Quantum State Tomography with Joint SIC POMs and Product SIC POMs

TL;DR

This paper uses random matrix theory to analyze the efficiency of SIC POMs and their product forms in quantum state tomography, revealing their relative performance and optimality in multipartite systems.

Contribution

It provides analytic formulas for measurement errors and proves the optimality of product SIC POMs among product measurements, highlighting their efficiency in multipartite quantum systems.

Findings

Product SIC POMs are optimal among all product measurements.

Joint SIC POMs have a marginal advantage in bipartite systems.

Efficiency advantage of joint SIC POMs grows exponentially with system size.

Abstract

We introduce random matrix theory to study the tomographic efficiency of a wide class of measurements constructed out of weighted 2-designs, including symmetric informationally complete (SIC) probability operator measurements (POMs). In particular, we derive analytic formulae for the mean Hilbert-Schmidt distance and the mean trace distance between the estimator and the true state, which clearly show the difference between the scaling behaviors of the two error measures with the dimension of the Hilbert space. We then prove that the product SIC POMs---the multipartite analogue of the SIC POMs---are optimal among all product measurements in the same sense as the SIC POMs are optimal among all joint measurements. We further show that, for bipartite systems, there is only a marginal efficiency advantage of the joint SIC POMs over the product SIC POMs. In marked contrast, for multipartite systems, the efficiency advantage of the joint SIC POMs increases exponentially with the number of parties.