An optimal estimator is established under the sole promise that one of the two states is pure, without knowing which one, under the sole promise of which state is pure.
Abstract
The Uhlmann fidelity ${\rm F}(\rho_0,\rho_1) = {\rm tr}|\sqrt{\rho_0}\sqrt{\rho_1}|$ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error $\varepsilon$ requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of $\rho_0$ and $\rho_1$. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on $1/\varepsilon$. However, the known optimal estimator for ${\rm F}(\rho,|\psi\rangle\!\langle\psi|)$ due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure. In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for ${\rm F}(\rho, |\psi\rangle\!\langle\psi|)$ under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of ${\rm tr}_{\sf A}(|\psi_0\rangle\!\langle\psi_1|)$ (or its inverse) to the reference register $\sf R$ of the purification $|\psi_1\rangle$ (or $|\psi_0\rangle$) on the registers $\sf A$ and $\sf R$, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.
We study the trace distance \[D_{\mathrm{tr}}(\rho,\sigma) =\frac12\|\rho-\sigma\|_1, \rho=\bigotimes_{i=1}^n\rho_i,\quad \sigma=\bigotimes_{i=1}^n\sigma_i, \] when the two exponentially large states are specified by their local factors. We give a deterministic approximation within a universal constant factor for rational product inputs. Its running time is polynomial in the number of factors, the local dimension, and the input bit length. In the opposite direction, exact computation is $\#\mathsf P$-hard even for diagonal qubit states, by the corresponding hardness of total variation distance between product distributions. The proof uses local Uhlmann-optimal purifications to reduce the problem to estimating the product-fidelity defect and the trace norm of a structured first-order operator. Although this operator acts on an exponentially large space, we approximate its trace norm by a local convex surrogate that admits a polynomial-size classical conic formulation. A square-function estimate shows that the surrogate upper-bounds this trace norm. Conversely, duality and local dephasing reduce the reverse comparison to a head--tail inequality for independent centered random variables, showing that the surrogate is at most a dimension-free constant times the same norm.
Kun He, Dimitrios Myrisiotis, Junhong Nie et al.· 0 citations
We settle the sample complexity of estimating the root Uhlmann fidelity $F(\rho,\sigma)=\operatorname{tr}\sqrt{\sqrt{\sigma}\rho\sqrt{\sigma}}$ between an unknown state $\rho$ and a known rank-$r$ reference state $\sigma$. Writing $S(r,\varepsilon)$ for the sample complexity at additive error $\varepsilon$, we resolve the open problem posed by Wang by closing, up to logarithmic factors, the gap between the previously known bounds $\Omega(r/\varepsilon^2)$ and $O(r^2/\varepsilon^2)$. We prove $S(r,\varepsilon)=\widetilde{\Theta}(r^2/\varepsilon^2)$ for all $0<\varepsilon\le\varepsilon_0$, where $\varepsilon_0>0$ is a universal constant. The lower bound already holds on a $2r$-dimensional system when $\sigma$ is maximally mixed on a fixed $r$-dimensional subspace, and for a hard family of states that do not commute with $\sigma$. The proof combines exact spectral moment matching, a radially size-biased doubly correlated Wishart model, and the Cauchy identity, reducing state indistinguishability to a long-cycle estimate for a weighted random permutation. A direct-sum embedding and binomial thinning yield the optimal $1/\varepsilon^2$ dependence. We also prove a near-quadratic lower bound $\widetilde{\Omega}(r^2)$ for quantum spectrum estimation at constant accuracy. Combined with the recent $O(r^2(\log\log r/\log r)^2)$ upper bound, this determines the polynomial order of the sample complexity in this regime and establishes a near-quadratic barrier.
We give an algorithm which, given $n = O(d^2 \cdot (\log\log(d)/\log(d))^2)$ copies of $\rho$, estimates the eigenvalues of $\rho$ to constant error in total variation distance. Thus, we can learn the eigenvalues of a quantum state with fewer copies than the $\Theta(d^2)$ needed to run full state tomography. This is the first improvement to spectrum estimation over the influential Keyl-Werner algorithm, which uses $n = \Theta(d^2)$ copies, thereby resolving a question raised by Keyl and Werner in 2001 and refuting a 2016 conjecture of Wright. Our main technical tool is a new tomography guarantee, where the error of tomography in a particular direction $|w\rangle$ scales with $\langle w | \rho |w\rangle$ for all directions simultaneously. From this stronger"relative-error"bound, we recover better algorithms for principal component analysis in Bures distance and tomography in $\chi^2$-divergence as corollaries.
Angelos Pelecanos, Jack Spilecki, Ewin Tang et al.· arXiv.org· 5 citations· ⚡5
We study the task of estimating a $d-$dimensional quantum state $\rho$ under the constraint that the measurement is $\alpha-$gentle. Such measurements $M$ do not collapse the state; they issue both a random variable $R^M = \omega$ containing statistical information and a post-measurement state $\rho_{M \to \omega}$ such that $\|\rho_{M\to \omega} - \rho\|_{Tr} \leq \alpha$. We describe gentle measurements and their connection to quantum differential privacy. Our results show that the optimal minimax estimation rate in Frobenius norm is of order $d^3/(n \alpha^2)$, instead of $d^2/n$ for general measurements. Moreover, for rank $r$ states with $r\leq d$ we prove that the optimal minimax rate is $rd^2/(n \alpha^2)$, instead of $rd/n$. Very surprisingly, the loss for gentleness $d/\alpha^2$ scales with the ambient dimension of the Hilbert space, rather than the number of parameters $rd$, typically seen in classical differential privacy. We propose optimal gentle measurements and indicate how they can be physically implemented using an ancillary state and a CNOT gate to entangle it with the initial state. We notice that the resulting random variable has a likelihood that satisfies local differential privacy. Lower bounds are proven through a new quantum information-theoretic inequality applied to well chosen families of states in the manifold of (small-rank) quantum states.
Cristina Butucea, Jan Johannes, Henning Stein· 0 citations
It is well-known that learning a pure $n$-qubit stabilizer state $|\psi\rangle$ both requires, and can be accomplished with, access to a number of copies of $|\psi\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_\delta(n)$, the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most $0<\delta<1/8$, satisfies $n+\lceil\log_2(1/\delta)\rceil-3\leq L_\delta(n)\leq n+\left\lceil\log_2(1/\delta)\right\rceil+4$. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown $n$-qubit Clifford unitary from $2n+\left\lceil\log_2(1/\delta)\right\rceil+4$ queries, the $n$-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group $\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}$, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.
Rebecca Chang, Matthias C. Caro, Martín Larocca et al.· 1 citation
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown $d$-dimensional quantum state $\rho$ and a known collection of observables ${E_1,\ldots,E_M}$, the goal is to estimate all expectation values $\{\text{Tr}(\rho E_i)\}_{i=1}^M$ to additive accuracy $\varepsilon$ with probability at least $1-\delta$. An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on $M$, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity \[ O\left( \frac{\log(M)\log(M/\delta)}{\varepsilon^2} \right), \] which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography.
F. G. Jeronimo, Qi-Zhao Huang, Le Liu· 1 citation
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