arxiv.org/pdf/2303.17765

5/5
Von Neumann projections connect to quantum logic, a generalization of classical logic in quantum mechanics. Projections represent propositions about physical systems, playing a role in #quantumcomputing, #quantuminformationscience, and #quantum#communication.

#qis #quantumlogic #logic #quantum

4/5
Murray and von Neumann introduced a theory of #equivalence for projections. Two projections are equivalent if a partial isometry maps one’s range to the other. This equivalence is foundational for the classification of von Neumann algebras.

3/5
Von Neumann algebras are generated by their projections, meaning any element can be approximated by linear combinations of projections. This is analogous to how simple
#functions
are dense in L∞ spaces in classical #measuretheory.

2/5
In a von Neumann algebra, a projection E satisfies two properties:

#Self-adjoint (E = E*).
#Idempotent (E = E²).
Geometrically, projections map vectors onto closed #subspaces of a #Hilbert space, with the subspace being the "range" of E.