Covariant catalysis requires correlations and good quantum reference frames degrade little

Lauritz van Luijk, Reinhard F. Werner, and Henrik Wilming

Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

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Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little.

In quantum mechanics, transitions between quantum states can be made possible in the presence of a third quantum system, which returns to its initial state at the end — just like a catalyst in Chemistry. We show that if all transitions have to respect conservation laws, then a catalyst has to build up correlations to the system, the environment, or both. Otherwise, it cannot enable new transitions. Therefore, pure states are useless for catalysis because they cannot be correlated to other systems.

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