Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements

arXiv:2609.10514v1 Announce Type: cross Abstract: We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small arepsilon, estimating an unknown state on C^d of rank at most r to trace norm error arepsilon with constant success probability requires, and is achievable with, $ Thetaleft( frac{dr}{arepsilon^2} maxleft{1,frac r{sqrt t}right} right) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most t samples improve the complexity of algorithms making single-sample measurements by at most a factor sqrt t. Further, measuring order r^2 samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

Source: arXiv cs.LG | 2026-09-10

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