Accessible information without disturbing partially known quantum states on a von Neumann algebra

TL;DR

This paper investigates the limits of extracting classical information from partially known quantum states on a von Neumann algebra without disturbance, defining the classical part of a statistical experiment and applying it to broadcasting.

Contribution

It introduces a formal definition of the classical part of a statistical experiment in the von Neumann algebra setting and characterizes the maximum extractable information without disturbance.

Findings

Maximum accessible classical information is characterized by the classical part of the experiment.

The classical part of a direct product of experiments equals the product of their classical parts.

The classical part corresponds to the maximal classical information extractable without disturbing the states.

Abstract

This paper addresses the problem of how much information we can extract without disturbing a statistical experiment, which is a family of partially known normal states on a von Neumann algebra. We define the classical part of a statistical experiment as the restriction of the equivalent minimal sufficient statistical experiment to the center of the outcome space, which, in the case of density operators on a Hilbert space, corresponds to the classical probability distributions appearing in the maximal decomposition by Koashi and Imoto [Phys.~Rev.~A $\textbf{66}$, 022318 (2002)]. We show that we can access by a Schwarz or completely positive channel at most the classical part of a statistical experiment if we do not disturb the states. We apply this result to the broadcasting problem of a statistical experiment. We also show that the classical part of the direct product of statistical experiments is the direct product of the classical parts of the statistical experiments. The proof of the latter result is based on the theorem that the direct product of minimal sufficient statistical experiments is also minimal sufficient.