Hilbert-space-valued states on quantum logics
The authors study finitely additive orthogonal vector states (=normed measures) with values in a real Hilbert space (so called \(h\)-states). It is shown that there is no \(h\)-state on a Hilbert logic \(L(M)\) of projections in \(H\), \(\dim H \geq 3\), to a Hilbert space of a lesser dimension. Further, a Hilbert space \(H\), \(\dim H \geq 3\), has a finite dimension iff every \(h\)-state on \(L(H)\) to \(H\) is, up to a unitary mapping, concentrated on a one-dimensional projection. Finally, a characterization of logics with extension property with respect to \(h\)-states to a finite dimensional Hilbert space is given.
finitely additive orthogonal vector states
scientific article; zbMATH DE number 58780
scientific article