# Convexity properties of superpositions of degenerate bipartite eigenstates

## TL;DR

This paper investigates the convexity properties of entanglement measures in superpositions of degenerate bipartite eigenstates, revealing predictable convexity behavior based on shared entropy, with exact analysis of specific two-particle systems.

## Contribution

It demonstrates the convexity properties of von Neumann entropy in superpositions of degenerate eigenstates and provides a method to predict these properties using shared entropy measures.

## Key findings

- Von Neumann entropy exhibits definite convexity or concavity as a function of superposition parameter.
- Shared entropy at superposition extremes predicts the convexity behavior.
- Exact analysis of two-particle systems confirms theoretical predictions.

## Abstract

The entanglement content of superpositions of pairs of degenerate eigenstates of a bipartite system are considered in the case that both are also eigenstates of the $z$ component of the total angular momentum. It is shown that the von Neumann entropy of the state that is obtained tracing out one of the parts of the system has a definite convexity (concavity) as a function of the superposition parameter and that its convexity (concavity) can be predicted using a quantity of information that measures the entropy shared by the states at the extremes of the superposition. Several examples of two particle system, whose eigenfunctions and density matrices can be obtained exactly, are analyzed thoroughly.