85 research outputs found
On Quantum Capacity of Compound Channels
In this paper we address the issue of universal or robust communication over
quantum channels. Specifically, we consider memoryless communication scenario
with channel uncertainty which is an analog of compound channel in classical
information theory. We determine the quantum capacity of finite compound
channels and arbitrary compound channels with informed decoder. Our approach in
the finite case is based on the observation that perfect channel knowledge at
the decoder does not increase the capacity of finite quantum compound channels.
As a consequence we obtain coding theorem for finite quantum averaged channels,
the simplest class of channels with long-term memory. The extension of these
results to quantum compound channels with uninformed encoder and decoder, and
infinitely many constituents remains an open problem.Comment: 16 pages, no figure
Asymptotic behaviour of random tridiagonal Markov chains in biological applications
Discrete-time discrete-state random Markov chains with a tridiagonal
generator are shown to have a random attractor consisting of singleton subsets,
essentially a random path, in the simplex of probability vectors. The proof
uses the Hilbert projection metric and the fact that the linear cocycle
generated by the Markov chain is a uniformly contractive mapping of the
positive cone into itself. The proof does not involve probabilistic properties
of the sample path and is thus equally valid in the nonautonomous deterministic
context of Markov chains with, say, periodically varying transitions
probabilities, in which case the attractor is a periodic path.Comment: 13 pages, 22 bibliography references, submitted to DCDS-B, added
references and minor correction
Entanglement transmission and generation under channel uncertainty: Universal quantum channel coding
We determine the optimal rates of universal quantum codes for entanglement
transmission and generation under channel uncertainty. In the simplest scenario
the sender and receiver are provided merely with the information that the
channel they use belongs to a given set of channels, so that they are forced to
use quantum codes that are reliable for the whole set of channels. This is
precisely the quantum analog of the compound channel coding problem. We
determine the entanglement transmission and entanglement-generating capacities
of compound quantum channels and show that they are equal. Moreover, we
investigate two variants of that basic scenario, namely the cases of informed
decoder or informed encoder, and derive corresponding capacity results.Comment: 45 pages, no figures. Section 6.2 rewritten due to an error in
equation (72) of the old version. Added table of contents, added section
'Conclusions and further remarks'. Accepted for publication in
'Communications in Mathematical Physics
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
Quantum channels and memory effects
Any physical process can be represented as a quantum channel mapping an
initial state to a final state. Hence it can be characterized from the point of
view of communication theory, i.e., in terms of its ability to transfer
information. Quantum information provides a theoretical framework and the
proper mathematical tools to accomplish this. In this context the notion of
codes and communication capacities have been introduced by generalizing them
from the classical Shannon theory of information transmission and error
correction. The underlying assumption of this approach is to consider the
channel not as acting on a single system, but on sequences of systems, which,
when properly initialized allow one to overcome the noisy effects induced by
the physical process under consideration. While most of the work produced so
far has been focused on the case in which a given channel transformation acts
identically and independently on the various elements of the sequence
(memoryless configuration in jargon), correlated error models appear to be a
more realistic way to approach the problem. A slightly different, yet
conceptually related, notion of correlated errors applies to a single quantum
system which evolves continuously in time under the influence of an external
disturbance which acts on it in a non-Markovian fashion. This leads to the
study of memory effects in quantum channels: a fertile ground where interesting
novel phenomena emerge at the intersection of quantum information theory and
other branches of physics. A survey is taken of the field of quantum channels
theory while also embracing these specific and complex settings.Comment: Review article, 61 pages, 26 figures; 400 references. Final version
of the manuscript, typos correcte
Recommended from our members
Leo Strauss and International Relations: The politics of modernity's abyss
This article argues that an engagement with the political philosophy of Leo Strauss is of considerable value in International Relations (IR), in relation to the study of both recent US foreign policy and contemporary IR theory. The question of Straussian activities within and close to the foreign policy-making establishment in the United States during the period leading up to the 2003 invasion of Iraq has been the focus of significant scholarly and popular attention in recent years. This article makes the case that several individuals influenced by Strauss exercised considerable influence in the fields of intelligence production, the media and think tanks, and traces the ways in which elements of Strauss’ thought are discernible in their interventions in these spheres. It further argues that Strauss’ political philosophy is of broader significance for IR insofar as it can be read as a securitising response to the dangers he associated with the foundationlessness of the modern condition. The article demonstrates that the politics of this response are of crucial importance for contemporary debates between traditional and critical IR theorists
- …