Generalized collective quantum tomography: algorithm design, optimization, and validation

TL;DR

This paper introduces a generalized framework for collective quantum tomography, developing algorithms for state, detector, and process estimation that improve accuracy and efficiency, validated through numerical and experimental results.

Contribution

It extends collective quantum tomography to a broader setting and proposes new algorithms with analytical complexity and error analysis, utilizing sum of squares techniques.

Findings

Algorithms achieve lower mean squared errors than existing methods.

Numerical and experimental validation demonstrates effectiveness.

Approaches approach the theoretical MSE bound by leveraging purity information.

Abstract

Quantum tomography is a fundamental technique for characterizing, benchmarking, and verifying quantum states and devices. It plays a crucial role in advancing quantum technologies and deepening our understanding of quantum mechanics. Collective quantum state tomography, which estimates an unknown state \r{ho} through joint measurements on multiple copies $\rho\otimes\cdots\otimes\rho$ of the unknown state, offers superior information extraction efficiency. Here we extend this framework to a generalized setting where the target becomes $S_1\otimes\cdots\otimes S_n$, with each $S_i$ representing identical or distinct quantum states, detectors, or processes from the same category. We formulate these tasks as optimization problems and develop three algorithms for collective quantum state, detector and process tomography, respectively, each accompanied by an analytical characterization of the computational complexity and mean squared error (MSE) scaling. Furthermore, we develop optimal solutions of these optimization problems using sum of squares (SOS) techniques with semi-algebraic constraints. The effectiveness of our proposed methods is demonstrated through numerical examples. Additionally, we experimentally demonstrate the algorithms using two-copy collective measurements, where entangled measurements directly provide information about the state purity. Compared to existing methods, our algorithms achieve lower MSEs and approach the collective MSE bound by effectively leveraging purity information.