Optimal fidelity estimation when one state is pure via algorithmic Uhlmann transform

Technical Summary: Optimal Fidelity Estimation when One State is Pure via Algorithmic Uhlmann Transform

Problem Statement
The paper addresses the problem of estimating the Uhlmann fidelity $F(\rho_0, \rho_1) = \text{tr}|\sqrt{\rho_0}\sqrt{\rho_1}|$ between two quantum states. While general fidelity estimation for mixed states suffers from complexity dependencies on the ranks of the states, the task simplifies significantly when at least one state is pure. In such cases, the fidelity reduces to $F(\rho, |\phi\rangle\langle\phi|) = \sqrt{\langle\phi|\rho|\phi\rangle}$.

Prior to this work, optimal estimators for the "one-pure-state" scenario required prior knowledge of which of the two states was pure. Without this knowledge, the best-known approach relied on the SWAP test combined with quantum amplitude estimation, resulting in suboptimal query complexity of $O(1/\varepsilon^2)$ and sample complexity of $O(1/\varepsilon^4)$. The authors aim to remove the requirement for knowing which state is pure while achieving the optimal query complexity of $\Theta(1/\varepsilon)$ and sample complexity of $\Theta(1/\varepsilon^2)$.

Methodology
The proposed solution leverages the Algorithmic Uhlmann Transform, specifically adapting the refined construction by Utsumi et al. (2025) to the case where one state is pure. The core technical insight relies on Uhlmann's theorem, which relates fidelity to the maximum overlap between purifications of the states.

1. Simplification of the Uhlmann Transform:
   When one state (e.g., $\rho_1 = |\phi\rangle\langle\phi|$) is pure, its purification is a product state. Consequently, the Uhlmann cross operator $X = \text{tr}_A(|\psi_0\rangle\langle\psi_1|)$ has rank at most one. In this specific setting, the optimal Uhlmann unitary $U^\star$ is proportional to the cross operator $X$ itself, with the proportionality constant being the fidelity $F(\rho_0, \rho_1)$. Specifically, $U^\star = X / F(\rho_0, \rho_1)$.

2. Unitary Dilation:
   The authors construct an exact unitary dilation $W$ of the cross operator $X$. This dilation is defined as $W = Q_1^\dagger (I_A \otimes \text{SWAP}_{R', S}) Q_0$, where $Q_0$ and $Q_1$ are state-preparation circuits for $\rho_0$ and $\rho_1$, and $S$ is an ancillary register. The operator $X$ is recovered by projecting the ancillary registers of $W$ onto the zero state.

3. Bidirectional Estimation Strategy:
   Since the algorithm does not know which state is pure, it cannot determine a priori whether to apply $W$ or $W^\dagger$ to recover the fidelity. The authors propose a symmetric approach:

   • Construct two quantum circuits, $U_1$ and $U_0$.
   • $U_1$ applies the dilation $W$ (forward) to the purification of $\rho_1$.
   • $U_0$ applies the inverse dilation $W^\dagger$ to the purification of $\rho_0$.
   • The fidelity is shown to be equal to $\max\{a_0, a_1\}$, where $a_j$ is the amplitude associated with the projection of the output of $U_j$ onto a specific subspace.

4. Amplitude Estimation:
   The algorithm estimates the amplitudes $a_0$ and $a_1$ using square-root amplitude estimation (a variant of quantum amplitude estimation). By taking the maximum of the two estimates, the algorithm recovers the fidelity regardless of which state is pure.

Key Contributions and Results

• Optimal Query Complexity: The paper establishes a quantum estimator that achieves query complexity $\Theta(1/\varepsilon)$ for estimating the fidelity of two states when at least one is pure, without requiring prior knowledge of which state is pure. This matches the lower bound and improves quadratically over the previous $O(1/\varepsilon^2)$ bound derived from SWAP-test-based methods.
• Optimal Sample Complexity: By applying quantum sample-to-query lifting techniques, the authors derive a sample-optimal estimator with complexity $\Theta(1/\varepsilon^2)$, again quadratically improving upon the previous $O(1/\varepsilon^4)$ bound.
• Removal of Prior Knowledge: The work removes the "mathematically unnecessary" prior-knowledge requirement found in the optimal estimator by Fang and Wang (ESA 2025), unifying the optimal estimation for the case where the pure state is known and where it is unknown.
• Technical Implementation: The method avoids the heavy machinery of Quantum Singular Value Transformation (QSVT) for this specific case by utilizing the rank-1 property of the cross operator to implement the Uhlmann transform via a direct unitary dilation.

Significance
The paper claims to establish the first optimal estimator for one-pure-state fidelity estimation that functions without knowing which state is pure. By specializing the refined algorithmic Uhlmann transform, the authors demonstrate that the rank dependence vanishes completely in the pure-state limit, and the "unknown pure side" scenario is no longer more complex than the "known pure side" scenario in terms of query complexity. The results close the gap between upper and lower bounds for this specific fidelity estimation task, providing a theoretically optimal solution for a fundamental problem in quantum information theory.