A Spectral Order for Infinite Dimensional Quantum Spaces: A Preliminary Report

In 2002 Coecke and Martin created a Bayesian order for the finite dimensional spaces of classical states in physics and used this to define a similar order, the spectral order on the finite dimensional quantum states. These orders gave the spaces a structure similar to that of a domain. This allows for measuring information content of states and for determining which partial states are approximations of which pure states. In a previous paper the author extended the Bayesian order to infinite dimensional spaces of classical states. The order on infinite dimensional spaces retains many of the characteristics important to physics, but loses the domain theoretic structure. It becomes impossible to measure information content in the same way that it is done for the finite dimensional spaces, and the sense of approximation is lost. In this paper, we will use the Bayesian order to define a spectral order on the infinite dimensional spaces of quantum states.