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Frontiers in Physics

Measures and operators associated with Parseval distribution frames

Camillo Trapanı, Francesco Tschinke

Published Oct 15, 2025• 4 citations
Publisher

Abstract

Continuing the study by Tschinke et al. (2019), we examine further aspects of distribution frames (namely, Gel’fand and Parseval), particularly regarding those that are more relevant for applications in quantum physics. Parseval distribution frames are, in particular, closely related to coherent states. Thus, POV measures, Naimark dilations, and operators defined by Parseval distribution frames are the main subjects of this paper. The main results are Theorems 2.2 and 3.1. Theorem 2.2 gives a sufficient conditions for the existence of such distribution coherent states for positive operator valued measures. Theorem 3.1 establishes conditions under which the distribution coherent states can be identified with the projections of some Gel’fand distribution basis in a larger Hilbert space (in Naimark's sense).

DOI
10.3389/fphy.2025.1640681

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Continuing the study by Tschinke et al. (2019), we examine further aspects of distribution frames (namely, Gel’fand and Parseval), particularly regarding those that are more relevant for applications in quantum physics. Parseval distribution frames are, in particular, closely related to coherent states. Thus, POV measures, Naimark dilations, and operators defined by Parseval distribution frames are the main subjects of this paper. The main results are Theorems 2.2 and 3.1. Theorem 2.2 gives a sufficient conditions for the existence of such distribution coherent states for positive operator valued measures. Theorem 3.1 establishes conditions under which the distribution coherent states can be identified with the projections of some Gel’fand distribution basis in a larger Hilbert space (in Naimark's sense).
The tight frames can be regarded as a particular case of POVMs (positive operator-valued measures describing generalized measurements), namely the case when all the operators are rank-one. Each orthonormal basis is a tight frame, and every tight frame, after the embedding into a higher-dimensional space, is the orthogonal projection of an orthonormal basis. There exist several POVM-based definitions of coherence, and they are well-investigated. Our aim is to identify properties specific to the particular case of tight frames, and to look for some applications. All the POVM-based definitions use a Naimark extension. The frame-dependent coherence can be regarded as a particular case of POVM-based coherence, but it can be defined directly, without to use a Naimark extension. Its definition is a direct generalization of the basis-dependent ℓ 1 -norm of coherence, and it offers a more accurate description because we can use a frame containing several orthogonal bases. A quasi frame-independent definition of coherence for qubit systems is presented.
Abstract It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a D{\mathcal{D}}D D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of EEE E -connection for observables. Several examples are discussed.
Abstract In this chapter we first review projective or von Neumann measurements from the point of view of the projection postulate and the Lüders rule, before turning to more general measurements modelled using positive operator-valued measures (POVMs). After examining their mathematical description, we discuss symmetric informationally complete (SIC) POVMs and Zauner’s conjecture, as well as the representation of POVMs as projective measurements on a larger Hilbert space, a result called Naimark dilation. We then a model for non-ideal projective measurements and analyse the properties of such measurements in relation to the properties unbiasedness, faithfulness, and non-invasiveness of ideal projective measurements. Finally, we consider the problem of distinguishing non-orthogonal quantum states and examine how the use of POVMs can be beneficial for this task.
In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.
© The Author(s) 2024. We present an operationally motivated treatment of quantum reference frames in the setting that the frame is a covariant positive operator valued measure (POVM) on a finite homogeneous space, generalising the principal homogeneous spaces studied in previous work. We focus on the case that the reference observable is the canonical covariant projection valued measure on the given space, and show that this gives rise to a rank-one covariant POVM on the group, which can be seen as a system of coherent states.