Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

It is well known that if a (finite-dimensional) density matrix ρ has smaller entropy than ρ0, then the tensor product of sufficiently many copies of ρ majorizes a quantum state arbitrarily close to the tensor product of correspondingly many copies of ρ0. In this short note I show that if additionally rank(ρ) ≤ rank(ρ0), then n copies of ρ also majorize a state where all single-body marginals are exactly identical to ρ0 but arbitrary correlations are allowed (for some sufficiently large n). An immediate application of this is an affirmative solution of the exact catalytic entropy conjecture introduced by Boes et al. [PRL 122, 210402 (2019)]: If H(ρ) < H(ρ0) and rank(ρ) ≤ rank(ρ0) there exists a finite dimensional density matrix σ and a unitary U such that Uρ⊗ σU has marginals ρ0 and σ exactly. All the results transfer to the classical setting of probability distributions over finite alphabets with unitaries replaced by permutations.