Hilbert-space-valued states on quantum logics

summary:We analyze finitely additive orthogonal states whose values lie in a real Hilbert space. We call them $h$-states. We first consider the important case of $h$-states on a standard Hilbert logic $L(H)$ of projectors in $H$-we describe the $h$-states $s$: $L(H_1) \rightarrow H_2$, where $\text {dim } H_2 \leq$ \text {dim} H_1 < \infty$. In particular, we show that, up to a unitary mapping, every $h$-state $s$: $L(H)\rightarrow H(3\leq \text {dim } H < \infty)$ has to be concentrated on a one-dimensional projection. We also study the $h$-states $s$: $L(H_1)\rightarrow H_2$ for the case of $\text {dim } H_1 = \infty$. The results of the first part complement the papers [10] and [13]. In the second part we investigate $h$-states on general logics. Being motivated by the quantum axiomatics, the main question we ask here is as follow: Given a Hilbert space $H$ with $\text {dim } H < \infty$, what is the class of such logics $L$ that, for any Boolean subalgebra $B$ of $L$, every $h$-states $s$: $B \rightarrow H$ extends over $L$? We answer this question by finding a simple condition characterizing the class (Theorem 3.4]. It turns out that the class is considerably large-it contains e.g. all concrete logics-but, on the other hand, it does not contain all finite logics (we construct a counterexample in the appendix)