A Note on the Estimation of Von Neumann and Relative Entropy via Quantum State Observers

TL;DR

This paper demonstrates that quantum state observers can reliably estimate the von Neumann and relative entropy of a quantum system, with exponential convergence guarantees, enhancing quantum information analysis.

Contribution

It introduces a method showing exponential convergence of von Neumann and relative entropy estimates via quantum state observers, a novel insight in quantum state estimation.

Findings

Von Neumann entropy estimate converges exponentially to the true entropy.

Relative entropy between system and observer's state converges exponentially to zero.

The method applies when the system starts in a full-rank state.

Abstract

An essential quantity in quantum information theory is the von Neumann entropy which depends entirely on the quantum density operator. Once known, the density operator reveals the statistics of observables in a quantum process, and the corresponding von Neumann Entropy yields the full information content. However, the state, or density operator, of a given system may be unknown. Quantum state observers have been proposed to infer the unknown state of a quantum system. In this note, we show (i) that the von Neumann entropy of the state estimate produced by our quantum state observer is exponentially convergent to that of the system's true state, and (ii) the relative entropy between the system and observer's state converges exponentially to zero as long as the system starts in a full-rank state.