64 research outputs found
Ergodic Classical-Quantum Channels: Structure and Coding Theorems
We consider ergodic causal classical-quantum channels (cq-channels) which
additionally have a decaying input memory. In the first part we develop some
structural properties of ergodic cq-channels and provide equivalent conditions
for ergodicity. In the second part we prove the coding theorem with weak
converse for causal ergodic cq-channels with decaying input memory. Our proof
is based on the possibility to introduce joint input-output state for the
cq-channels and an application of the Shannon-McMillan theorem for ergodic
quantum states. In the last part of the paper it is shown how this result
implies coding theorem for the classical capacity of a class of causal ergodic
quantum channels.Comment: 19 pages, no figures. Final versio
Coding Theorem for a Class of Quantum Channels with Long-Term Memory
In this paper we consider the transmission of classical information through a
class of quantum channels with long-term memory, which are given by convex
combinations of product channels. Hence, the memory of such channels is given
by a Markov chain which is aperiodic but not irreducible. We prove the coding
theorem and weak converse for this class of channels. The main techniques that
we employ, are a quantum version of Feinstein's Fundamental Lemma and a
generalization of Helstrom's Theorem.Comment: Some typos correcte
Quantum Channels with Memory
We present a general model for quantum channels with memory, and show that it
is sufficiently general to encompass all causal automata: any quantum process
in which outputs up to some time t do not depend on inputs at times t' > t can
be decomposed into a concatenated memory channel. We then examine and present
different physical setups in which channels with memory may be operated for the
transfer of (private) classical and quantum information. These include setups
in which either the receiver or a malicious third party have control of the
initializing memory. We introduce classical and quantum channel capacities for
these settings, and give several examples to show that they may or may not
coincide. Entropic upper bounds on the various channel capacities are given.
For forgetful quantum channels, in which the effect of the initializing memory
dies out as time increases, coding theorems are presented to show that these
bounds may be saturated. Forgetful quantum channels are shown to be open and
dense in the set of quantum memory channels.Comment: 21 pages with 5 EPS figures. V2: Presentation clarified, references
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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
Entanglement Assisted Classical Capacity for a Class of Quantum Channels with Long-Term Memory
In this paper we evaluate the entanglement assisted classical capacity of a class of quantum channels with long-term memory, which are convex combinations of memoryless channels. The memory of such channels can be considered to be given by a Markov chain which is aperiodic but not irreducible. This class of channels was introduced in [7] and its product state capacity was evaluated
Informationsübertragung durch Quantenkanäle
This PhD thesis represents work done between Aug. 2003 and Dec. 2006 in Reinhard F. Werner's quantum information theory group at Technische Universität Braunschweig, and Artur Ekert's Centre for Quantum Computation at the University of Cambridge. Quantum information science combines ideas from physics, computer science and information theory to investigate how quintessentially quantum mechanical effects such as superposition and entanglement can be employed for the handling and transfer of information. My thesis falls into the field of abstract quantum information theory, which is concerned with the fundamental resources for quantum information processing and their interconversion and tradeoffs. Every such processing of quantum information can be represented as a quantum channel: a completely positive and trace-preserving map between observable algebras associated to physical systems. This work investigates both fundamental properties of quantum channels (mostly in Chs. 3 and 4) and their asymptotic capacities for classical as well as quantum information transfer (in Chs. 5 through 8).Diese Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) entstand zwischen August 2003 und Dezember 2006 in Prof. Reinhard F. Werners Arbeitsgruppe Quanteninformationstheorie an der Technischen Universität Braunschweig und Prof. Artur Ekerts Centre for Quantum Computation an der Universität Cambridge. Die Quanteninformationswissenschaft untersucht mit den Ideen und Methoden der Physik, der Informatik und der Informationstheorie, wie sich charakteristisch quantenphysikalische Effekte, beispielsweise Superposition und Verschränkung, zur Verarbeitung und Übertragung von Information nutzbar machen lassen. Die vorliegende Dissertation fällt in das Gebiet der abstrakten Quanteninformationstheorie, die die grundlegenden Ressourcen für die Verarbeitung von Quanteninformation sowie deren Wechselbeziehungen und Abhängigkeiten untersucht. Eine jede solche Verarbeitung von Quanteninformation läßt sich mathematisch beschreiben als sogenannter Quantenkanal, eine vollständig positive und spurerhaltende Abbildung zwischen den physikalischen Systemen zugeordneten Observablen-Algebren. In dieser Arbeit werden sowohl grundlegende Eigenschaften solcher Quantenkanäle (vor allem in den Kap. 3 und Kap. 4) als auch ihre asymptotischen Kapazitäten für die Übertragung von klassischer Information und Quanteninformation (in Kap. 5 bis 8) untersucht
On the Szeg\"o-Asymptotics for Doubly-Dispersive Gaussian Channels
We consider the time-continuous doubly-dispersive channel with additive
Gaussian noise and establish a capacity formula for the case where the channel
correlation operator is represented by a symbol which is periodic in time and
fulfills some further integrability and smoothness conditions. The key to this
result is a new Szeg\"o formula for certain pseudo-differential operators. The
formula justifies the water-filling principle along time and frequency in terms
of the time--continuous time-varying transfer function (the symbol).Comment: 5 pages, to be presented at ISIT 2011, minor typos corrected,
references update
Quantum reservoir computing in finite dimensions
Most existing results in the analysis of quantum reservoir computing (QRC)
systems with classical inputs have been obtained using the density matrix
formalism. This paper shows that alternative representations can provide better
insights when dealing with design and assessment questions. More explicitly,
system isomorphisms are established that unify the density matrix approach to
QRC with the representation in the space of observables using Bloch vectors
associated with Gell-Mann bases. It is shown that these vector representations
yield state-affine systems (SAS) previously introduced in the classical
reservoir computing literature and for which numerous theoretical results have
been established. This connection is used to show that various statements in
relation to the fading memory (FMP) and the echo state (ESP) properties are
independent of the representation, and also to shed some light on fundamental
questions in QRC theory in finite dimensions. In particular, a necessary and
sufficient condition for the ESP and FMP to hold is formulated using standard
hypotheses, and contractive quantum channels that have exclusively trivial
semi-infinite solutions are characterized in terms of the existence of
input-independent fixed points.Comment: 19 pages, 4 figure
Classical noise in quantum systems.
Ph. D. University of KwaZulu-Natal, Durban, 2013.Quantum mechanics contains a fresh and mysterious view of reality. Besides the philosophical intrigue, it has also produced and continues to inspire tantalizing new technological innovations. In any technological system, the designers must contend with the
problem of noise. This thesis studies classical noise in two different quantum settings.
The first is the classical capacity of a quantum channel with memory. Adding forgetful-memory, attempts to push the boundaries of our understanding of how best to transmit
information in the presence of correlated noise. We study the noise within two different
frameworks; Algebraic Measure theory and Monte Carlo simulations. Both tools are
used to calculate the capacity of the channel as correlations in the noise are increased.
The second classical-quantum system investigated is atomic clocks. Using power spectral
density methods we study aliasing noise induced by periodic-correction which includes
the Dick Effect. We propose a novel multi-window scheme that extends the standard
method of noise correction and exhibits better anti-aliasing properties.
A uniting thread that emerges is that correlations can be put to good use. In the classical
capacity setting, correlations occur between uses of the quantum channel. We show that
stronger correlations increase the classical capacity. The benefits of correlation are even
seen at a meta-level within the framework of Monte Carlo simulations. Correlations are
designed into the algorithm which have nothing to do with real-world correlations, but
are abstract correlations created by a Markov chain employed in the algorithm to help
efficiently sample from a distribution of exponential size. Finally, in the atomic clock
setting, correlations in the measured noise are used to help predict and cancel noise on
a short time-scale while trying to limit aliasing.
Channel capacity and precise time-keeping are distinct topics and require very different
approaches to study. However, common to both topics is their application to com-
munication and other tasks, the need to overcome noise and the benefits of exploiting
correlations in the noise
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