Notes on Some Questions in Mathematical Physics and Quantum Information

摘要

This is a set of notes on some unrelated topics in mathematical physics, atvarying levels of detail. First, I consider certain questions relating to thedecay of correlation functions in matrix product states, in particular thosegenerated by quantum expanders. This is discussed in relation to recent resultsof Brandao and Horodecki on area laws on systems with exponentially decayingcorrelation function\cite{areaexp}. Second, I consider some difficulties intrying to construct a tensor product state (or PEPS) describing atwo-dimensional fermionic system with non-vanishing Hall conductance. Third, Ipresent some relations between the theory of almost commuting matrices and thatof vector bundles, making the connection between the classifications moreexplicit. Fourth, I present an open question about quantum channels, and somepartial results.