The Uhlmann fidelity ${\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_1}|$ 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 $ρ_0$ and $ρ_1$ . 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 ${\rm F}(ρ,|ψ\rangle\!\langleψ|)$ 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 ${\rm F}(ρ, |ψ\rangle\!\langleψ|)$ 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 ${\rm tr}_{\sf A}(|ψ_0\rangle\!\langleψ_1|)$ (or its inverse) to the reference register $\sf R$ of the purification $|ψ_1\rangle$ (or $|ψ_0\rangle$ ) 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.The Uhlmann fidelity ${\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_1}|$ 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 $ρ_0$ and $ρ_1$. 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 ${\rm F}(ρ,|ψ\rangle\!\langleψ|)$ 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 ${\rm F}(ρ, |ψ\rangle\!\langleψ|)$ 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 ${\rm tr}_{\sf A}(|ψ_0\rangle\!\langleψ_1|)$ (or its inverse) to the reference register $\sf R$ of the purification $|ψ_1\rangle$ (or $|ψ_0\rangle$) 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.

quant-pharXiv:2608.10674Aug 12, 20260 citations

Optimal Estimation of Qubit States with Continuous Time Measurements

Mădălin Guţǎ, Bas Janssens, Jonas Kahn

arXiv:quant-ph/0608074Communications in Mathematical PhysicsOct 26, 200736 citations

Reexamination of optimal quantum state estimation of pure states (5 pages)

A. Hayashi, T. Hashimoto

Physical Review AJan 1, 20050 citations

Quantum Wasserstein distance and its relation to several types of fidelities

Géza Tóth, József Pitrik

We consider several definitions of the quantum Wasserstein distance based on an optimization over general bipartite quantum states with given marginals. Then, we examine the quantities obtained after the optimization is carried out over bipartite separable states instead. We prove that several of these quantities are equal to each other. Thus, we connect several approaches in the literature. We prove the triangle inequality for some of these quantities for the case of one of the three states being pure. As a byproduct, we show that the square root of the Uhlmann-Jozsa quantum fidelity can also be written as an optimization over separable states with given marginals. We use this to prove that some of these quantities equal the Uhlmann-Jozsa quantum fidelity for qubits. We also find relations with the superfidelity.We consider several definitions of the quantum Wasserstein distance based on an optimization over general bipartite quantum states with given marginals. Then, we examine the quantities obtained after the optimization is carried out over bipartite separable states instead. We prove that several of these quantities are equal to each other. Thus, we connect several approaches in the literature. We prove the triangle inequality for some of these quantities for the case of one of the three states being pure. As a byproduct, we show that the square root of the Uhlmann-Jozsa quantum fidelity can also be written as an optimization over separable states with given marginals. We use this to prove that some of these quantities equal the Uhlmann-Jozsa quantum fidelity for qubits. We also find relations with the superfidelity.

arXiv:2506.14523Jun 17, 20250 citations

The optimal approximation of qubit states with limited quantum states

Li‐qiang Zhang, Deng-hui Yu, Chang‐shui Yu

arXiv:2203.07813Physics Letters AMar 18, 20215 citations

The Uhlmann fidelity

{\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_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

ρ_0

and

ρ_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}(ρ,|ψ\rangle\!\langleψ|)

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}(ρ, |ψ\rangle\!\langleψ|)

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}(|ψ_0\rangle\!\langleψ_1|)

(or its inverse) to the reference register

\sf R

of the purification

|ψ_1\rangle

(or

|ψ_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.