A Partial Order on Classical and Quantum States

Abstract

We introduce a partial order on classical and quantum mixed states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the partial orders involved conclusively establishes that physical information has a natural domain theoretic structure. For example, the set of maximal elements in the domain of quantum states is precisely the set of pure states, while the completely mixed ensemble $I/n$ is the order theoretic least element ⊥.

In the same way that the order on a domain provides a rigorous qualitative defini- tion of information, a special type of mapping on a domain called a measurement provides a formal account of the intuitive notion “information content.” Not only is physical information domain theoretic, but so too is physical entropy: Shannon entropy is a measurement on the domain of classical states; von Neumann entropy is a measurement on the domain of quantum states.

These results yield a foundation from which one can (a) reason qualitatively about probability, (b) derive the lattices of Birkhoff and von Neumann in a unified manner, suggesting that domains may provide a formalism for the logic of partial belief, and (c) develop new techniques for studying phenomena like noise and entanglement. Along the way, new lines of investigation open up within various subdisciplines of physics, mathematics and theoretical computer science.