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

Henrik Wilming

Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

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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).

The entropy of a quantum state or probability distribution is an important quantity in physics. Among others, its applications range from being a central quantity of interest in statistical mechanics and thermodynamics, over quantifying how much one can compress a signal in (quantum) information theory to quantifying the amount of entanglement contained in a quantum state. However, typically it attains its physical meaning only in an "asymptotic limit" where many weakly correlated copies of the state are available such as in a large, thermodynamic system or when long messages need to be compressed.

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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