Quantum State Tomography with incomplete data: Maximum Entropy and Variational Quantum Tomography

TL;DR

This paper compares and connects Maximum Entropy and Variational Quantum Tomography methods for estimating quantum states from incomplete data, showing their equivalence in certain cases and highlighting VQT's computational advantages.

Contribution

It introduces a variant of Variational Quantum Tomography, establishes its relationship with MaxEnt, and demonstrates their equivalence and efficiency in quantum state estimation.

Findings

VQT and MaxEnt are equivalent for eigenbasis measurements.

The modified VQT provides unbiased quantum state estimates.

VQT can be solved more efficiently via linear semidefinite programs.

Abstract

Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state can not be uniquely determined. In this case, among the density matrices compatible with the available data, it is commonly preferred that one which is the most uncommitted with the missing information. This is the purpose of the Maximum Entropy estimation (MaxEnt) and the Variational Quantum Tomography (VQT). Here, we propose a variant of Variational Quantum Tomography and show its relationship with Maximum Entropy methods in quantum tomographies with incomplete set of measurements. We prove their equivalence in case of eigenbasis measurements, and through numerical simulations we stress their similar behavior. Hence, in the modified VQT formulation we have an estimate of a quantum state as unbiased as in MaxEnt and with the benefit that VQT can be more efficiently solved by means of linear semidefinite programs.