Introduction
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 sigmaalgebra 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 nonseperable)
Hilbert Spaces, interesting features of this logic and probability theory already appear in finite dimensions
(e.g. entanglement, Bell's inequality, GHZ states, the KochenSpecker theorem,
the algorithms of quantum computing, etc.).
(Closed) linear subspaces can be described by orthogonal projection operators P
(= P^{2} = P*).
Quantum designs are sets of orthogonal projection matrices in ddimensional 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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