Decision Effects, the Hahn–Jordan Decomposition for States, and Their Connection to Ludwig’s Axiomatic Approach to Quantum Mechanics

Abstract

We begin with a quantum logic carrying a large collection of states. We then form a dual pair of Banach spaces—base normed and order unit normed—containing the states and the logic, respectively. A Galois connection on the face lattices of the states and the dual positive order unit interval is introduced. The elements of the logic are connected to a dense subset of the extreme points of this order interval in the order unit space using a generalized form of the Hahn–Jordan decomposition theorem. Decision effects are defined and identified with the elements of the original logic. Finally, an important axiom of Ludwig is introduced which ties together all the lattices of Galois closed faces of states, Galois closed order intervals of the positive order unit interval, decision effects, and the original quantum logic. The emphasis here is on the consequences of functional analytic assumptions. The paper concludes with a simple example where Ludwig’s axiom does not hold and we see parts of the theory dissolve.