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  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]].\nThe *quantum logic* of $H$ is the set $\\mathcal{L}(H)$ of closed linear subspaces of $H$, with the following structure:\n\n1. the order is inclusion $\\subseteq$;\n2. the meet $M \\wedge N$ is the intersection $M \\cap N$;\n3. the join $M \\vee N$ is the closure of the sum $M + N$;\n4. the negation $M^\\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $M$.\n\nThe elements of $\\mathcal{L}(H)$ are called *propositions*.\nMore abstractly, a *quantum logic* is an [[orthomodular lattice|6FPKEFX0X4]], whose elements are the propositions.\n\n## Remarks\n\nWith this structure $\\mathcal{L}(H)$ is an orthomodular lattice with least element $\\set{0}$ and greatest element $H$.\nAuthors differ on the abstract notion: some use orthomodular partially ordered sets, often with countable joins of orthogonal elements.\n\nExample: in $H = \\C^2$, let $A$, $B$ and $C$ be the lines spanned by $(1, 0)$, $(0, 1)$ and $(1, 1)$.\nThen $A \\wedge (B \\vee C) = A \\wedge \\C^2 = A$, while $(A \\wedge B) \\vee (A \\wedge C) = \\set{0}$.\nSo the distributive law of classical logic fails in $\\mathcal{L}(\\C^2)$.\n\nNon-example: the hexagon ortholattice $O_6$, with $0 \u003c a \u003c b \u003c 1$, $0 \u003c b^\\perp \u003c a^\\perp \u003c 1$ and no other relations, is orthocomplemented.\nIt 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.\n",
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