Tomography by Design: An Algebraic Approach to Low-Rank Quantum States

TL;DR

This paper introduces an algebraic quantum state tomography method that efficiently reconstructs low-rank quantum states using observable measurements and matrix completion techniques, offering deterministic guarantees and broad applicability.

Contribution

It develops a novel algebraic matrix completion framework for low-rank quantum state tomography, improving computational efficiency and providing deterministic recovery guarantees.

Findings

Efficient algebraic algorithm for low-rank quantum state reconstruction

Applicable to a broad class of mixed quantum states

Provides deterministic recovery guarantees

Abstract

We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.