We study transition probabilities of generalized functions as introduced by Colombeau and Gsponer. We formally introduz the study of H. Bohr almost periodic functions in the generalized context and use them to give exact values of transition probabilities in dimension 2. We also prove the existence of transition probabilities, in the generalized sense, for any moderate net $T = (T_{\varepsilon})$ with $T_{\varepsilon} : \mathbb H \rightarrow \mathbb H$ , $\mathbb H$ a Hilbert space. In particular, we consider the case of selfadjoint Hilbert-Schmidt operators. This alludes to the possibility of existence of transition probabilities in the Fock space, in the sense introduced by Colombeau and Gsponer, if the spectrum of the operators involved are pure infinities or infinitesimals. This was already indicated in two papers coauthored by J. Aragona and J.F. Colombeau et. al.We study transition probabilities of generalized functions as introduced by Colombeau and Gsponer. We formally introduz the study of H. Bohr almost periodic functions in the generalized context and use them to give exact values of transition probabilities in dimension 2. We also prove the existence of transition probabilities, in the generalized sense, for any moderate net $T = (T_{\varepsilon})$ with $T_{\varepsilon} : \mathbb H \rightarrow \mathbb H$, $\mathbb H$ a Hilbert space. In particular, we consider the case of selfadjoint Hilbert-Schmidt operators. This alludes to the possibility of existence of transition probabilities in the Fock space, in the sense introduced by Colombeau and Gsponer, if the spectrum of the operators involved are pure infinities or infinitesimals. This was already indicated in two papers coauthored by J. Aragona and J.F. Colombeau et. al.

arXiv:2305.08862Jan 1, 20230 citations

A generic approach to the quantum mechanical transition probability

Gerd Niestegge

In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these elements. A probabilistically motivated and more general definition of this transition probability was introduced in a preceding paper and is extended here to a general type of quantum logics: the orthomodular partially ordered sets. A very general version of the quantum no-cloning theorem, creating promising new opportunities for quantum cryptography, is presented and an interesting relationship between the transition probability and Jordan algebras is highlighted.In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these elements. A probabilistically motivated and more general definition of this transition probability was introduced in a preceding paper and is extended here to a general type of quantum logics: the orthomodular partially ordered sets. A very general version of the quantum no-cloning theorem, creating promising new opportunities for quantum cryptography, is presented and an interesting relationship between the transition probability and Jordan algebras is highlighted.

arXiv:2110.12754Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesApr 1, 20222 citations

T = (T_{\varepsilon})

with

T_{\varepsilon} : \mathbb H \rightarrow \mathbb H

\mathbb H

a Hilbert space. In particular, we consider the case of selfadjoint Hilbert-Schmidt operators. This alludes to the possibility of existence of transition probabilities in the Fock space, in the sense introduced by Colombeau and Gsponer, if the spectrum of the operators involved are pure infinities or infinitesimals. This was already indicated in two papers coauthored by J. Aragona and J.F. Colombeau et. al.