The Quantum Logic of Direct-Sum Decompositions

Abstract

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space-which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space-which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over Z2 where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF (q) is analyzed with computations for the case of QM/Sets where q = 2. 1 2 (3 2 −1−2 2 ) = 7×3 3 2 1 2 (4) = 7 × 4 = 28. DSDs in DSD Z 3 2 = all binary DSDs for Z 3 2 .

Quantum logical entropy: fundamentals and general properties

“…Every projection-valued measure (PVM) defines a partition p in a Hilbert space in the sense that it provides a directsum decomposition of the space [24,25]. In fact, recall that a PVM is characterized by parts B i = |b i ihb i | such that P i B i ¼ I.…”

Quantum logical entropy: fundamentals and general properties

“…Every projection-valued measure (PVM) defines a partition π in a Hilbert space in the sense that it provides a direct-sum decomposition of the space [24,25]. In fact, recall that a PVM is characterized by parts B i = |b i b i | such that i B i = I.…”