Condensed Matter Journal Club: "Von Neumann Algebras with Microscopic Data"

We consider a coarse graining map for an N-site Hamiltonian in the large-N limit. If the Hamiltonian meets sufficient conditions, the outcome of this map — rather than reducing the number of sites or the energy scale — is an algebra of observables which is not the default type for an infinitely entangled infinite dimensional Hilbert space (not type III but type II). Sufficient conditions for this to happen include that the map between default and corrected algebra is completely positive (a quantum channel). It should further (the following are equivalent): attain the equality case in the monotonicity of relative entropy; conserve the Bogoliubov–Kubo–Mori covariance between pairs of observables; and be the adjoint of the Petz recovery map.  In von Neumann algebra terms, we identify sufficient conditions for the existence of a type II-infty subfactor of a type III factor. Contrary to Witten's crossed product construction in holography, the type II-infty candidate is a conditional expectation of the type III parent (shrink instead of enlarge). Innerness of the modular automorphism and the existence of a faithful normal semifinite trace depend on the underlying Hamiltonian. In physics where both make sense, the shrinking and enlarging constructions may give dual algebras.