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Life sciences · Preprint
arXiv · September 9, 2026
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This is a theoretical contribution deriving optimal sample complexity bounds for low-rank quantum state tomography under joint measurement constraints. The work establishes that joint measurements on at most t samples improve single-sample complexity by at most a factor √t, with matching upper and lower bounds achieved via Fisher information analysis and Gaussian measurement construction. It has no demonstrated application to clinical or life-science practice.
Preprint.
Optimal sample complexity is Θ(dr/ε² max{1, r/√t}) for rank-r d-dimensional quantum states to trace norm error ε Joint measurements on at most t samples improve complexity by at most factor √t relative to single-sample measurements Order r² samples measured jointly are necessary and sufficient to attain unrestricted collective rate
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This is a theoretical computer science result on optimal sample complexity for quantum state tomography, lacking empirical validation or clinical application.
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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 $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $$ Θ\left( \frac{dr}{\varepsilon^2} \max\left\{1,\frac r{\sqrt t}\right\} \right)$$ 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.
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