Posted

Thilo Scharnhorst, Jack Spilecki, John Wright (Oct 10 2025).
Abstract: We show that n=Ω(rd/ε2)n = \Omega(rd/\varepsilon^2) copies are necessary to learn a rank rr mixed state ρCd×d\rho \in \mathbb{C}^{d \times d} up to error ε\varepsilon in trace distance. This matches the upper bound of n=O(rd/ε2)n = O(rd/\varepsilon^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 ρ\rho is promised to be of the form ρ=P/r\rho = P/r, where PCd×dP \in \mathbb{C}^{d \times d} is a rank rr 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 ε\varepsilon in trace distance to an algorithm which learns to error O(ε)O(\varepsilon) in the more stringent Bures distance.

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