46 research outputs found
Geodesic distances on density matrices
We find an upper bound for geodesic distances associated to monotone
Riemannian metrics on positive definite matrices and density matrices.Comment: 10 page
Quantum hypothesis testing and sufficient subalgebras
We introduce a new notion of a sufficient subalgebra for quantum states: a
subalgebra is 2- sufficient for a pair of states if it
contains all Bayes optimal tests of against . In classical
statistics, this corresponds to the usual definition of sufficiency. We show
this correspondence in the quantum setting for some special cases. Furthermore,
we show that sufficiency is equivalent to 2 - sufficiency, if the latter is
required for , for all .Comment: 12 page
Smearing of Observables and Spectral Measures on Quantum Structures
An observable on a quantum structure is any -homomorphism of quantum
structures from the Borel -algebra of the real line into the quantum
structure which is in our case a monotone -complete effect algebras
with the Riesz Decomposition Property. We show that every observable is a
smearing of a sharp observable which takes values from a Boolean
-subalgebra of the effect algebra, and we prove that for every element
of the effect algebra there is its spectral measure
How sharp are PV measures?
Properties of sharp observables (normalized PV measures) in relation to
smearing by a Markov kernel are studied. It is shown that for a sharp
observable defined on a standard Borel space, and an arbitrary observable
, the following properties are equivalent: (a) the range of is contained
in the range of ; (b) is a function of ; (c) is a smearing of
.Comment: 9 page
Entropy on Spin Factors
Recently it has been demonstrated that the Shannon entropy or the von Neuman
entropy are the only entropy functions that generate a local Bregman
divergences as long as the state space has rank 3 or higher. In this paper we
will study the properties of Bregman divergences for convex bodies of rank 2.
The two most important convex bodies of rank 2 can be identified with the bit
and the qubit. We demonstrate that if a convex body of rank 2 has a Bregman
divergence that satisfies sufficiency then the convex body is spectral and if
the Bregman divergence is monotone then the convex body has the shape of a
ball. A ball can be represented as the state space of a spin factor, which is
the most simple type of Jordan algebra. We also study the existence of recovery
maps for Bregman divergences on spin factors. In general the convex bodies of
rank 2 appear as faces of state spaces of higher rank. Therefore our results
give strong restrictions on which convex bodies could be the state space of a
physical system with a well-behaved entropy function.Comment: 30 pages, 6 figure
Sharp and fuzzy observables on effect algebras
Observables on effect algebras and their fuzzy versions obtained by means of
confidence measures (Markov kernels) are studied. It is shown that, on effect
algebras with the (E)-property, given an observable and a confidence measure,
there exists a fuzzy version of the observable. Ordering of observables
according to their fuzzy properties is introduced, and some minimality
conditions with respect to this ordering are found. Applications of some
results of classical theory of experiments are considered.Comment: 23 page
Relations for certain symmetric norms and anti-norms before and after partial trace
Changes of some unitarily invariant norms and anti-norms under the operation
of partial trace are examined. The norms considered form a two-parametric
family, including both the Ky Fan and Schatten norms as particular cases. The
obtained results concern operators acting on the tensor product of two
finite-dimensional Hilbert spaces. For any such operator, we obtain upper
bounds on norms of its partial trace in terms of the corresponding
dimensionality and norms of this operator. Similar inequalities, but in the
opposite direction, are obtained for certain anti-norms of positive matrices.
Through the Stinespring representation, the results are put in the context of
trace-preserving completely positive maps. We also derive inequalities between
the unified entropies of a composite quantum system and one of its subsystems,
where traced-out dimensionality is involved as well.Comment: 11 pages, no figures. A typo error in Eq. (5.15) is corrected. Minor
improvements. J. Stat. Phys. (in press