For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.

Jul 9, 20190 citations

Phase operators on Hilbert space

John A. Vaccaro

A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ . This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ and so the correponding limit points are ordinary operators on ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ . This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ and so the correponding limit points are ordinary operators on ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.

Physical Review AApr 1, 199525 citations

Linear Vector Spaces

V. Balakrishnan

Springer eBooksJan 1, 20200 citations

Vector Spaces and Linear Transformations

Ralph E. Christoffersen

Springer advanced texts in chemistryJan 1, 19890 citations

Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space

Vladimir Mikhailovich Busovikov, Alexander Nikolaevich Pechen, Vsevolod Zhanovich Sakbaev

We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.

arXiv:2406.10699Jun 15, 20241 citation

A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

and so the correponding limit points are ordinary operators on

{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}

. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.