12 research outputs found
Effective complexity of stationary process realizations
The concept of effective complexity of an object as the minimal description
length of its regularities has been initiated by Gell-Mann and Lloyd. The
regularities are modeled by means of ensembles, that is probability
distributions on finite binary strings. In our previous paper we propose a
definition of effective complexity in precise terms of algorithmic information
theory. Here we investigate the effective complexity of binary strings
generated by stationary, in general not computable, processes. We show that
under not too strong conditions long typical process realizations are
effectively simple. Our results become most transparent in the context of
coarse effective complexity which is a modification of the original notion of
effective complexity that uses less parameters in its definition. A similar
modification of the related concept of sophistication has been suggested by
Antunes and Fortnow.Comment: 14 pages, no figure
Process Dimension of Classical and Non-Commutative Processes
We treat observable operator models (OOM) and their non-commutative
generalisation, which we call NC-OOMs. A natural characteristic of a stochastic
process in the context of classical OOM theory is the process dimension. We
investigate its properties within the more general formulation, which allows to
consider process dimension as a measure of complexity of non-commutative
processes: We prove lower semi-continuity, and derive an ergodic decomposition
formula. Further, we obtain results on the close relationship between the
canonical OOM and the concept of causal states which underlies the definition
of statistical complexity. In particular, the topological statistical
complexity, i.e. the logarithm of the number of causal states, turns out to be
an upper bound to the logarithm of process dimension.Comment: 8 page
A quantum version of Sanov's theorem
We present a quantum extension of a version of Sanov's theorem focussing on a
hypothesis testing aspect of the theorem: There exists a sequence of typical
subspaces for a given set of stationary quantum product states
asymptotically separating them from another fixed stationary product state.
Analogously to the classical case, the exponential separating rate is equal to
the infimum of the quantum relative entropy with respect to the quantum
reference state over the set . However, while in the classical case the
separating subsets can be chosen universal, in the sense that they depend only
on the chosen set of i.i.d. processes, in the quantum case the choice of the
separating subspaces depends additionally on the reference state.Comment: 15 page
Typical support and Sanov large deviations of correlated states
Discrete stationary classical processes as well as quantum lattice states are
asymptotically confined to their respective typical support, the exponential
growth rate of which is given by the (maximal ergodic) entropy. In the iid case
the distinguishability of typical supports can be asymptotically specified by
means of the relative entropy, according to Sanov's theorem. We give an
extension to the correlated case, referring to the newly introduced class of
HP-states.Comment: 29 pages, no figures, references adde
Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno's Theorem
In classical information theory, entropy rate and Kolmogorov complexity per
symbol are related by a theorem of Brudno. In this paper, we prove a quantum
version of this theorem, connecting the von Neumann entropy rate and two
notions of quantum Kolmogorov complexity, both based on the shortest qubit
descriptions of qubit strings that, run by a universal quantum Turing machine,
reproduce them as outputs.Comment: 26 pages, no figures. Reference to publication added: published in
the Communications in Mathematical Physics
(http://www.springerlink.com/content/1432-0916/
Chained Typical Subspaces -- a Quantum Version of Breiman's Theorem
We give an equivalent finitary reformulation of the classical Shannon-McMillan-Breiman theorem which has an immediate translation to the case of ergodic quantum lattice systems. This version of a quantum Breiman theorem can be derived from the proof of the quantum ShannonMcMillan theorem presented in [2]