2,019 research outputs found
Continuity bounds on the quantum relative entropy
The quantum relative entropy is frequently used as a distance, or
distinguishability measure between two quantum states. In this paper we study
the relation between this measure and a number of other measures used for that
purpose, including the trace norm distance. More precisely, we derive lower and
upper bounds on the relative entropy in terms of various distance measures for
the difference of the states based on unitarily invariant norms. The upper
bounds can be considered as statements of continuity of the relative entropy
distance in the sense of Fannes. We employ methods from optimisation theory to
obtain bounds that are as sharp as possible.Comment: 13 pages (ReVTeX), 3 figures, replaced with published versio
Witnessing entanglement by proxy
Entanglement is a ubiquitous feature of low temperature systems and believed
to be highly relevant for the dynamics of condensed matter properties and
quantum computation even at higher temperatures. The experimental certification
of this paradigmatic quantum effect in macroscopic high temperature systems is
constrained by the limited access to the quantum state of the system. In this
paper we show how macroscopic observables beyond the energy of the system can
be exploited as proxy witnesses for entanglement detection. Using linear and
semi-definite relaxations we show that all previous approaches to this problem
can be outperformed by our proxies, i.e. entanglement can be certified at
higher temperatures without access to any local observable. For an efficient
computation of proxy witnesses one can resort to a generalized grand canonical
ensemble, enabling entanglement certification even in complex systems with
macroscopic particle numbers.Comment: 22 pages, 8 figure
Effective interactions and large deviations in stochastic processes
We discuss the relationships between large deviations in stochastic systems,
and "effective interactions" that induce particular rare events. We focus on
the nature of these effective interactions in physical systems with many
interacting degrees of freedom, which we illustrate by reviewing several recent
studies. We describe the connections between effective interactions, large
deviations at "level 2.5", and the theory of optimal control. Finally, we
discuss possible physical applications of variational results associated with
those theories.Comment: 12 page
First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
We investigate the dynamics of kinetically constrained models of glass
formers by analysing the statistics of trajectories of the dynamics, or
histories, using large deviation function methods. We show that, in general,
these models exhibit a first-order dynamical transition between active and
inactive dynamical phases. We argue that the dynamical heterogeneities
displayed by these systems are a manifestation of dynamical first-order phase
coexistence. In particular, we calculate dynamical large deviation functions,
both analytically and numerically, for the Fredrickson-Andersen model, the East
model, and constrained lattice gas models. We also show how large deviation
functions can be obtained from a Landau-like theory for dynamical fluctuations.
We discuss possibilities for similar dynamical phase-coexistence behaviour in
other systems with heterogeneous dynamics.Comment: 29 pages, 7 figs, final versio
Utility indifference pricing with market incompleteness
Utility indifference pricing and hedging theory is presented, showing
how it leads to linear or to non-linear pricing rules for contingent
claims. Convex duality is first used to derive probabilistic
representations for exponential utility-based prices, in a general
setting with locally bounded semi-martingale price processes. The
indifference price for a finite number of claims gives a non-linear
pricing rule, which reduces to a linear pricing rule as the number of
claims tends to zero, resulting in the so-called marginal
utility-based price of the claim. Applications to basis risk models
with lognormal price processes, under full and partial information
scenarios are then worked out in detail. In the full information case,
a claim on a non-traded asset is priced and hedged using a correlated
traded asset. The resulting hedge requires knowledge of the drift
parameters of the asset price processes, which are very difficult to
estimate with any precision. This leads naturally to a further
application, a partial information problem, with the drift parameters
assumed to be random variables whose values are revealed to the hedger
in a Bayesian fashion via a filtering algorithm. The indifference
price is given by the solution to a non-linear PDE, reducing to a
linear PDE for the marginal price when the number of claims becomes
infinitesimally small
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