Well-posedness and instability of free electron quantum tomography

TL;DR

This paper investigates the mathematical properties of reconstructing quantum states of free electrons from experimental data, revealing that while the problem is well-posed, it suffers from severe instability and slow convergence of estimators.

Contribution

It demonstrates the well-posedness of the quantum tomography inverse problem using positive semi-definiteness, and uncovers its inherent instability and lack of stability estimates, with implications for practical reconstruction.

Findings

The inverse problem is well-posed due to positive semi-definiteness.

No global stability estimates exist, leading to arbitrarily slow convergence.

Numerical experiments confirm instability and constrained problem stability.

Abstract

Recent advancements in photon induced near-field electron microscopy (PINEM) enable the preparation, coherent manipulation and characterization of free-electron quantum states. The available measurement consists of electron energy spectrograms and the goal is the reconstruction of a density matrix representing the quantum state. This requires the solution of a constrained linear inverse problem, where a positive semi-definite trace-class operator is reconstructed given its diagonal in different bases. We show the well-posedness of this problem by exploiting the regularizing effect of the positive semi-definiteness constraint. Unusually, well-posedness in this case does not imply any stability estimates. We show that no global stability estimates exist and any estimator converges arbitrarily slowly. We also provide further bounds on the instability generally complementing the analysis done in [arXiv:1907.03438]. Furthermore, we derive a decomposition of the discretized operator which allows us to study its injectivity and stability properties. It also leads to a faster implementation which we exploit in numerical experiments validating the instability estimates and the stability of the constrained problem.