281 research outputs found
Asymptotic Redundancies for Universal Quantum Coding
Clarke and Barron have recently shown that the Jeffreys' invariant prior of
Bayesian theory yields the common asymptotic (minimax and maximin) redundancy
of universal data compression in a parametric setting. We seek a possible
analogue of this result for the two-level {\it quantum} systems. We restrict
our considerations to prior probability distributions belonging to a certain
one-parameter family, , . Within this setting, we are
able to compute exact redundancy formulas, for which we find the asymptotic
limits. We compare our quantum asymptotic redundancy formulas to those derived
by naively applying the classical counterparts of Clarke and Barron, and find
certain common features. Our results are based on formulas we obtain for the
eigenvalues and eigenvectors of (Bayesian density) matrices,
. These matrices are the weighted averages (with respect to
) of all possible tensor products of identical density
matrices, representing the two-level quantum systems. We propose a form of {\it
universal} coding for the situation in which the density matrix describing an
ensemble of quantum signal states is unknown. A sequence of signals would
be projected onto the dominant eigenspaces of \ze_n(u)
Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
We report the results of certain integrations of quantum-theoretic interest,
relying, in this regard, upon recently developed parameterizations of Boya et
al of the n x n density matrices, in terms of squared components of the unit
(n-1)-sphere and the n x n unitary matrices. Firstly, we express the normalized
volume elements of the Bures (minimal monotone) metric for n = 2 and 3,
obtaining thereby "Bures prior probability distributions" over the two- and
three-state systems. Then, as an essential first step in extending these
results to n > 3, we determine that the "Hall normalization constant" (C_{n})
for the marginal Bures prior probability distribution over the
(n-1)-dimensional simplex of the n eigenvalues of the n x n density matrices
is, for n = 4, equal to 71680/pi^2. Since we also find that C_{3} = 35/pi, it
follows that C_{4} is simply equal to 2^{11} C_{3}/pi. (C_{2} itself is known
to equal 2/pi.) The constant C_{5} is also found. It too is associated with a
remarkably simple decompositon, involving the product of the eight consecutive
prime numbers from 2 to 23.
We also preliminarily investigate several cases, n > 5, with the use of
quasi-Monte Carlo integration. We hope that the various analyses reported will
prove useful in deriving a general formula (which evidence suggests will
involve the Bernoulli numbers) for the Hall normalization constant for
arbitrary n. This would have diverse applications, including quantum inference
and universal quantum coding.Comment: 14 pages, LaTeX, 6 postscript figures. Revised version to appear in
J. Phys. A. We make a few slight changes from the previous version, but also
add a subsection (III G) in which several variations of the basic problem are
newly studied. Rather strong evidence is adduced that the Hall constants are
related to partial sums of denominators of the even-indexed Bernoulli
numbers, although a general formula is still lackin
Quantum and Fisher Information from the Husimi and Related Distributions
The two principal/immediate influences -- which we seek to interrelate here
-- upon the undertaking of this study are papers of Zyczkowski and
Slomczy\'nski (J. Phys. A 34, 6689 [2001]) and of Petz and Sudar (J. Math.
Phys. 37, 2262 [1996]). In the former work, a metric (the Monge one,
specifically) over generalized Husimi distributions was employed to define a
distance between two arbitrary density matrices. In the Petz-Sudar work
(completing a program of Chentsov), the quantum analogue of the (classically
unique) Fisher information (montone) metric of a probability simplex was
extended to define an uncountable infinitude of Riemannian (also monotone)
metrics on the set of positive definite density matrices. We pose here the
questions of what is the specific/unique Fisher information metric for the
(classically-defined) Husimi distributions and how does it relate to the
infinitude of (quantum) metrics over the density matrices of Petz and Sudar? We
find a highly proximate (small relative entropy) relationship between the
probability distribution (the quantum Jeffreys' prior) that yields quantum
universal data compression, and that which (following Clarke and Barron) gives
its classical counterpart. We also investigate the Fisher information metrics
corresponding to the escort Husimi, positive-P and certain Gaussian probability
distributions, as well as, in some sense, the discrete Wigner
pseudoprobability. The comparative noninformativity of prior probability
distributions -- recently studied by Srednicki (Phys. Rev. A 71, 052107 [2005])
-- formed by normalizing the volume elements of the various information
metrics, is also discussed in our context.Comment: 27 pages, 10 figures, slight revisions, to appear in J. Math. Phy
Bures Metrics for Certain High-Dimensional Quantum Systems
Hubner's formula for the Bures (statistical distance) metric is applied to
both a one-parameter and a two-parameter series (n=2,...,7) of sets of 2^n x
2^n density matrices. In the doubly-parameterized series, the sets are
comprised of the n-fold tensor products --- corresponding to n independent,
identical quantum systems --- of the 2 x 2 density matrices with real entries.
The Gaussian curvatures of the corresponding Bures metrics are found to be
constants (4/n). In the second series of 2^n x 2^n density matrices studied,
the singly-parameterized sets are formed --- following a study of Krattenthaler
and Slater --- by averaging with respect to a certain Gibbs distribution, the
n-fold tensor products of the 2 x 2 density matrices with complex entries. For
n = 100, we are also able to compute the Bures distance between two arbitrary
(not necessarily neighboring) density matrices in this particular series,
making use of the eigenvalue formulas of Krattenthaler and Slater, together
with the knowledge that the 2^n x 2^n density matrices in this series commute.Comment: 8 pages, LaTeX, 4 postscript figures, minor changes, to appear in
Physics Letters
A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression
The causal structure of a stochastic process can be more efficiently
transmitted via a quantum channel than a classical one, an advantage that
increases with codeword length. While previously difficult to compute, we
express the quantum advantage in closed form using spectral decomposition,
leading to direct computation of the quantum communication cost at all encoding
lengths, including infinite. This makes clear how finite-codeword compression
is controlled by the classical process' cryptic order and allows us to analyze
structure within the length-asymptotic regime of infinite-cryptic order (and
infinite Markov order) processes.Comment: 21 pages, 13 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/eqc.ht
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