Verification of state and entanglement with incomplete tomography

TL;DR

The paper introduces a new estimation scheme using semidefinite programming to verify quantum entanglement with incomplete measurement data, improving robustness and accuracy in quantum state analysis.

Contribution

It presents a novel method combining maximum-likelihood and maximum-entropy estimators for quantum states from incomplete data, enabling entanglement verification.

Findings

Effective estimation of quantum states with incomplete data

Ability to verify entanglement using entanglement witnesses

Enhanced computational robustness in likelihood and entropy maximization

Abstract

There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numerical scheme that is more computationally robust for the sole purpose of maximizing the likelihood and entropy.