The general objective of this DFG-funded project is to provide concepts and methods solving basic state identification and decision problems within the field of quantum statistics. These naturally appear in various scenarios while quantum information processing tasks are addressed. Following an operational viewpoint adopted in quantum statistics we assume that the missing information about the quantum state should be inferred from outcomes of measurements performed on the considered system. This is in correspondence to the approach of mathematical statistics where statistical decision, inference or estimation problems are typically formulated with respect to observed data considered as actual values of a random variable with unknown distribution.
We use a -algebraic formalism which allows to treat classical random variables and stochastic processes, as well as quantum states -with their intrinsic randomness- in a mathematically unified way. The emphasis is on the notion of state space of a C-algebra which models the algebra of observables associated to a physical system, either classical or quantum. Assuming that there is an arbitrary large number of copies of the system of interest available for observation, solutions of classical statistical problems are typically related to some entropic quantities. This in general turns out to be true as well in quantum mechanics. Although, it is non-trivial to find quantum counterparts of classical results and their proofs usually require new mathematical methods.
We focus on problems of asymptotic (multiple) state discrimination, testing compound quantum hypotheses and quantum maximum-entropy inference, and corresponding entropic distances on state spaces being Umegaki relative entropy and generalizations of quantum Chernoff distance. Our mathematical methods mainly combine results of matrix analysis and mathematical statistics as well as (non-commutative) ergodic theory.

People:

Arleta Szkoła (principal investigator)

Stephan Weis (post-doc)

Yuri Campbell-Borges (PhD student)

Sajad Saeedinaeeni (master student)

Collaborations:

Michael Nussbaum

Further Information:

Supported by the DFG