Quantum state measurements need fewer samples with joint testing
Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements
Data Structures and AlgorithmsInformation TheoryMachine Learning
Summary
Measuring the state of a quantum system usually requires many repeated tests. This paper finds the exact minimum number of tests needed when each measurement can look at a small group of samples together instead of one at a time. The authors show that combining up to t samples in a joint measurement can only reduce the total number of samples needed by about the square root of t. They also explain the mathematical reasons behind this limit using advanced tools, and provide a practical method to achieve this best possible number of samples.
quantum state tomographysample complexityjoint measurementtrace norm errorlow-rank quantum statesadaptive measurementFisher informationvan Trees inequalityGaussian measurement
Authors
Ashwin Nayak, Xingyu Zhou
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 $\varepsilon$, 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.