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

It is well known that if a (finite-dimensional) density matrix rho has smaller entropy than rho 0, then the tensor product of sufficiently many copies of rho majorizes a quantum state arbitrarily close to the tensor product of correspondingly many copies of rho 0. In this short note I show that if additionally rank(rho) < rank(rho 0), then n copies of rho also majorize a state where all single-body marginals are exactly identical to rho 0 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(rho) < H(rho 0) and rank(rho) < rank(rho 0) there exists a finite dimensional density matrix sigma and a unitary U such that U rho (R) sigma U & DAG; has marginals rho 0 and sigma exactly. All the results transfer to the classical setting of probability distributions over finite alphabets with unitaries replaced by permutations.

Automated summary

This short note discusses a property of quantum states and density matrices. It states that if a density matrix ρ has a smaller entropy than ρ0, then the tensor product of sufficiently many copies of ρ can approximate a quantum state that is very close to the tensor product of the corresponding number of copies of ρ0. Additionally, if the rank of ρ is smaller than the rank of ρ0, then the tensor product of n copies of ρ can dominate a state with identical single-body marginals as ρ0 but allowing for arbitrary correlations.