Optimal lower bounds for quantum state tomography

Abstract

We show that $n=\hat{I}\text{c◯}(rd/{\epsilon }^2)$ copies are necessary to learn a rank $r$ mixed state $\ddot{I}\in {\mathbb{C}}^{d\times d}$ up to error $\epsilon$ in trace distance. This matches the upper bound of $n=O(rd/{\epsilon }^2)$ from prior work, and therefore settles the sample complexity of mixed state tomography. We prove this lower bound by studying a special case of full state tomography that we refer to as projector tomography, in which $\ddot{I}$ is promised to be of the form $\ddot{I}=P/r$, where $P\in {\mathbb{C}}^{d\times d}$ is a rank $r$ projector. A key technical ingredient in our proof, which may be of independent interest, is a reduction which converts any algorithm for projector tomography which learns to error $\epsilon$ in trace distance to an algorithm which learns to error $O(\epsilon )$ in the more stringent Bures distance.

41 pages