Noncommutativity and transition probability in quantum mechanics (Q1308999)

For a `desirable' quantum logic a notion of transition probability between two pure states is defined with the help of the ampersand \& given by \(a \& b = (a \vee b^ \perp) \wedge b\). The supremum over the absolute value of all states evaluated at the ampersand commutator \(a \& b = b \& a\) serves as a measure of the `commutativity gap' between \(a\) and \(b\).