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

Özetle: We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples.

Özet

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, $$Θleft(frac{dr}{varepsilon^2}mathop{mathrm{max}}left{1,frac{r}{sqrt{t}}right}right)$$ 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.

Orijinal Özet (İngilizce)

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, $$Θleft(frac{dr}{varepsilon^2}mathop{mathrm{max}}left{1,frac{r}{sqrt{t}}right}right)$$ 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.

Kaynak: arXiv:2609.10514 · PDF