Instance-Optimal Quantum State Certification with Entangled Measurements

2025-07-08

AI Summary

This work addresses the instance-optimal sample complexity of quantum state certification when the tester is allowed arbitrary (including entangled) measurements—a long-standing open problem. Prior results established instance-optimal bounds only for unentangled measurements; the entangled case remained unresolved. We resolve this by deriving tight instance-optimal upper and lower bounds for quantum state certification under general measurements, showing that the optimal sample complexity fundamentally depends on the fidelities between the unknown state, the hypothesized state, and the maximally mixed state. Technically, we introduce a quantum extension of the Ingster–Suslina method to prove the lower bound and design a matching entangled measurement protocol achieving the upper bound. Our results not only settle the open problem but also recover, within a unified framework, the classical lower bound for quantum mixedness testing—thereby providing new theoretical benchmarks and analytical tools for quantum state verification.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We consider the task of quantum state certification: given a description of a hypothesis state $σ$ and multiple copies of an unknown state $ρ$, a tester aims to determine whether the two states are equal or $ε$-far in trace distance. It is known that $Θ(d/ε^2)$ copies of $ρ$ are necessary and sufficient for this task, assuming the tester can make entangled measurements over all copies [CHW07,OW15,BOW19]. However, these bounds are for a worst-case $σ$, and it is not known what the optimal copy complexity is for this problem on an instance-by-instance basis. While such instance-optimal bounds have previously been shown for quantum state certification when the tester is limited to measurements unentangled across copies [CLO22,CLHL22], they remained open when testers are unrestricted in the kind of measurements they can perform. We address this open question by proving nearly instance-optimal bounds for quantum state certification when the tester can perform fully entangled measurements. Analogously to the unentangled setting, we show that the optimal copy complexity for certifying $σ$ is given by the worst-case complexity times the fidelity between $σ$ and the maximally mixed state. We prove our lower bounds using a novel quantum analogue of the Ingster-Suslina method, which is likely to be of independent interest. This method also allows us to recover the $Ω(d/ε^2)$ lower bound for mixedness testing [OW15], i.e., certification of the maximally mixed state, with a surprisingly simple proof.

Problem

• Determining optimal copy complexity for quantum state certification with entangled measurements
• Establishing instance-specific bounds for certifying quantum states
• Developing a quantum analogue of the Ingster-Suslina method for lower bounds

Innovation

• Uses entangled measurements for state certification
• Proves instance-optimal bounds with fidelity
• Novel quantum Ingster-Suslina method