23 research outputs found
Quantum Data Compression and Relative Entropy Revisited
B. Schumacher and M. Westmoreland have established a quantum analog of a
well-known classical information theory result on a role of relative entropy as
a measure of non-optimality in (classical) data compression. In this paper, we
provide an alternative, simple and constructive proof of this result by
constructing quantum compression codes (schemes) from classical data
compression codes. Moreover, as the quantum data compression/coding task can be
effectively reduced to a (quasi-)classical one, we show that relevant results
from classical information theory and data compression become applicable and
therefore can be extended to the quantum domain.Comment: 7 pages, no figures, minor revisio
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/
Artificial Sequences and Complexity Measures
In this paper we exploit concepts of information theory to address the
fundamental problem of identifying and defining the most suitable tools to
extract, in a automatic and agnostic way, information from a generic string of
characters. We introduce in particular a class of methods which use in a
crucial way data compression techniques in order to define a measure of
remoteness and distance between pairs of sequences of characters (e.g. texts)
based on their relative information content. We also discuss in detail how
specific features of data compression techniques could be used to introduce the
notion of dictionary of a given sequence and of Artificial Text and we show how
these new tools can be used for information extraction purposes. We point out
the versatility and generality of our method that applies to any kind of
corpora of character strings independently of the type of coding behind them.
We consider as a case study linguistic motivated problems and we present
results for automatic language recognition, authorship attribution and self
consistent-classification.Comment: Revised version, with major changes, of previous "Data Compression
approach to Information Extraction and Classification" by A. Baronchelli and
V. Loreto. 15 pages; 5 figure
Reexamination of Quantum Data Compression and Relative Entropy
Schumacher and Westmoreland [Phys. Rev. A 64, 42304 (2001)] have established a quantum analog of a well-known classical information theory result on a role of relative entropy as a measure of nonoptimality in classical data compression. In this paper, we provide an alternative simple and constructive proof of this result by constructing quantum compression codes schemes from classical data compression codes. Moreover, as the quantum data compression or coding task can be effectively reduced to a quasi classical one, we show that relevant results from classical information theory and data compression become applicable and therefore can be extended to the quantum domain
Classical Complexity of Unitary Transformations
We discuss a classical complexity of finite-dimensional unitary
transformation, which can been seen as a computable approximation of
descriptional complexity of a unitary transformation acting on a set of qubits.Comment: 5 page