The power of random measurements: measuring Tr(\rho^n) on single copies of \rho

TL;DR

This paper demonstrates that random measurements on single copies of a quantum state can directly estimate the trace of its nth power, Tr( ho^n), without the need for joint measurements or full state reconstruction.

Contribution

It introduces a method to estimate Tr( ho^n) using only random measurements on individual copies, simplifying quantum state analysis.

Findings

Random measurements on single copies suffice for estimating Tr( ho^n)

Averaging over random measurements yields accurate estimates

Method does not require prior knowledge of the measurements performed

Abstract

While it is known that Tr(\rho^n) can be measured directly (i.e., without first reconstructing the density matrix) by performing joint measurements on n copies of the same state rho, it is shown here that random measurements on single copies suffice, too. Averaging over the random measurements directly yields estimates of Tr(\rho^n), even when it is not known what measurements were actually performed (so that one cannot reconstruct \rho).