Quantum state reconstruction from dynamical systems theory

TL;DR

This paper introduces a new method using the physical imposition operator to efficiently find all solutions, called Pauli partners, for quantum state reconstruction with incomplete measurements, significantly advancing computational techniques in quantum information.

Contribution

The paper presents a novel algorithm leveraging fixed points of the physical imposition operator to efficiently identify all Pauli partners in quantum state reconstruction.

Findings

Found 24 mutually unbiased bases in dimension 23 in less than 30 seconds

Demonstrated the algorithm's efficiency in identifying solutions for quantum state reconstruction

Suggested potential applications to construct MU Constellations, SIC-POVMs, and quantum t-Designs

Abstract

When an informationally incomplete set of observables is considered there are several solutions to the quantum state reconstruction problem using von Neumann measurements. The set of solutions are known as Pauli partners, which are not easy to find even numerically. We present, in a self-contained paper, a new way to find this solutions using the physical imposition operator. We show that every Pauli partner is an attractive fixed point of this operator, which means that we can find complete sets of Pauli partners very efficiently. As a particular case, we found numerically 24 mutually unbiased bases in dimension N=23 in less than 30 seconds in a standard PC. We hope that the algorithm presented can be adapted to construct MU Constellations, SIC-POVMs, Equiangular Tight Frames and Quantum t-Designs, which could open new possibilities to find numerical solutions to these open problems related with quantum information theory.