Minimum number of experimental settings required to verify bipartite pure states and unitaries

TL;DR

This paper systematically studies the minimal number of experimental settings needed to verify bipartite pure states and unitaries, introducing efficient protocols and exploring entanglement-free methods.

Contribution

It establishes the minimal settings for verifying bipartite states and unitaries, including entanglement-free protocols, and analyzes properties of two-qubit unitaries.

Findings

Bipartite pure states verified with only two measurement settings.

Bipartite unitaries in dimension d verified with 2d settings.

Two-qubit unitaries verified with at most five settings.

Abstract

Efficient verification of quantum states and gates is crucial to the development of quantum technologies. Although the sample complexities of quantum state verification and quantum gate verification have been studied by many researchers, the number of experimental settings has received little attention and is poorly understood. In this work we study systematically quantum state verification and quantum gate verification with a focus on the number of experimental settings. We show that any bipartite pure state can be verified by only two measurement settings based on local projective measurements. Any bipartite unitary in dimension $d$ can be verified by $2d$ experimental settings based on local operations. In addition, we introduce the concept of entanglement-free verification and clarify its connection with minimal-setting verification. Finally, we show that any two-qubit unitary can be verified with at most five experimental settings; moreover, a generic two-qubit unitary (except for a set of measure zero) can be verified by an entanglement-free protocol based on four settings. In the course of study we clarify the properties of Schmidt coefficients of two-qubit unitaries, which are of independent interest.