785 research outputs found
Brussels-Austin Nonequilibrium Statistical Mechanics in the Early Years: Similarity Transformations between Deterministic and Probabilistic Descriptions
The fundamental problem on which Ilya Prigogine and the Brussels-Austin Group
have focused can be stated briefly as follows. Our observations indicate that
there is an arrow of time in our experience of the world (e.g., decay of
unstable radioactive atoms like Uranium, or the mixing of cream in coffee).
Most of the fundamental equations of physics are time reversible, however,
presenting an apparent conflict between our theoretical descriptions and
experimental observations. Many have thought that the observed arrow of time
was either an artifact of our observations or due to very special initial
conditions. An alternative approach, followed by the Brussels-Austin Group, is
to consider the observed direction of time to be a basics physical phenomenon
and to develop a mathematical formalism that can describe this direction as
being due to the dynamics of physical systems. In part I of this essay, I
review and assess an attempt to carry out an approach that received much of
their attention from the early 1970s to the mid 1980s. In part II, I will
discuss their more recent approach using rigged Hilbert spaces.Comment: 22 pages, Part I of two parts; updated institutional affiliatio
Finite Computational Structures and Implementations
What is computable with limited resources? How can we verify the correctness
of computations? How to measure computational power with precision? Despite the
immense scientific and engineering progress in computing, we still have only
partial answers to these questions. In order to make these problems more
precise, we describe an abstract algebraic definition of classical computation,
generalizing traditional models to semigroups. The mathematical abstraction
also allows the investigation of different computing paradigms (e.g. cellular
automata, reversible computing) in the same framework. Here we summarize the
main questions and recent results of the research of finite computation.Comment: 12 pages, 3 figures, will be presented at CANDAR'16 and final version
published by IEEE Computer Societ
The Algebraic View of Computation
We argue that computation is an abstract algebraic concept, and a computer is
a result of a morphism (a structure preserving map) from a finite universal
semigroup.Comment: 13 pages, final version will be published elsewher
Impact of positivity and complete positivity on accessibility of Markovian dynamics
We consider a two-dimensional quantum control system evolving under an
entropy-increasing irreversible dynamics in the semigroup form. Considering a
phenomenological approach to the dynamics, we show that the accessibility
property of the system depends on whether its evolution is assumed to be
positive or completely positive. In particular, we characterize the family of
maps having different accessibility and show the impact of that property on
observable quantities by means of a simple physical model.Comment: 11 pages, to appear in J. Phys.
General fixed points of quasi-local frustration-free quantum semigroups: from invariance to stabilization
We investigate under which conditions a mixed state on a finite-dimensional
multipartite quantum system may be the unique, globally stable fixed point of
frustration-free semigroup dynamics subject to specified quasi-locality
constraints. Our central result is a linear-algebraic necessary and sufficient
condition for a generic (full-rank) target state to be frustration-free
quasi-locally stabilizable, along with an explicit procedure for constructing
Markovian dynamics that achieve stabilization. If the target state is not
full-rank, we establish sufficiency under an additional condition, which is
naturally motivated by consistency with pure-state stabilization results yet
provably not necessary in general. Several applications are discussed, of
relevance to both dissipative quantum engineering and information processing,
and non-equilibrium quantum statistical mechanics. In particular, we show that
a large class of graph product states (including arbitrary thermal graph
states) as well as Gibbs states of commuting Hamiltonians are frustration-free
stabilizable relative to natural quasi-locality constraints. Likewise, we
provide explicit examples of non-commuting Gibbs states and non-trivially
entangled mixed states that are stabilizable despite the lack of an underlying
commuting structure, albeit scalability to arbitrary system size remains in
this case an open question.Comment: 44 pages, main results are improved, several proofs are more
streamlined, application section is refine
Quantum and classical dynamical semigroups of superchannels and semicausal channels
Quantum devices are subject to natural decay. We propose to study these decay processes as the Markovian evolution of quantum channels, which leads us to dynamical semigroups of superchannels. A superchannel is a linear map that maps quantum channels to quantum channels while satisfying suitable consistency relations. If the input and output quantum channels act on the same space, then we can consider dynamical semigroups of superchannels. No useful constructive characterization of the generators of such semigroups is known. We characterize these generators in two ways: First, we give an efficiently checkable criterion for whether a given map generates a dynamical semigroup of superchannels. Second, we identify a normal form for the generators of semigroups of quantum superchannels, analogous to the Gorini-Kossakowski-Lindblad-Sudarshan form in the case of quantum channels. To derive the normal form, we exploit the relation between superchannels and semicausal completely positive maps, reducing the problem to finding a normal form for the generators of semigroups of semicausal completely positive maps. We derive a normal for these generators using a novel technique, which applies also to infinite-dimensional systems. Our work paves the way for a thorough investigation of semigroups of superchannels: Numerical studies become feasible because admissible generators can now be explicitly generated and checked. Analytic properties of the corresponding evolution equations are now accessible via our normal form
Engineering Stable Discrete-Time Quantum Dynamics via a Canonical QR Decomposition
We analyze the asymptotic behavior of discrete-time, Markovian quantum
systems with respect to a subspace of interest. Global asymptotic stability of
subspaces is relevant to quantum information processing, in particular for
initializing the system in pure states or subspace codes. We provide a
linear-algebraic characterization of the dynamical properties leading to
invariance and attractivity of a given quantum subspace. We then construct a
design algorithm for discrete-time feedback control that allows to stabilize a
target subspace, proving that if the control problem is feasible, then the
algorithm returns an effective control choice. In order to prove this result, a
canonical QR matrix decomposition is derived, and also used to establish the
control scheme potential for the simulation of open-system dynamics.Comment: 12 pages, 1 figur
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