Statistical analysis of low rank tomography with compressive random measurements

TL;DR

This paper analyzes the statistical efficiency of low rank quantum state tomography using compressive random measurements, showing optimal error rates for Frobenius norm but limitations for infidelity in nearly pure states.

Contribution

It demonstrates that low rank states can be estimated efficiently with fewer measurements in Frobenius norm, but highlights limitations in infidelity for nearly pure states.

Findings

Optimal Frobenius norm error rate of rd/N with O(r log d) measurements.

Concentration of Frobenius error around the optimal for all states.

Lack of uniform concentration in quantum infidelity for nearly pure states.

Abstract

We consider the statistical problem of `compressive' estimation of low rank states with random basis measurements, where the estimation error is expressed terms of two metrics - the Frobenius norm and quantum infidelity. It is known that unlike the case of general full state tomography, low rank states can be identified from a reduced number of observables' expectations. Here we investigate whether for a fixed sample size $N$, the estimation error associated to a `compressive' measurement setup is `close' to that of the setting where a large number of bases are measured. In terms of the Frobenius norm, we demonstrate that for all states the error attains the optimal rate $rd/N$ with only $O(r \log{d})$ random basis measurements. We provide an illustrative example of a single qubit and demonstrate a concentration in the Frobenius error about its optimal for all qubit states. In terms of the quantum infidelity, we show that such a concentration does not exist uniformly over all states. Specifically, we show that for states that are nearly pure and close to the surface of the Bloch sphere, the mean infidelity scales as $1/\sqrt{N}$ but the constant converges to zero as the number of settings is increased. This demonstrates a lack of `compressive' recovery for nearly pure states in this metric.