Jack Spilecki

In the problem of quantum state tomography, one is given $n$ copies of an unknown rank-$r$ mixed state $\rho \in \C^{d \times d}$ and asked to produce an estimator $\widehat{\brho}$ of $\rho$ which is close to it. One of the most basic and useful properties a statistical estimator can have is that of being \emph{unbiased}, meaning that $\widehat{\brho}$ is equal to the true state $\rho$ in expectation. Unfortunately, of the three estimators for tomography which are known to be either sample-optimal or almost sample-optimal, namely Keyl's algorithm~\cite{Key06} and the two tomography algorithms from~\cite{HHJ+16}, none of them produce unbiased estimators. This situation has recently been highlighted as a bottleneck in several important tomographic applications~\cite{CLL24a,CLL24b}. In this work, we present the \emph{debiased Keyl's algorithm}, the first estimator for full state tomography which is both unbiased and sample-optimal. Our algorithm is the result of taking Keyl's algorithm and carefully modifying it in order to correct for its bias. In addition, we derive an explicit formula for the second moment of our estimator which is simple and easy to use. Using it, we show the following applications. \begin{itemize} \item[$\circ$] We show that $n = O(rd/\epsilon^2)$ copies are sufficient to learn a rank-$r$ mixed state $\rho \in \C^{d \times d}$ to trace distance error $\epsilon$. This gives a new proof of the main result of~\cite{OW16} and matches the lower bound of~\cite{SSW25}. \item[$\circ$] We then improve this result and show that $n = O(rd/\epsilon^2)$ copies are sufficient to learn to error $\epsilon$ in the more challenging Bures distance. This is optimal due to the lower bound of~\cite{Yue23} and improves on the best known bound of $n = \widetilde{O}(rd/\epsilon^2)$ from prior work~\cite{HHJ+16,OW17a,FO24}. \item[$\circ$] We consider the scenario of full state tomography with limited entanglement, in which one is given $n$ copies of $\rho$ but only allowed to perform entangled measurements across $k$ of them at a time. We show that \begin{equation*} n =O\left(\max \left(\frac{d^3}{\sqrt{k}\epsilon^2}, \frac{d^2}{\epsilon^2} \right) \right) \end{equation*} copies of $\rho$ suffice to learn $\rho$ to trace distance error $\epsilon$ in this scenario. This improves on the prior work of \cite{CLL24a} and matches their lower bound which holds for $k \leq 1/\epsilon^c$, where $c$ is a constant. \item[$\circ$] In the problem of shadow tomography, one is given $m$ observables $O_1, \ldots, O_m$, and the goal is to estimate each quantity $\tr(O_i \cdot \rho)$ up to $\epsilon$ error. If $\Vert O_i \Vert_{\infty} \leq 1$ for all $i$, we show that $O(\log(m)/\epsilon^2)$ copies are sufficient to solve this task in the ``high accuracy regime'' when $\epsilon = O(1/d)$. This improves a result of~\cite{CLL24b}, who showed this bound holds when $\epsilon = O(1/d^{12})$. More generally, we show that if $\tr(O_i^2) \leq F$ for all $i$, then \begin{equation*} n = O\Big(\log(m) \cdot \Big(\min\Big\{\frac{\sqrt{r F}}{\epsilon}, \frac{F^{2/3}}{\epsilon^{4/3}}\Big\} + \frac{1}{\epsilon^2}\Big)\Big) \end{equation*} copies suffice to solve this task. This improves on a result of~\cite{GLM24}, who showed a bound of $n = O(\log(m) \cdot \sqrt{rF}/\epsilon^2)$, and disproves a conjecture of~\cite{CLL24b}. \item[$\circ$] Our final application is to the field of quantum metrology. In quantum metrology, one is provided copies of a mixed state $\rho_{\theta}$ parameterized by a vector $\theta \in \R^n$, and the goal is to estimate the parameter $\theta$. An algorithm is \emph{locally unbiased} at a parameter $\theta^*$ if, roughly, it is unbiased given $\rho_{\theta}$ when $\theta$ is in a small neighborhood around $\theta^*$. The quantum Cramér–Rao bound states that the mean squared error matrix (MSEM) of any locally unbiased algorithm is lower bounded by the inverse of the Quantum Fisher Information (QFI) matrix. We give a locally unbiased algorithm whose MSEM is \emph{upper bounded} by twice the inverse of the QFI matrix in the asymptotic limit of large $n$, which is optimal. \end{itemize}

We show that $n = \Omega(rd/\epsilon^2)$ copies are necessary to learn a rank $r$ mixed state $\rho \in \C^{d \times d}$ up to error~$\epsilon$ in trace distance. This matches the upper bound of $n = O(rd/\epsilon^2)$ from~\cite{OW16} and therefore settles the sample complexity of mixed state tomography. We prove this lower bound by studying a special case of full state tomography that we refer to as \emph{projector tomography}, in which $\rho$ is promised to be of the form $\rho = P/r$, where $P \in \C^{d \times d}$ is a rank $r$ projector. A key technical ingredient in our proof, which may be of independent interest, is a reduction which converts any algorithm for projector tomography which learns to error $\epsilon$ in trace distance to an algorithm which learns to error $O(\epsilon)$ in the more stringent Bures distance.