Divergence of Lubkin's series for a quantum subsystem's mean entropy

TL;DR

This paper investigates Lubkin's series approximation for the mean entropy of a quantum subsystem, proving it converges only for systems of dimension two or less, despite the approximation's overall correctness.

Contribution

We derive an exact formula for mean traces and establish the convergence conditions of Lubkin's series, clarifying its limitations.

Findings

Lubkin's series converges only when subsystem dimension m ≤ 2.

The series provides correct entropy approximation despite convergence issues.

Exact mean trace formulas enable precise convergence analysis.

Abstract

In 1978, Lubkin proposed a method of approximating the mean von Neumann entropy for a subsystem of a finite-dimensional quantum system in an overall pure state by expanding the entropy as a series in terms of the mean trace of powers of the system's reduced density operator, but the convergence of this series was never established. We find an exact closed form expression for the mean traces, which enables us to prove that the series converges if and only if the system's dimension $m\le2$, in spite of the fact that Lubkin's proposed approximation for the entropy is now known to be correct.