Anna Jenčová

Sufficiency in Quantum Statistical Inference

This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only in finite Hilbert space dimension, but proved here in generality: the Koashi-Imoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity.

Local Asymptotic Normality in Quantum Statistics

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family

REVERSIBILITY CONDITIONS FOR QUANTUM OPERATIONS

\varphi_{\theta_{0}+u/\sqrt{n}}^{n}

Geometry of quantum states: dual connections and divergence functions

consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state ϕu of an algebra of canonical commutation relations. The convergence holds for all “local parameters”

SUFFICIENCY IN QUANTUM STATISTICAL INFERENCE: A SURVEY WITH EXAMPLES

u\in {\mathbb{R}}^{m}

Base norms and discrimination of generalized quantum channels

such that