Partitions: The Logical Concept to Describe Indefiniteness and Definiteness

TheRohrlich, Fritz basic non-classical notion in quantum mechanics (QM) is the notion of superposition; entanglement is a particularly vexing special case. This chapter focuses on slowly developing the notion of a superposition state starting at the logical level of superposition applied to the notion of a set. To mathematically represent a superposition set, one must go beyond the one-dimensional notion of a 0, 1-vector representing which elements of the universe set are in the subset. A two-dimensional matrix will do the job with the 0, 1-vector along the diagonal and the off-diagonal elements of 1 representing which elements are superposed in the sense of being in the same equivalence class of an equivalence relation (which is just another way to look at a partition). Those 0, 1-matrices are the incidence matrices of equivalence relations. When such logical representations of equivalence relations are normalized by dividing through by the trace (sum of the diagonal elements), one arrives at a density matrix–which is the partitional way to represent a quantum state. The three main distinctive elements of QM math are the notions of a quantum state, a quantum observable, and a (projective) measurement applying an observable to a state. This chapter gives the partition-based treatment of quantum states via density matrices.