Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
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Abstract
I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC).
Featured image: Three conditions on a pair of finite-dimensional density matrices $(rho,rho’)$ are equivalent. Top: Entropy is strictly increasing and rank non-decreasing. Middle: There exists a finite-dimensional auxiliary system in state $sigma$ and a joint-unitary $U$ such that the transformed state $Urhootimes sigma U^dagger$ has reduced density matrices $rho’$ and $sigma$. This provides a single-shot characterization of entropy. Bottom: There exists a finite number $n$ such that $n$ copies of $rho$ majorize a state whose marginals all exactly coincide with $rho’$.
Popular summary
In the paper, a conjecture is solved in the affirmative which implies that one can think of entropy without an asymptotic limit. Instead it is asked when it is the case that a system’s statistical state (density matrix) can be transformed to a different one using unitary dynamics if one has access to a finite auxiliary system whose statistical state must not change in the process. The auxiliary system is refereed to as catalyst, as it enables state-transitions otherwise impossible while not changing its own state. The results of the paper show that a system’s state can be transformed from one state to another using a suitable catalyst if and only if the entropy increases (and the rank of the density matrix does not decrease).
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