Coherent quantum tomography

TL;DR

This paper introduces a quantum tomography method using weighted X-ray transforms to recover a position-dependent Hamiltonian matrix from quantum state evolution, with proofs of injectivity and discussions of physical applications.

Contribution

It provides a rigorous mathematical framework for a quantum tomography technique based on weighted X-ray transforms, including proofs of injectivity and analysis of physical relevance.

Findings

Weighted X-ray transforms are injective with rough weights.

The method can solve several inverse problems in quantum mechanics.

Some physically relevant problems remain open.

Abstract

We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previously described in the physical literature.