The debiased Keyl's algorithm: a new unbiased estimator for full state tomography

In the problem of quantum state tomography, one is given n copies of an unknown rank-r mixed state ρin mathbb{C}^{d times d} and asked to produce an estimator of ρ. In this work, we present the debiased Keyl's algorithm, the first estimator for full state tomography which is both unbiased and sample-optimal. We derive an explicit formula for the second moment of our estimator, with which we show the following applications. (1) We give a new proof that n = O\(rd/varepsilon²\) copies are sufficient to learn a rank-r mixed state to trace distance error varepsilon, which is optimal. (2) We further show that n = O\(rd/varepsilon²\) copies are sufficient to learn to error varepsilon in the more challenging Bures distance, which is also optimal. (3) We consider full state tomography when one is only allowed to measure k copies at once. We show that n =Oleft\(max left(frac{d³}{sqrt{k}varepsilon²}, frac{d²}{varepsilon²} right\) right) copies suffice to learn in trace distance. This improves on the prior work of Chen et al. and matches their lower bound. (4) For shadow tomography, we show that O\(log(m\)/varepsilon²) copies are sufficient to learn m given observables O₁, dots, O_m in the "high accuracy regime", when varepsilon = O(1/d), improving on a result of Chen et al. More generally, we show that if tr\(O_i²\) leq F for all i, then n = OBig\(log(m\) cdot Big\(minBig\{frac{sqrt{r F}}{varepsilon}, frac{F^2/3}{varepsilon^4/3}Big\} + frac{1}{varepsilon²}Big\)Big) copies suffice, improving on existing work. (5) For quantum metrology, we give a locally unbiased algorithm whose mean squared error matrix is upper bounded by twice the inverse of the quantum Fisher information matrix in the asymptotic limit of large n, which is optimal.