Spectral properties of observables and convex mappings in quantum mechanics. (English) Zbl 0790.46054

Cassinelli, Gianni; Lahti, Pekka J.

J. Math. Phys. 34, No. 12, 5468-5475 (1993).

Observables of the physical system are represented as normalized positive-operator-valued measures \(E\) on some measurable space \((\Omega,{\mathcal B}(\Omega))\), whereas its states are represented as positive trace class operators \(T\) of trace one. Observables \(E\) can be identified with the following convex mappings \(V_ E\) sending states into probability measures: \[ E:T\to V_ E(T)(X):= \text{tr}[TE(X)], \qquad X\in {\mathcal B}(\Omega). \] The authors obtain here the following theorem on the PV observables represented as normalized projection- operator-valued measure \(E\). Let \(E_ q\) be the set of states. If \(E\) is a PV observable on \((\Omega,{\mathcal B}(\Omega))\), then the following three conditions are equivalent:
\((\alpha)\) \(V_ E\) is surjective;
\((\beta)\) \(\text{supp}(E)= \{\omega\in\Omega\mid E(\{\omega\})\neq 0\}\);
\((\gamma)\) \(V_ E(K_ q)\) has the Krein-Milman property. That is, the closed convex hull of the extreme elements of \(V_ E(K_ q)\) is equal to \(V_ E(K_ q)\).

Reviewer: H.Yamagata (Osaka)

MSC:

46N50	Applications of functional analysis in quantum physics
47N50	Applications of operator theory in the physical sciences
81T05	Axiomatic quantum field theory; operator algebras

Keywords:

observables; normalized positive-operator-valued measures; states; positive trace class operators; Krein-Milman property

References:

[1]	DOI: 10.1007/BF01889303 · doi:10.1007/BF01889303
[2]	DOI: 10.1007/BF00671008 · doi:10.1007/BF00671008
[3]	DOI: 10.1007/BF00731904 · doi:10.1007/BF00731904