Canonical Quantum Coarse-Graining and Surfaces of Ignorance

In this paper we introduce a canonical quantum coarse-graining and use negentropy to connect ignorance as measured by quantum information entropy and ignorance related to quantum coarse-graining. For our procedure, macro-states are the set of purifications $\{|\bar{\Gamma}^{\rho}\rangle\}$ associated with density operator $\rho$ and micro-states are elements of $\{|\bar{\Gamma}^{\rho}\rangle\}$. Unlike other quantum coarse-graining procedures, ours always gives a well-defined unique coarse-graining of phase space. Our coarse-graining is also unique in that the volumes of phase space associated with macro-states are computed from differential manifolds whose metric components are constructed from the Lie group symmetries that generate $\{|\bar{\Gamma}^{\rho}\rangle\}$. We call these manifolds surfaces of ignorance, and their volumes quantify the lack of information in $\rho$ as measured by quantum information entropies. To show that these volumes behave like information entropies, we compare them to the von Neumann and linear entropies for states whose symmetries are given by $SO(3)$, $SU(2)$, and $SO(N)$. We also show that our procedure reproduces features of Boltzmann's original coarse-graining by showing that the majority of phase space consists of states near or at equilibrium. As a consequence of this coarse-graining, it is shown that an inherent flag variety structure underlies composite Hilbert spaces.