Nearly tight lower bounds for estimating quantum functionals: Uhlmann fidelity, trace distance, and von Neumann entropy

2026-08-03

AI Summary

This study addresses the problem of establishing lower bounds on the sample complexity required to estimate fundamental quantum state functionals—such as Uhlmann fidelity, trace distance, and von Neumann entropy. By integrating techniques from quantum information theory, complexity analysis, and sample-to-query lifting, the authors develop a unified framework for proving such lower bounds. Their work yields nearly tight bounds for the first time: estimating these quantities requires $\widetilde{\Omega}(N^2)$ copies of the quantum state, which translates to a $\widetilde{\Omega}(N)$ lower bound on the number of quantum queries. These results match the best-known upper bounds, thereby confirming the near-optimality of more than a dozen quantum algorithms developed since 2016 for these estimation tasks.

Abstract

In this paper, we present a unified framework for proving lower bounds for estimating functionals of quantum states. We therefore resolve several open problems by establishing lower bounds that match known upper bounds: we show that it requires $\widetildeΩ(N^2)$ samples to estimate the Uhlmann fidelity, trace distance, and von Neumann entropy. Moreover, they immediately imply matching query lower bounds of $\widetildeΩ(N)$ by quantum sample-to-query lifting. These lower bounds imply the near-optimality of a dozen quantum algorithms since 2016.

Problem

• quantum functionals
• Uhlmann fidelity
• trace distance
• von Neumann entropy
• sample complexity

Innovation

• quantum lower bounds
• Uhlmann fidelity
• trace distance
• von Neumann entropy
• sample complexity