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

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 \(V_ E(K_ q)\).

\((\alpha)\) \(V_ E\) 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 |

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