Quantum Theory lives in complex Hilbert Spaces. The closed linear subspaces of seperable Hilbert Spaces
form a so called orthocomplemented and quasimodular lattice,
featuring - in opposition to classical (boolean) logic - an own "Logic of Quantum Mechanics"
(G.Birkhoff, J.v.Neumann, 1936).
These subspaces also replace the sigma-algebra of classical probability theory with probability measures
on it fixed via the Born Rule, resp. Gleason's theorem.
Allthough most of (current) quantum theory uses infinite dimensional (often even non-seperable)
Hilbert Spaces, interesting features of this logic and probability theory already appear in finite dimensions
(e.g. entanglement, Bell's inequality, GHZ states, the Kochen-Specker theorem,
the algorithms of quantum computing, etc.).
(Closed) linear subspaces can be described by orthogonal projection operators P
(= P2 = P*).
Quantum designs are sets of orthogonal projection matrices in d-dimensional Hilbert Spaces,
with certain features: A fundamental differentiation is whether the quantum design is regular,
this means that all assigned
subspaces have same dimension r, and here the special case r=1 (this means that the subspaces
are spanned by single (unit) vectors).
Here is an overview of the most important special cases
(as explained on the next pages):
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