Quantum states as countable convex combination of pure states with bounded energy

TL;DR

This paper investigates the structure of quantum states with bounded energy in infinite-dimensional systems, showing such states can be expressed as countable convex combinations of pure states, and explores implications for entropy continuity.

Contribution

It demonstrates that in infinite dimensions, states with bounded energy can be decomposed into pure states with bounded energy, and applies this to entropy continuity bounds.

Findings

States with bounded energy can be expressed as countable convex combinations of pure states.

Such states do not exist under Gibbs' hypothesis for certain Hamiltonians.

The von Neumann entropy's continuity is analyzed using the Alicki-Fannes-Winter technique.

Abstract

We give response to the question: in infinite dimension states,given a state with energy bounded by E, we can write the state as a countable convex combination of pure states with energy bounded by E. We review the Alicki-Fannes-Winter technique to obtain a uniform continuity bound for the von Neumann entropy in states that are a mix of pure states with bounded energy, using this bound we conclude that for a Hamiltonian satisfying the Gibb's hypothesis such states cannot exist.