# The quantum Shannon-McMillan theorem and rank of spectral projections of macroscopic observables

# The quantum Shannon-McMillan theorem and rank of spectral projections of macroscopic observables

The classical Shannon-McMillan theorem states that an ergodic system has typical sets satisfying the asymptotic equipartition property. This theorem demonstrates the significance of entropy which gives the size of the typical sets. There has recently been great progress in the quantum version of the Shannon-McMillan theorem .In particular, Bjelakovic et al. proved Shannon-McMillan theorem for ergodic quantum spin systems, by reducing the quantum setting to a classical one, and applying the classical Shannon-McMillan theorem. In this talk, I would like to introduce a direct proof of the quantum Shannon-McMillan theorem, without relying on the classical theory. Our proof is based on the variational principle, which is a well-known thermodynamic property of quantum spin systems. As an application, I would like to show that the quantum Shannon-McMillan theorem gives an estimate on rank of spectral projections of macroscopic observables. Using this estimate, we can show that a set of macroscopic observables can be approximated by commuting ones in the norm topology.