Fisher-symmetric informationally complete measurements for pure states

TL;DR

This paper introduces a new quantum measurement that is symmetric in Fisher information, locally informationally complete, and reduces the number of outcomes needed for pure state parameter estimation in high-dimensional systems.

Contribution

It proposes a Fisher-symmetric measurement that is locally informationally complete with fewer outcomes than traditional global measurements for pure states.

Findings

Reduces measurement outcomes from near 4d-3 to 2d-1 for pure states.

Ensures uniform Fisher information across parameters.

Provides a measurement optimal in the multi-parameter quantum Cramer-Rao bound.

Abstract

We introduce a new kind of quantum measurement that is defined to be symmetric in the sense of uniform Fisher information across a set of parameters that injectively represent pure quantum states in the neighborhood of a fiducial pure state. The measurement is locally informationally complete---i.e., it uniquely determines these parameters, as opposed to distinguishing two arbitrary quantum states---and it is maximal in the sense of a multi-parameter quantum Cramer-Rao bound. For a $d$-dimensional quantum system, requiring only local informational completeness allows us to reduce the number of outcomes of the measurement from a minimum close to but below $4d-3$, for the usual notion of global pure-state informational completeness, to $2d-1$.