Quantum logic

Remarks

With this structure $\mathcal{L}(H)$ is an orthomodular lattice with least element $\set{0}$ and greatest element $H$. Authors differ on the abstract notion: some use orthomodular partially ordered sets, often with countable joins of orthogonal elements.

Example: in $H = \C^2$, let $A$, $B$ and $C$ be the lines spanned by $(1, 0)$, $(0, 1)$ and $(1, 1)$. Then $A \wedge (B \vee C) = A \wedge \C^2 = A$, while $(A \wedge B) \vee (A \wedge C) = \set{0}$. So the distributive law of classical logic fails in $\mathcal{L}(\C^2)$.

Non-example: the hexagon ortholattice $O_6$, with $0 < a < b < 1$, $0 < b^\perp < a^\perp < 1$ and no other relations, is orthocomplemented. It is not orthomodular, since $a \le b$ but $a \vee (a^\perp \wedge b) = a \vee 0 = a \ne b$; so it is not a quantum logic in the abstract sense.