the finite block length trade-off between communication cost and error for information processing tasks in the asymptotic regime, where the communication cost approaches a capacity-like quantity and the error vanishes at the same time. We find exact characterisations of these trade-offs for a variety of fully quantum communication tasks, including quantum source coding, quantum state splitting, entanglement-assisted quantum channel coding, and entanglement-assisted quantum channel simulation. The main technical tool we derive is a tight relation between the partially smoothed max-information and the hypothesis testing relative entropy. This allows us to obtain the expansion of the partially smoothed max-information for i.i.d. states in the moderate deviation regime.
We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel E mapping ρ1⊗n into ρ2⊗Rnn with an error ϵn (measured by trace distance) and σ1⊗n into σ2⊗Rnn exactly, for a large number n . We derive second-order asymptotic expressions for the optimal transformation rate Rn in the small-, moderate-, and large-deviation error regimes, as well as the zero-error regime, for an arbitrary pair (ρ1,σ1) of initial states and a commuting pair (ρ2,σ2) of final states. We also prove that for σ1 and σ2 given by thermal Gibbs states, the derived optimal transformation rates in the first three regimes can be attained by thermal operations. This allows us, for the first time, to study the second-order asymptotics of thermodynamic state interconversion with fully general initial states that may have coherence between different energy eigenspaces. Thus, we discuss the optimal performance of thermodynamic protocols with coherent inputs and describe three novel resonance phenomena allowing one to significantly reduce transformation errors induced by finite-size effects. What is more, our result on quantum dichotomies can also be used to obtain, up to second-order asymptotic terms, optimal conversion rates between pure bipartite entangled states under local operations and classical communication. Published by the American Physical Society 2024