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Bounds for the $$C^*$$-algebraic transition probability yield best lower and upper bounds to the overlap. (English) Zbl 0674.46039
Summary: Bounds are proved for the $$C^*$$-algebraic transition probability $$P_ A(\omega,\nu)$$ between the abstract ground state $$\nu$$ with respect to a symmetric subspace N of a unital $$C^*$$ algebra A and a state $$\omega$$ with the restriction $$\omega | N=\sigma | N$$ to N for an arbitrarily given, but fixed state $$\sigma$$. A is assumed to be the unital $$C^*$$-algebra generated by N. The results are specified in the case where A is a subalgebra of a v.N. algebra is standard form and N is dimensionally finite. Under these assumptions, the relationships of the algebraic transition probability to the notion of the (square of the) overlap integral known in quantum physics are clearly established. The general resuls are used to treat the standard problem of finding upper and lower bounds to the overlap in a quantum mechanical context. The best bounds are found and their properties discussed.

##### MSC:
 46L60 Applications of selfadjoint operator algebras to physics 46L30 States of selfadjoint operator algebras 46N99 Miscellaneous applications of functional analysis
##### Keywords:
$$C^*$$-algebraic transition probability; state
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##### References:
 [1] DOI: 10.1016/0034-4877(76)90060-4 · Zbl 0355.46040 [2] DOI: 10.1002/andp.19854970419 · Zbl 0571.60007 [3] Alberti P. M., Wiss. Z. Karl-Marx-Univ. Leipzig MNR 34 pp 572– (1985) [4] DOI: 10.1063/1.1665372 [5] DOI: 10.1088/0305-4470/9/1/007 [6] DOI: 10.1007/BF00398708 · Zbl 0538.46051
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