Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2305.08862

Transition Probabilities and Almost Periodic Functions

S. O. Juriaans, Paulo Queiroz

Published Jan 1, 2023• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.13140/rg.2.2.13388.08320

Related papers

Transition probabilities and almost periodic functions in colombeau algebras

S. O. Juriaans, Paulo Queiroz

Monatshefte für Mathematik•Sep 6, 2025•0 citations

Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

Tatsuya Tate

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.

arXiv:1411.4215•Infinite Dimensional Analysis Quantum Probability and Related Topics•Jun 1, 2019•9 citations

Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects

M. Ruzzi, Marcelo A. Marchiolli, Eduardo Costa da Silva +1 more

An approach featuring $s$-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.

arXiv:quant-ph/0602216•Feb 26, 2006•27 citations

Transition probabilities for non self-adjoint Hamiltonians in infinite dimensional Hilbert spaces

Fabio Bagarello

arXiv:1508.02572•Annals of Physics•Aug 14, 2015•16 citations

Are almost-symmetries almost linear?

Javier Cuesta, Michael M. Wolf

It d-pends. Wigner’s symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4εd–close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.

arXiv:1812.10019•Journal of Mathematical Physics•Aug 1, 2019•2 citations

Generalization of Gisin’s theorem to quantum fields

Konrad Schlichtholz, Marcin Markiewicz

Abstract We generalize Gisin’s theorem on the relation between the entanglement of pure states and Bell non-classicality to the case of mode entanglement of separated groups of modes of quantum fields extending the theorem to cover also states with undefined particle number. We show that any pure state of the field which contains entanglement between two groups of separated modes violates some Clauser–Horne (CH) inequality. In order to construct the observables leading to a violation in the first step, we show an isomorphism between the Fock space built from a single-particle space involving two separated groups of modes and a tensor product of two abstract separable Hilbert spaces spanned by formal monomials of creation operators. In the second step, we perform a Schmidt decomposition of a given entangled state mapped to this tensor product space and then we map back the obtained Schmidt decomposition to the original Fock space of the system under consideration. Such obtained Schmidt decomposition in Fock space allows for construction of observables leading to a violation of the CH inequality. We also show that our generalization of Gisin’s theorem holds for the case of states on non-separable Hilbert spaces, which physically represent states with actually infinite number of particles. Such states emerge, for example, in the discussion of quantum phase transitions. Finally, we discuss the experimental feasibility of constructed Bell test and provide a necessary condition for realizability of this test within the realm of passive linear optics.

arXiv:2308.14913•New Journal of Physics•Feb 1, 2024•3 citations
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=(Tε)T = (T_{\varepsilon})T=(Tε​) with Tε:H→HT_{\varepsilon} : \mathbb H \rightarrow \mathbb HTε​:H→H, H\mathbb HH 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.
The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.
An approach featuring sss-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.
It d-pends. Wigner’s symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4εd–close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.
Abstract We generalize Gisin’s theorem on the relation between the entanglement of pure states and Bell non-classicality to the case of mode entanglement of separated groups of modes of quantum fields extending the theorem to cover also states with undefined particle number. We show that any pure state of the field which contains entanglement between two groups of separated modes violates some Clauser–Horne (CH) inequality. In order to construct the observables leading to a violation in the first step, we show an isomorphism between the Fock space built from a single-particle space involving two separated groups of modes and a tensor product of two abstract separable Hilbert spaces spanned by formal monomials of creation operators. In the second step, we perform a Schmidt decomposition of a given entangled state mapped to this tensor product space and then we map back the obtained Schmidt decomposition to the original Fock space of the system under consideration. Such obtained Schmidt decomposition in Fock space allows for construction of observables leading to a violation of the CH inequality. We also show that our generalization of Gisin’s theorem holds for the case of states on non-separable Hilbert spaces, which physically represent states with actually infinite number of particles. Such states emerge, for example, in the discussion of quantum phase transitions. Finally, we discuss the experimental feasibility of constructed Bell test and provide a necessary condition for realizability of this test within the realm of passive linear optics.