Quantum state tomography as a numerical optimization problem

TL;DR

This paper introduces a numerical optimization framework for quantum state tomography applicable to various systems, including subsystems, demonstrated on a six-dimensional qubit-qutrit system, improving measurement schemes beyond qubits-only configurations.

Contribution

It formulates quantum state tomography as a flexible optimization problem, extending optimal measurement schemes to higher-dimensional systems like qubit-qutrit setups.

Findings

Optimized measurement schemes for six-dimensional systems were developed.

Mutually unbiased subspaces can be approximated in dimension six with negligible deviation.

The method surpasses previous qubits-only approaches by generalizing to subsystems.

Abstract

We present a framework that formulates the quest for the most efficient quantum state tomography scheme as an optimization problem which can be solved numerically. This approach can be applied to a broad spectrum of relevant setups including measurements restricted to a subsystem. To illustrate the power of this method we present results for the six-dimensional Hilbert space constituted by a qubit-qutrit system, which could be realized e.g. by the N-14 nuclear spin-1 and two electronic spin states of a nitrogen-vacancy center in diamond. Measurements of the qubit subsystem are expressed by projectors of rank three, i.e., projectors on half-dimensional subspaces. For systems consisting only of qubits, it was shown analytically that a set of projectors on half-dimensional subspaces can be arranged in an informationally optimal fashion for quantum state tomography, thus forming so-called mutually unbiased subspaces. Our method goes beyond qubits-only systems and we find that in dimension six such a set of mutually-unbiased subspaces can be approximated with a deviation irrelevant for practical applications.