693 research outputs found
Quantum dichotomies and coherent thermodynamics beyond first-order asymptotics
We address the problem of exact and approximate transformation of quantum
dichotomies in the asymptotic regime, i.e., the existence of a quantum channel
mapping into with an
error (measured by trace distance) and into
exactly, for a large number . We derive
second-order asymptotic expressions for the optimal transformation rate
in the small, moderate, and large deviation error regimes, as well as the
zero-error regime, for an arbitrary pair of initial states
and a commuting pair of final states. We also prove that
for and 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.Comment: 51 pages, 6 figures, comments welcom
Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols
The quantum relative entropy is known to play a key role in determining the
asymptotic convertibility of quantum states in general resource-theoretic
settings, often constituting the unique monotone that is relevant in the
asymptotic regime. We show that this is no longer the case when one allows
stochastic protocols that may only succeed with some probability, in which case
the quantum relative entropy is insufficient to characterize the rates of
asymptotic state transformations, and a new entropic quantity based on a
regularization of Hilbert's projective metric comes into play. Such a scenario
is motivated by a setting where the cost associated with transformations of
quantum states, typically taken to be the number of copies of a given state, is
instead identified with the size of the quantum memory needed to realize the
protocol. Our approach allows for constructing transformation protocols that
achieve strictly higher rates than those imposed by the relative entropy.
Focusing on the task of resource distillation, we give broadly applicable
strong converse bounds on the asymptotic rates of probabilistic distillation
protocols, and show them to be tight in relevant settings such as entanglement
distillation with non-entangling operations. This generalizes and extends
previously known limitations that only apply to deterministic protocols. Our
methods are based on recent results for probabilistic one-shot transformations
as well as a new asymptotic equipartition property for the projective relative
entropy.Comment: 7+16 pages, 2 figure
Sufficiency of R\'enyi divergences
A set of classical or quantum states is equivalent to another one if there
exists a pair of classical or quantum channels mapping either set to the other
one. For dichotomies (pairs of states) this is closely connected to (classical
or quantum) R\'enyi divergences (RD) and the data-processing inequality: If a
RD remains unchanged when a channel is applied to the dichotomy, then there is
a recovery channel mapping the image back to the initial dichotomy. Here, we
prove for classical dichotomies that equality of the RDs alone is already
sufficient for the existence of a channel in any of the two directions and
discuss some applications. We conjecture that equality of the minimal quantum
RDs is sufficient in the quantum case and prove it for special cases. We also
show that neither the Petz quantum nor the maximal quantum RDs are sufficient.
As a side-result of our techniques we obtain an infinite list of inequalities
fulfilled by the classical, the Petz quantum, and the maximal quantum RDs.
These inequalities are not true for the minimal quantum RDs.Comment: Comments welcome, 31 pages, v2: Removed insignifcant error; v3:
Misupload; v4: Significantly improved presentatio
Relative Entropy and Catalytic Relative Majorization
Given two pairs of quantum states, a fundamental question in the resource
theory of asymmetric distinguishability is to determine whether there exists a
quantum channel converting one pair to the other. In this work, we reframe this
question in such a way that a catalyst can be used to help perform the
transformation, with the only constraint on the catalyst being that its reduced
state is returned unchanged, so that it can be used again to assist a future
transformation. What we find here, for the special case in which the states in
a given pair are commuting, and thus quasi-classical, is that this catalytic
transformation can be performed if and only if the relative entropy of one pair
of states is larger than that of the other pair. This result endows the
relative entropy with a fundamental operational meaning that goes beyond its
traditional interpretation in the setting of independent and identical
resources. Our finding thus has an immediate application and interpretation in
the resource theory of asymmetric distinguishability, and we expect it to find
application in other domains.Comment: v3: 18 pages, 1 figure, accepted for publication in Physical Review
Researc
The Semiring of Dichotomies and Asymptotic Relative Submajorization
We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an asymptotic variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum. These strong constraints allow us to classify and explicitly describe all such monotones, leading to a rate formula expressed as an optimization involving sandwiched Rényi divergences. As an application we give a new derivation of the strong converse error exponent in quantum hypothesis testing
The semiring of dichotomies and asymptotic relative submajorization
We study quantum dichotomies and the resource theory of asymmetric
distinguishability using a generalization of Strassen's theorem on preordered
semirings. We find that an asymptotic variant of relative submajorization,
defined on unnormalized dichotomies, is characterized by real-valued monotones
that are multiplicative under the tensor product and additive under the direct
sum. These strong constraints allow us to classify and explicitly describe all
such monotones, leading to a rate formula expressed as an optimization
involving sandwiched R\'enyi divergences. As an application we give a new
derivation of the strong converse error exponent in quantum hypothesis testing.Comment: 21 page
Variance of Relative Surprisal as Single-Shot Quantifier
The variance of (relative) surprisal, also known as varentropy, so far mostly plays a role in information theory as quantifying the leading-order corrections to asymptotic independent and identically distributed (IID) limits. Here, we comprehensively study the use of it to derive single-shot results in (quantum) information theory. We show that it gives genuine sufficient and necessary conditions for approximate state transitions between pairs of quantum states in the single-shot setting, without the need for further optimization. We also clarify its relation to smoothed min and max entropies, and construct a monotone for resource theories using only the standard (relative) entropy and variance of (relative) surprisal. This immediately gives rise to enhanced lower bounds for entropy production in random processes. We establish certain properties of the variance of relative surprisal, which will be useful for further investigations, such as uniform continuity and upper bounds on the violation of subadditivity. Motivated by our results, we further derive a simple and physically appealing axiomatic single-shot characterization of (relative) entropy, which we believe to be of independent interest. We illustrate our results with several applications, ranging from interconvertibility of ergodic states, over Landauer erasure to a bound on the necessary dimension of the catalyst for catalytic state transitions and Boltzmann’s H theorem
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
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