47 research outputs found
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
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
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
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