A thermofield-double model of Uhlmann’s anholonomy

A bstract Recently it has been shown that certain aspects of changing states in holography can be studied by considering the Berry Phase properties of simple entangled qubit systems. In this context the term wormhole quantum mechanics has been coined. In particular an interesting interplay between entanglement and Berry Phase was found. Here we present a mixed state generalization of this connection illustrated for a simple parametrized family of entangled quantum systems in the thermofield double state. Our state is purifying two subsystems dubbed the left and the right ones and evolves in a purely geometric manner based on a parallel transport condition due to Uhlmann. We show that different interpretations of this evolution relative to observers either coupled to the left or to the right exist. When the evolution on say the left manifests itself via local operations of non-unitary optimal measurements then on the right the evolutionary steps are organized into a sequence of unitary operations of a holonomic quantum computation. For closed curves featuring geodesic segments with respect to the Bures metric we calculate the anholonomy of Uhlmann’s connection which turns out to be related to the one of higher dimensional instantons. The initial and final states are having the same entanglement but different Uhlman’s Phase. We show that by conducting an interference experiment one can observe the physical effects of the resulting anholonomic quantum computation. The consequences of our findings in the holographic context are discussed.