114 research outputs found
Guaranteed energy-efficient bit reset in finite time
Landauer's principle states that it costs at least kTln2 of work to reset one
bit in the presence of a heat bath at temperature T. The bound of kTln2 is
achieved in the unphysical infinite-time limit. Here we ask what is possible if
one is restricted to finite-time protocols. We prove analytically that it is
possible to reset a bit with a work cost close to kTln2 in a finite time. We
construct an explicit protocol that achieves this, which involves changing the
system's Hamiltonian avoiding quantum coherences, and thermalising. Using
concepts and techniques pertaining to single-shot statistical mechanics, we
further develop the limit on the work cost, proving that the heat dissipated is
close to the minimal possible not just on average, but guaranteed with high
confidence in every run. Moreover we exploit the protocol to design a quantum
heat engine that works near the Carnot efficiency in finite time.Comment: 5 pages + 5 page technical appendix. 5 figures. Author accepted
versio
Maximum one-shot dissipated work from Renyi divergences
Thermodynamics describes large-scale, slowly evolving systems. Two modern
approaches generalize thermodynamics: fluctuation theorems, which concern
finite-time nonequilibrium processes, and one-shot statistical mechanics, which
concerns small scales and finite numbers of trials. Combining these approaches,
we calculate a one-shot analog of the average dissipated work defined in
fluctuation contexts: the cost of performing a protocol in finite time instead
of quasistatically. The average dissipated work has been shown to be
proportional to a relative entropy between phase-space densities, to a relative
entropy between quantum states, and to a relative entropy between probability
distributions over possible values of work. We derive one-shot analogs of all
three equations, demonstrating that the order-infinity Renyi divergence is
proportional to the maximum possible dissipated work in each case. These
one-shot analogs of fluctuation-theorem results contribute to the unification
of these two toolkits for small-scale, nonequilibrium statistical physics.Comment: 8 pages. Close to published versio
Introducing one-shot work into fluctuation relations
Two approaches to small-scale and quantum thermodynamics are fluctuation
relations and one-shot statistical mechanics. Fluctuation relations (such as
Crooks' Theorem and Jarzynski's Equality) relate nonequilibrium behaviors to
equilibrium quantities such as free energy. One-shot statistical mechanics
involves statements about every run of an experiment, not just about averages
over trials.
We investigate the relation between the two approaches. We show that both
approaches feature the same notions of work and the same notions of probability
distributions over possible work values. The two approaches are alternative
toolkits with which to analyze these distributions. To combine the toolkits, we
show how one-shot work quantities can be defined and bounded in contexts
governed by Crooks' Theorem. These bounds provide a new bridge from one-shot
theory to experiments originally designed for testing fluctuation theorems.Comment: 37 pages, 6 figure
Entropic equality for worst-case work at any protocol speed
We derive an equality for non-equilibrium statistical mechanics in
finite-dimensional quantum systems. The equality concerns the worst-case work
output of a time-dependent Hamiltonian protocol in the presence of a Markovian
heat bath. It has has the form "worst-case work = penalty - optimum". The
equality holds for all rates of changing the Hamiltonian and can be used to
derive the optimum by setting the penalty to 0. The optimum term contains the
max entropy of the initial state, rather than the von Neumann entropy, thus
recovering recent results from single-shot statistical mechanics. Energy
coherences can arise during the protocol but are assumed not to be present
initially. We apply the equality to an electron box.Comment: 4 page + 14 page appendix; 8 figures; AA
A measure of majorisation emerging from single-shot statistical mechanics
The use of the von Neumann entropy in formulating the laws of thermodynamics
has recently been challenged. It is associated with the average work whereas
the work guaranteed to be extracted in any single run of an experiment is the
more interesting quantity in general. We show that an expression that
quantifies majorisation determines the optimal guaranteed work. We argue it
should therefore be the central quantity of statistical mechanics, rather than
the von Neumann entropy. In the limit of many identical and independent
subsystems (asymptotic i.i.d) the von Neumann entropy expressions are recovered
but in the non-equilbrium regime the optimal guaranteed work can be radically
different to the optimal average. Moreover our measure of majorisation governs
which evolutions can be realized via thermal interactions, whereas the
nondecrease of the von Neumann entropy is not sufficiently restrictive. Our
results are inspired by single-shot information theory.Comment: 54 pages (15+39), 9 figures. Changed title / changed presentation,
same main results / added minor result on pure bipartite state entanglement
(appendix G) / near to published versio
The chain rule implies Tsirelson's bound: an approach from generalized mutual information
In order to analyze an information theoretical derivation of Tsirelson's
bound based on information causality, we introduce a generalized mutual
information (GMI), defined as the optimal coding rate of a channel with
classical inputs and general probabilistic outputs. In the case where the
outputs are quantum, the GMI coincides with the quantum mutual information. In
general, the GMI does not necessarily satisfy the chain rule. We prove that
Tsirelson's bound can be derived by imposing the chain rule on the GMI. We
formulate a principle, which we call the no-supersignalling condition, which
states that the assistance of nonlocal correlations does not increase the
capability of classical communication. We prove that this condition is
equivalent to the no-signalling condition. As a result, we show that
Tsirelson's bound is implied by the nonpositivity of the quantitative
difference between information causality and no-supersignalling.Comment: 23 pages, 8 figures, Added Section 2 and Appendix B, result
unchanged, Added reference
Entanglement of random vectors
We analytically calculate the average value of i-th largest Schmidt
coefficient for random pure quantum states. Schmidt coefficients, i.e.,
eigenvalues of the reduced density matrix, are expressed in the limit of large
Hilbert space size and for arbitrary bipartite splitting as an implicit
function of index i.Comment: 8 page
Large Deviation Bounds for k-designs
We present a technique for derandomising large deviation bounds of functions
on the unitary group. We replace the Haar distribution with a pseudo-random
distribution, a k-design. k-designs have the first k moments equal to those of
the Haar distribution. The advantage of this is that (approximate) k-designs
can be implemented efficiently, whereas Haar random unitaries cannot. We find
large deviation bounds for unitaries chosen from a k-design and then illustrate
this general technique with three applications. We first show that the von
Neumann entropy of a pseudo-random state is almost maximal. Then we show that,
if the dynamics of the universe produces a k-design, then suitably sized
subsystems will be in the canonical state, as predicted by statistical
mechanics. Finally we show that pseudo-random states are useless for
measurement based quantum computation.Comment: 20 page
The thermodynamic meaning of negative entropy
Landauer's erasure principle exposes an intrinsic relation between
thermodynamics and information theory: the erasure of information stored in a
system, S, requires an amount of work proportional to the entropy of that
system. This entropy, H(S|O), depends on the information that a given observer,
O, has about S, and the work necessary to erase a system may therefore vary for
different observers. Here, we consider a general setting where the information
held by the observer may be quantum-mechanical, and show that an amount of work
proportional to H(S|O) is still sufficient to erase S. Since the entropy H(S|O)
can now become negative, erasing a system can result in a net gain of work (and
a corresponding cooling of the environment).Comment: Added clarification on non-cyclic erasure and reversible computation
(Appendix E). For a new version of all technical proofs see the Supplementary
Information of the journal version (free access
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