A Note on the Set of After-measurement States in Generalized Quantum Measurement

TL;DR

This paper compares after-measurement state sets in standard and generalized quantum measurements, revealing that for SIC-POVMs, the set's volume ratio diminishes rapidly as dimensions increase, with implications for finite system realizations.

Contribution

It analyzes the volume ratio of after-measurement states in generalized quantum measurements, highlighting the behavior for SIC-POVMs as dimensions grow.

Findings

Volume ratio tends to zero with increasing dimensions for SIC-POVMs.

The study uses Hilbert-Schmidt norm-based volume measures.

Implications for finite system measurements are discussed.

Abstract

The sets of after-measurement states for standard and generalized quantum measurements are compared. It is shown that for a SIC-POVM generalized measurement, the ratio of the volume of the set of after-measurement states and the volume of the simplex generated by individual outcomes quickly tends to zero with increase of the number of dimensions. The volumes used are based on the Hilbert-Schmidt norm. Some consequences on actual realizations, having finite collections of systems are discussed.