Ellerman, David
(2017)
The Quantum Logic of DirectSum Decompositions:
The Dual to the Quantum Logic of Subspaces.
[Preprint]
Abstract
ince 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 spacewhich is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition (or quotient set or equivalence relation) is dual (in a categorytheoretic 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 set partitions to the quantum logic of directsum decompositions (i.e., the vector space version of a set partition) of a general vector spacewhich can then be specialized to the directsum decompositions of a Hilbert space. This allows the quantum logic of directsum decompositions to express measurement by any selfadjoint operators. The quantum logic of directsum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.
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