484 research outputs found
Structure of sufficient quantum coarse-grainings
Let H and K be Hilbert spaces and T be a coarse-graining from B(H) to B(K).
Assume that density matrices D_1 and D_2 acting on H are given. In the paper
the consequences of the existence of a coarse-graining S from B(K) to B(H)
satisfying ST(D_1)=D_1 and ST(D_2)=D_2 are given. (This condition means the
sufficiency of T for D_1 and D_2.) Sufficiency implies a particular
decomposition of the density matrices. This decomposition allows to deduce the
exact condition for equality in the strong subadditivity of the von Neumann
entropy.Comment: 13 pages, LATE
Maps on density operators preserving quantum f-divergences
For an arbitrary strictly convex function f defined on the
non-negative real line we determine the structure of all transformations
on the set of density operators which preserve the quantum f-divergence
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
Monotonicity of quantum relative entropy revisited
Monotonicity under coarse-graining is a crucial property of the quantum
relative entropy. The aim of this paper is to investigate the condition of
equality in the monotonicity theorem and in its consequences such as the strong
sub-additivity of the von Neumann entropy, the Golden-Thompson trace inequality
and the monotonicity of the Holevo quantity.The relation to quantum Markovian
states is briefly indicated.Comment: 13 pages, LATEX fil
An entropic uncertainty principle for positive operator valued measures
Extending a recent result by Frank and Lieb, we show an entropic uncertainty
principle for mixed states in a Hilbert space relatively to pairs of positive
operator valued measures that are independent in some sense. This yields
spatial-spectral uncertainty principles and log-Sobolev inequalities for
invariant operators on homogeneous spaces, which are sharp in the compact case.Comment: 14 pages. v2: a technical assumption removed in main resul
Extension of Information Geometry to Non-statistical Systems: Some Examples
Our goal is to extend information geometry to situations where statistical
modeling is not obvious. The setting is that of modeling experimental data.
Quite often the data are not of a statistical nature. Sometimes also the model
is not a statistical manifold. An example of the former is the description of
the Bose gas in the grand canonical ensemble. An example of the latter is the
modeling of quantum systems with density matrices. Conditional expectations in
the quantum context are reviewed. The border problem is discussed: through
conditioning the model point shifts to the border of the differentiable
manifold.Comment: 8 pages, to be published in the proceedings of GSI2015, Lecture Notes
in Computer Science, Springe
Local asymptotic normality for qubit states
We consider n identically prepared qubits and study the asymptotic properties
of the joint state \rho^{\otimes n}. We show that for all individual states
\rho situated in a local neighborhood of size 1/\sqrt{n} of a fixed state
\rho^0, the joint state converges to a displaced thermal equilibrium state of a
quantum harmonic oscillator. The precise meaning of the convergence is that
there exist physical transformations T_{n} (trace preserving quantum channels)
which map the qubits states asymptotically close to their corresponding
oscillator state, uniformly over all states in the local neighborhood.
A few consequences of the main result are derived. We show that the optimal
joint measurement in the Bayesian set-up is also optimal within the pointwise
approach. Moreover, this measurement converges to the heterodyne measurement
which is the optimal joint measurement of position and momentum for the quantum
oscillator. A problem of local state discrimination is solved using local
asymptotic normality.Comment: 16 pages, 3 figures, published versio
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
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