Equivalence of the catalytic entropy conjecture and the transitivity conjecture is plausible

Abstract

In essence, the catalytic entropy conjecture states that for a given initial state and a target state with higher entropy there exists a unitary operator acting on the tensor product of the initial state and a state of an auxiliary system such that the marginal states after the action of the unitary are precisely the target state and the same auxiliary state respectively. Since the state of the auxiliary system remains unchanged, we refer to the auxiliary system as the catalyst and to the whole operation as a catalytic transition. The primary aim of this work is to show that the catalytic entropy is equivalent to the transitivity conjecture: if there exist two catalytic transitions leading from one state to another through an intermediate state, then there exists a compound transition from the initial state directly to the target state. This goal is achieved partially: we outline a two-part proof of equivalence of the catalytic entropy conjecture and the transitivity conjecture, rigorously prove the first step, and provide numerical evidence of plausibility of the second step in the $3$-dimensional case for precision-based versions of the two conjectures. In addition, we generalize the family of catalytic transitions introduced in a paper by Sparaciari, Jennings, and Oppenheim to an arbitrary dimension, as well as show that the transitivity conjecture is a nontrivial statement by proving that a naive approach to proving it fails.