Entropy of Quantum Measurement

Hanna Podsȩdkowska

doi:10.3390/e17031181

Abstract

A notion of entropy of a normal state on a finite von Neumann algebra in Segal’s sense is considered, and its superadditivity is proven together with a necessary and sufficient condition for its additivity. Bounds on the entropy of the state after measurement are obtained, and it is shown that a weakly repeatable measurement gives minimal entropy and that a minimal state entropy measurement satisfying some natural additional conditions is repeatable.

Highlights

Segal’s sense is considered, and its superadditivity is proven together with a necessary and sufficient condition for its additivity
The notion of the entropy of a state of a physical system was introduced by John von Neumann in the setup that is classical for quantum mechanics
We show the superadditivity of the entropy considered, together with a necessary and sufficient condition of its additivity and give bounds on the entropy of the state after measurement

Summary

Introduction

The notion of the entropy of a state of a physical system was introduced by John von Neumann (see [1]) in the setup that is classical for quantum mechanics In this approach, the observables of a physical system are identified with self-adjoint operators on a separable Hilbert space, and the states of the system, with the positive operators of trace one on this space. The observables of a physical system are identified with self-adjoint operators on a separable Hilbert space, and the states of the system, with the positive operators of trace one on this space This setting has been generalized in more modern theories, in particular in the so-called algebraic approach to quantum physics in which the bounded observables of a physical system form the self-adjoint part of a C*-, or von Neumann, algebra (see [2,3,4,5]). We show that a weakly repeatable measurement gives minimal entropy and that a minimal state entropy measurement satisfying some natural additional conditions is repeatable

Preliminaries and Notation

Instruments in Quantum Measurement Theory

Concept of Entropy

Entropy of Measurement

Conclusions