Quantum Nonlinear Properties from a Single Measurement Setting

TL;DR

This paper introduces a universal, single-measurement protocol called collision-based nonlinear estimation (CBNE) for efficiently measuring nonlinear properties of quantum states, significantly simplifying experimental requirements.

Contribution

The authors develop CBNE, a method that enables nonlinear quantum state measurements with only one measurement setting, contrasting with traditional multi-setting approaches.

Findings

CBNE requires only a single measurement setting for large systems or with ancillary qubits.

It allows simultaneous estimation of multiple nonlinear functions of quantum states.

The method extends to estimating principal components and partial-transpose moments.

Abstract

Nonlinear properties of quantum states are essential to quantum information and many-body physics, but assessing them experimentally is challenging, as it typically requires multi-copy operations or a large number of measurement settings. To address this challenge, we develop a universal framework, collision-based nonlinear estimation (CBNE), for efficiently measuring nonlinear quantities of a quantum state $\rho$, such as the higher-order expectation value ${\rm tr}(O\rho^t)$ for some observable $O$, using single-copy randomized measurements. Strikingly, our protocol requires only a single measurement setting, provided that the system dimension is sufficiently large or a few ancillary qubits are available; this contrasts with the conventional expectation that multiple measurement bases are necessary for nonlinear estimation. In addition, CBNE is observable-independent at the experimental stage, which enables simultaneous estimation of multiple nonlinear functions. It further extends to broader tasks, including the estimation of principal component properties and partial-transpose moments of quantum states. Our results provide a practical and scalable route for measuring nonlinear state properties on near-term quantum devices.