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

TL;DR

The paper develops the debiased Keyl's algorithm, the first unbiased and sample-optimal estimator for full quantum state tomography, by applying a novel donation operation on Young diagrams to correct Keyl’s bias while preserving optimal entangled measurements. It provides explicit first- and second-moment formulas via Schur-Weyl duality and Clebsch-Gordan analysis, and then leverages these to obtain tight sample complexities for multiple tomography tasks, including trace- and Bures-distance tomography, tomography with limited entanglement, shadow tomography, and quantum metrology. Across applications, the results match or improve known lower bounds and remove logarithmic factors in several regimes, establishing unbiasedness as a practical and theoretically robust property for entangled tomography. The work also introduces a general family of unbiased estimators based on legal Young diagram transformations and develops rich representation-theoretic machinery (Schur-Weyl duality, WSS, Clebsch-Gordan) that open avenues for spectrum estimation and higher-moment analyses in quantum learning contexts.

Abstract

In the problem of quantum state tomography, one is given $n$ copies of an unknown rank-$r$ mixed state $ρ\in \mathbb{C}^{d \times d}$ 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 $n = O(rd/\varepsilon^2)$ 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.

Figures (25)

• Figure 1: The Young diagram corresponding to the partition $\lambda = (6,6,2,1)$. In this case, $n = 15$ and $d = 4$. This resembles the histogram, sorted from highest to lowest and then turned on its side, of $n = 15$ samples ${\boldsymbol{x}}_1, \ldots, {\boldsymbol{x}}_{15}$ drawn from a discrete distribution $p = (p_1, p_2, p_3, p_4)$ whose probability values, when sorted from highest to lowest, are approximately $6/15$, $6/15$, $2/15$, and $1/15$.
• Figure 2: Let $d = 5$. On the left is the Young diagram $\lambda = (4,4,2,1,0)$. On the right is $\mathrm{donate}(\lambda) = (7,7,2, -1,-4)$. The gray-colored boxes correspond to rows with a negative length. Note that $\mathrm{donate}(\lambda)$ is not necessarily a Young diagram, since it may have negative entries.
• Figure 3: The Young diagram above corresponds to the partition $\lambda = (6,6,2,1)$, with $|\lambda| = 15$ and $\ell(\lambda) = 4$.
• Figure 4: Let $\lambda = (4,2,1)$. On the left, a standard Young tableau of shape $\lambda$. On the right, a semistandard Young tableau of shape $\lambda$ and alphabet $[3]$.
• Figure 5: Four Young diagrams. On the left is $\lambda = (3,2,1)$. The remaining Young diagrams are $\mu_i$, for $i = 1, 2, 3$. Each $\mu_i$ satisfies $\lambda \subseteq \mu_i$, with the boxes of $\mu\setminus \lambda$ shaded. We have $\lambda \nearrow \mu_1$, with $\mu_1 = \lambda + e_2$, since the additional box is in the second row. The shaded box in $\mu_1$ is $(2,3)$, and the content of this cell is $\mathrm{cont}(2,3) = 1$. Both $\mu_1$ and $\mu_2$ satisfy $\lambda \preceq \mu_i$, since there is at most one added box per column. However, $\lambda \not \precsim \mu_3$, since there are two additional boxes in the third column.

Theorems & Definitions (202)

• Definition 1.1: Young diagram box donation
• Definition 1.2: Debiased Keyl's algorithm
• Theorem 1.3: Debiased Keyl's algorithm is an unbiased estimator
• Theorem 1.4: Second moment of the debiased Keyl's algorithm
• Theorem 1.5: Trace distance tomography
• Theorem 1.6: Bures distance tomography
• Theorem 1.7: Learning bipartite pure states
• Theorem 1.8: Tomography with limited entanglement
• Theorem 1.9: Classical shadows
• Theorem 1.10: Quantum metrology