Editing Quantum logic (section)

=== Quantum probability measures ===
{{Main|Gleason's theorem|Quantum statistical mechanics}}
A ''quantum probability measure'' is a function P defined on ''Q'' with values in [0,1] such that  P("⊥)=0, P(⊤)=1 and if {''E''<sub>''i''</sub>}<sub>''i''</sub> is a sequence of pairwise-orthogonal elements of ''Q'' then
: <math> \operatorname{P}\!\left(\bigvee_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i). </math>

Every quantum probability measure on the closed subspaces of a Hilbert space is induced by a [[density matrix]]&nbsp;&mdash; a [[Positive operator|nonnegative operator]] of [[trace (linear algebra)#Generalizations|trace]] 1.  Formally,
: '''Theorem'''.<ref>[[Andrew Gleason|A. Gleason]], "Measures on the Closed Subspaces of a Hilbert Space", ''Indiana University Mathematics Journal'', vol.&nbsp;6, no.&nbsp;4, 1957.  pp.&nbsp;885-893.  DOI:&nbsp;[http://dx.doi.org/10.1512/iumj.1957.6.56050 10.1512/iumj.1957.6.56050].  Reprinted in ''The Logico-Algebraic Approach to Quantum Mechanics'', University of Western Ontario Series in Philosophy of Science 5a, ed.&nbsp;C.&nbsp;A. Hooker; D. Riedel, c.&nbsp;1975-1979.  pp.&nbsp;123-133.</ref> Suppose ''Q'' is the lattice of closed subspaces of a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure ''P'' on ''Q'' there exists a unique [[trace class]] operator ''S'' such that <math display=block>\operatorname{P}(E) = \operatorname{Tr}(S E)</math> for any self-adjoint projection ''E'' in ''Q''.