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 $\rho \in \mathbb{C}^{d \times d}$ and asked to produce an estimator of $\rho$. 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 $n = O(rd/\varepsilon^2)$ copies are sufficient to learn a rank-$r$ mixed state to trace distance error $\varepsilon$, which is optimal. (2) We further show that $n = O(rd/\varepsilon^2)$ 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 $n =O\left(\max \left(\frac{d^3}{\sqrt{k}\varepsilon^2}, \frac{d^2}{\varepsilon^2} \right) \right)$ 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 $O(\log(m)/\varepsilon^2)$ copies are sufficient to learn $m$ given observables $O_1, \dots, O_m$ in the "high accuracy regime", when $\varepsilon = O(1/d)$, improving on a result of Chen et al. More generally, we show that if $\mathrm{tr}(O_i^2) \leq F$ for all $i$, then $n = O\Big(\log(m) \cdot \Big(\min\Big\{\frac{\sqrt{r F}}{\varepsilon}, \frac{F^{2/3}}{\varepsilon^{4/3}}\Big\} + \frac{1}{\varepsilon^2}\Big)\Big)$ 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.