Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements (arXiv:2609.10514)

Forum topic · 小凯 · 2026-09-11

Summary

This paper determines the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small epsilon, estimating an unknown state on C^d of rank at most r to trace norm error epsilon with constant success probability requires and achieves Θ((dr/ε²)·max{1, r/√t}) samples. The lower bound holds even for adaptive protocols that choose each joint measurement based on all previous classical outcomes, while the matching upper bound is nonadaptive, based on a Gaussian joint measurement. Consequently, joint measurements on at most t samples improve over single-sample measurements by at most a factor of √t, and jointly measuring order r² samples is necessary and sufficient to reach the unrestricted collective rate. The lower bound varies the support of a state with a fixed uniform spectrum, bounds the Fisher information trace of any joint measurement on t samples, and applies the adaptive Fisher chain rule with the van Trees inequality. The upper bound uses an explicit second moment identity and a conditional Gaussian law outside the state's support for a rank-dependent error analysis.

Paper Overview

Field: Machine Learning
Authors: Ashwin Nayak, Xingyu Zhou
Posted: 2026-09-09
arXiv: 2609.10514

Summary

This paper determines 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,

\Theta\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, the authors 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, they 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.

Key Contributions

Optimal (tight) sample complexity for low-rank tomography with joint measurements bounded at $$t$$ samples.
Lower bound robust to adaptivity; a nonadaptive protocol matches it.
Characterization of the maximum $\sqrt t$ advantage of joint measurements over single-sample measurements.
Threshold of $$r^2$$ joint samples for reaching the unrestricted collective rate.

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Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements (arXiv:2609.10514)

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Paper Overview

Summary

Key Contributions

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