206 research outputs found
Equilibrium states and their entropy densities in gauge-invariant C*-systems
A gauge-invariant C*-system is obtained as the fixed point subalgebra of the
infinite tensor product of full matrix algebras under the tensor product
unitary action of a compact group. In the paper, thermodynamics is studied on
such systems and the chemical potential theory developed by Araki, Haag,
Kastler and Takesaki is used. As a generalization of quantum spin system, the
equivalence of the KMS condition, the Gibbs condition and the variational
principle is shown for translation-invariant states. The entropy density of
extremal equilibrium states is also investigated in relation to macroscopic
uniformity.Comment: 20 pages, revised in March 200
Quantum hypothesis testing with group symmetry
The asymptotic discrimination problem of two quantum states is studied in the
setting where measurements are required to be invariant under some symmetry
group of the system. We consider various asymptotic error exponents in
connection with the problems of the Chernoff bound, the Hoeffding bound and
Stein's lemma, and derive bounds on these quantities in terms of their
corresponding statistical distance measures. A special emphasis is put on the
comparison of the performances of group-invariant and unrestricted
measurements.Comment: 33 page
Positive contraction mappings for classical and quantum Schrodinger systems
The classical Schrodinger bridge seeks the most likely probability law for a
diffusion process, in path space, that matches marginals at two end points in
time; the likelihood is quantified by the relative entropy between the sought
law and a prior, and the law dictates a controlled path that abides by the
specified marginals. Schrodinger proved that the optimal steering of the
density between the two end points is effected by a multiplicative functional
transformation of the prior; this transformation represents an automorphism on
the space of probability measures and has since been studied by Fortet,
Beurling and others. A similar question can be raised for processes evolving in
a discrete time and space as well as for processes defined over non-commutative
probability spaces. The present paper builds on earlier work by Pavon and
Ticozzi and begins with the problem of steering a Markov chain between given
marginals. Our approach is based on the Hilbert metric and leads to an
alternative proof which, however, is constructive. More specifically, we show
that the solution to the Schrodinger bridge is provided by the fixed point of a
contractive map. We approach in a similar manner the steering of a quantum
system across a quantum channel. We are able to establish existence of quantum
transitions that are multiplicative functional transformations of a given Kraus
map, but only for the case of uniform marginals. As in the Markov chain case,
and for uniform density matrices, the solution of the quantum bridge can be
constructed from the fixed point of a certain contractive map. For arbitrary
marginal densities, extensive numerical simulations indicate that iteration of
a similar map leads to fixed points from which we can construct a quantum
bridge. For this general case, however, a proof of convergence remains elusive.Comment: 27 page
A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality
We introduce a generalization of relative entropy derived from the
Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is
convex. Moreover, special cases yield the joint convexity of relative entropy,
and for the map (A,B) --> Tr K^* A^p K B^{1-p} Lieb's joint concavity for 0 < p
< 1 and Ando's joint convexity for 1 < p < 2. This approach allows us to obtain
conditions for equality in these cases, as well as conditions for equality in a
number of inequalities which follow from them. These include the monotonicity
under partial traces, and some Minkowski type matrix inequalities proved by
Lieb and Carlen for mixed (p,q) norms. In all cases the equality conditions are
independent of p; for extensions to three spaces they are identical to the
conditions for equality in the strong subadditivity of relative entropy.Comment: Final version to appear in Rev. Math. Phys. with many typos and minor
errors correcte
The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states
Hypothesis testing is a fundamental issue in statistical inference and has
been a crucial element in the development of information sciences. The Chernoff
bound gives the minimal Bayesian error probability when discriminating two
hypotheses given a large number of observations. Recently the combined work of
Audenaert et al. [Phys. Rev. Lett. 98, 160501] and Nussbaum and Szkola
[quant-ph/0607216] has proved the quantum analog of this bound, which applies
when the hypotheses correspond to two quantum states. Based on the quantum
Chernoff bound, we define a physically meaningful distinguishability measure
and its corresponding metric in the space of states; the latter is shown to
coincide with the Wigner-Yanase metric. Along the same lines, we define a
second, more easily implementable, distinguishability measure based on the
error probability of discrimination when the same local measurement is
performed on every copy. We study some general properties of these measures,
including the probability distribution of density matrices, defined via the
volume element induced by the metric, and illustrate their use in the
paradigmatic cases of qubits and Gaussian infinite-dimensional states.Comment: 16 page
Free energy density for mean field perturbation of states of a one-dimensional spin chain
Motivated by recent developments on large deviations in states of the spin
chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the
variational expression of free energy density in the presence of a mean field
type perturbation. We extend their results from the product state case to the
Gibbs state case in the setting of translation-invariant interactions of finite
range. In the special case of a locally faithful quantum Markov state, we
clarify the relation between two different kinds of free energy densities (or
pressure functions).Comment: 29 pages, Section 5 added, to appear in Rev. Math. Phy
Exponents of quantum fixed-length pure state source coding
We derive the optimal exponent of the error probability of the quantum
fixed-length pure state source coding in both cases of blind coding and visible
coding. The optimal exponent is universally attained by Jozsa et al. (PRL, 81,
1714 (1998))'s universal code. In the direct part, a group representation
theoretical type method is essential. In the converse part, Nielsen and Kempe
(PRL, 86, 5184 (2001))'s lemma is essential.Comment: LaTeX2e and revetx4 with
aps,twocolumn,superscriptaddress,showpacs,pra,amssymb,amsmath. The previous
version has a mistak
Fundamental properties of Tsallis relative entropy
Fundamental properties for the Tsallis relative entropy in both classical and
quantum systems are studied. As one of our main results, we give the parametric
extension of the trace inequality between the quantum relative entropy and the
minus of the trace of the relative operator entropy given by Hiai and Petz. The
monotonicity of the quantum Tsallis relative entropy for the trace preserving
completely positive linear map is also shown without the assumption that the
density operators are invertible.
The generalized Tsallis relative entropy is defined and its subadditivity is
shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov
inequality is also proven
Information geometry of Gaussian channels
We define a local Riemannian metric tensor in the manifold of Gaussian
channels and the distance that it induces. We adopt an information-geometric
approach and define a metric derived from the Bures-Fisher metric for quantum
states. The resulting metric inherits several desirable properties from the
Bures-Fisher metric and is operationally motivated from distinguishability
considerations: It serves as an upper bound to the attainable quantum Fisher
information for the channel parameters using Gaussian states, under generic
constraints on the physically available resources. Our approach naturally
includes the use of entangled Gaussian probe states. We prove that the metric
enjoys some desirable properties like stability and covariance. As a byproduct,
we also obtain some general results in Gaussian channel estimation that are the
continuous-variable analogs of previously known results in finite dimensions.
We prove that optimal probe states are always pure and bounded in the number of
ancillary modes, even in the presence of constraints on the reduced state input
in the channel. This has experimental and computational implications: It limits
the complexity of optimal experimental setups for channel estimation and
reduces the computational requirements for the evaluation of the metric:
Indeed, we construct a converging algorithm for its computation. We provide
explicit formulae for computing the multiparametric quantum Fisher information
for dissipative channels probed with arbitrary Gaussian states, and provide the
optimal observables for the estimation of the channel parameters (e.g. bath
couplings, squeezing, and temperature).Comment: 19 pages, 4 figure
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