Quantum relative entropy regularization for quantum state tomography

TL;DR

This paper introduces a regularization method using quantum relative entropy for quantum state tomography, providing theoretical foundations and practical algorithms for reconstructing quantum states from measurements.

Contribution

It establishes the regularizing properties of quantum relative entropy in state tomography and develops computational tools for practical implementation.

Findings

Proved lower semi-compactness of the quantum relative entropy functional.

Derived subgradient, proximal operator, and conjugate functional for quantum relative entropy.

Applied the method successfully to PINEM and optical homodyne tomography examples.

Abstract

The density matrix is a positive semidefinite operator of trace 1 characterizing the state of a quantum system. We consider the inverse problem to reconstruct such density matrices from indirect measurements, also known as quantum state tomography. To solve such inverse problems in high or infinite dimensional settings, we study variational regularization using the quantum relative entropy as penalty functional. Quantum relative entropy is an analog of the well-known maximum entropy functional with compositions of functions replaced by the spectral functional calculus. The main aim of this paper is to establish the regularizing property of this scheme. As a crucial intermediate step, we establish lower semi-compactness of the penalty functional with respect to the weak-$*$-topology. Moreover, we compute the subgradient, proximal operator, and conjugate functional of the quantum relative entropy on finite dimensional spaces. This enables us to apply iterative algorithms from convex optimization to solve the regularized problems numerically. To show the validity and practical value of our results, we apply our theory to the examples of Photon-Induced Near-field Electron Microscopy (PINEM) and to optical homodyne tomography.