Symmetric minimal quantum tomography by successive measurements

TL;DR

This paper presents a two-step measurement scheme to implement symmetric informationally complete measurements in quantum systems, revealing a link between mutually unbiased bases and SIC POMs, with practical optical experiment proposals.

Contribution

It introduces a novel two-step measurement process for SIC POMs and uncovers a surprising operational connection with mutually unbiased bases in certain dimensions.

Findings

Any Heisenberg-Weyl covariant SIC POM can be realized with a Fourier basis measurement.

Mutually unbiased bases are used to construct SIC POMs in dimensions 2, 3, 4, and 8.

Feasible optical experiments are proposed for implementing SIC POMs.

Abstract

We consider the implementation of a symmetric informationally complete probability-operator measurement (SIC POM) in the Hilbert space of a d-level system by a two-step measurement process: a diagonal-operator measurement with high-rank outcomes, followed by a rank-1 measurement in a basis chosen in accordance with the result of the first measurement. We find that any Heisenberg-Weyl group-covariant SIC POM can be realized by such a sequence where the second measurement is simply a measurement in the Fourier basis, independent of the result of the first measurement. Furthermore, at least for the particular cases studied, of dimension 2, 3, 4, and 8, this scheme reveals an unexpected operational relation between mutually unbiased bases and SIC POMs; the former are used to construct the latter. As a laboratory application of the two-step measurement process, we propose feasible optical experiments that would realize SIC POMs in various dimensions.