Optimal lower bounds for quantum state tomography

We show that n = Omega(rd/varepsilon2) copies are necessary to learn a rank r mixed state ρ ∈ mathbbCd × d up to error varepsilon in trace distance. This matches the upper bound of n = O(rd/varepsilon2) 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 ρ is promised to be of the form ρ = P/r, where P ∈ mathbbCd × 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 varepsilon in trace distance to an algorithm which learns to error O(varepsilon) in the more stringent Bures distance.