Basis Of Hilbert Space Research Articles

Coherent Generation of the Complete High-Dimensional Bell Basis by Adaptive Pump Modulation

The Bell basis, a set of maximally entangled biphoton state, is a critical prerequisite towards quantum information processing, and many quantum applications have highlighted the requirement for the manipulation of high-dimensional Bell basis. While the Bell states can be created by using ingenious single-photon quantum gates, its implementation complexity in higher dimensions is significantly increased. Here we present an elaborate approach to show that the adaptive pump modulation enable the efficient preparation of Bell basis in arbitrary-dimensional Hilbert space. A complete set of four-dimensional orbital angular momentum Bell states are experimentally created, yielding high fidelities for certifying the entanglement dimensionality. Our strategy can be simply generalized to prepare more complex forms of quantum states even exploiting other physical degrees of freedom. Also, it can facilitate the use of high-dimensional entanglement in a variety of quantum protocols, in particular those requiring quantum dense coding.

Computing the q-Numerical Range of Differential Operators

A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of thez Hilbert space. In this paper, we establish an approximation of the q-numerical range of bounded and unbounnded operator matrices by variational methods. Application to Schrödinger operator, Stokes operator, and Hain-Lüst operator is given.

Some results on U-cross Gram matrices by using K-frames

U-cross Gram matrices are produced by frames and Riesz bases. In this paper, we represent bounded operators as matrix operators using K-frames. We study the invertibility matrices respect to K-frames. Moreover, we apply the concept of K-Riesz bases in Hilbert space $${\mathcal {H}}$$ to the concept of matrix induced by U with respect to K-Riesz bases.

Generalization properties of neural network approximations to frustrated magnet ground states

Neural quantum states (NQS) attract a lot of attention due to their potential to serve as a very expressive variational ansatz for quantum many-body systems. Here we study the main factors governing the applicability of NQS to frustrated magnets by training neural networks to approximate ground states of several moderately-sized Hamiltonians using the corresponding wave function structure on a small subset of the Hilbert space basis as training dataset. We notice that generalization quality, i.e. the ability to learn from a limited number of samples and correctly approximate the target state on the rest of the space, drops abruptly when frustration is increased. We also show that learning the sign structure is considerably more difficult than learning amplitudes. Finally, we conclude that the main issue to be addressed at this stage, in order to use the method of NQS for simulating realistic models, is that of generalization rather than expressibility.

Nonlinear Coherent States Associated with a Measure on the Positive Real Half Line

We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated Bargmann-type transform is discussed. A polynomials realization of the basis of the quantum states Hilbert space is also obtained. Here, the entire structure owes its existence to a certain measure on the positive real half line, of finite total mass, together with all its moments. We illustrate this method with the measure $$r^\beta e^{-r}dr$$, where $$\beta $$ is a non-negative constant, which leads to a new generalization of the true-polyanalytic Bargmann transform.

Mutually unbiased unitary bases of operators on d-dimensional hilbert space

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].

An introduction to S Bases in S Linear Algebra

In this lecture note we define the S bases for the spaces of tempered distributions.These new bases are the analogous of Hilbert bases of separable Hilbert spaces for the continuous case (they are indexed by m-dimensional Euclidean spaces) and enjoy properties similar to those shown by algebraic bases in the finite dimensional case.The S bases are one possible rigorous and extremely manageable mathematical model for the "physical" bases used in Quantum Mechanics.

Orthogonal function series formulae for inversion of the conical Radon transform with a fixed central axis

In this study, we derive two new inversion formulae for the n + 1-dimensional conical Radon transform that integrates a given n + 1-dimensional function on the upper half space over all conical surfaces with vertices on a hyperplane and a central axis orthogonal to this hyperplane: both formulae are based on the orthonormal bases of certain Hilbert spaces. One formula is derived from a singular value decomposition, a valuable tool in the study of ill-posed problems, of the conical Radon transform.

Mutually Unbiased Bases and Their Symmetries

We present and generalize the basic ideas underlying recent work aimed at the construction of mutually unbiased bases in finite dimensional Hilbert spaces with the help of group and graph theoretical concepts. In this approach finite groups are used to construct maximal sets of mutually unbiased bases. Thus the prime number restrictions of previous approaches are circumvented and this construction principle sheds new light onto the intricate relation between mutually unbiased bases and characteristic geometrical structures of Hilbert spaces.

Modular Riesz bases versus Riesz bases in Hilbert C∗-modules

In this paper, we give new characterizations of modular Riesz bases in Hilbert [Formula: see text]-modules. We prove that modular Riesz bases share many properties with Riesz bases in Hilbert spaces. Moreover, we show that there are also important differences; for example, there exist exact frames that are not modular Riesz bases.

Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion

Nonlocal quantum field theory (QFT) of one-component scalar field φ in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions Z as a functional of external source j, coupling constant g and spatial measure d μ is studied. An expression for GF Z in terms of the abstract integral over the primary field φ is given. An expression for GF Z in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator L ^ over the separable HS basis. The classification of functional integration measures D φ is formulated, according to which trivial and two nontrivial versions of GF Z are obtained. Nontrivial versions of GF Z are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure D φ over the primary field is suggested. In the 0-norm case, the definition and the meaning of 0-norm are given in terms of the replica-functional Taylor series. The definition of the 0-norm generator Ψ is suggested. Simple cases of sharp and smooth generators are considered. An alternative derivation of GF Z in terms of 0-norm is also given. All these definitions allow to calculate corresponding functional integrals over φ in quadratures. Expressions for GF Z in terms of integrals over the separable HS, aka the basis functions representation, with new integrands are obtained. For polynomial theories φ 2 n , n = 2 , 3 , 4 , … , and for the nonpolynomial theory sinh 4 φ , integrals over the separable HS in terms of a power series over the inverse coupling constant 1 / g for both norms (1-norm and 0-norm) are calculated. Thus, the strong coupling expansion in all theories considered is given. “Phase transitions” and critical values of model parameters are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated—GF Z for an arbitrary QFT and the strong coupling expansion for the theory φ 4 are derived. Finally a comparison of two GFs Z , one on the continuous lattice of functions and one obtained using the Parseval–Plancherel identity, is given.

Generating Functions for Bases in Hilbert Spaces of Entire Functions

We prove that unconditional bases in a functional Hilbert space H have a generating function if and only if the space H is stable. Necessary and sufficient conditions for the stability of spaces adjoint to weighted spaces on an interval are obtained.

Groups, Special Functions and Rigged Hilbert Spaces

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider S O ( 2 ) and functions on the unit circle, S U ( 2 ) and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, S O ( 3 , 2 ) and spherical harmonics, s u ( 1 , 1 ) and Laguerre functions, s u ( 2 , 2 ) and algebraic Jacobi functions and, finally, s u ( 1 , 1 ) ⊕ s u ( 1 , 1 ) and Zernike functions on a circle.

Riesz basis of exponential family for a hyperbolic system

Abstract This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C 0 {C_{0}} -semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system { e λ n ⁢ t } n ≥ 1 {\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in L 2 ⁢ [ 0 , T ] {L^{2}[0,T]} . Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

On mutually unbiased unitary bases in prime-dimensional Hilbert spaces

Akin to the idea of complete sets of Mutually Unbiased Bases for prime dimensional Hilbert spaces, $\mathcal{H}_d$, we study its analogue for a $d$ dimensional subspace of $M (d,\mathbb{C})$, i.e. Mutually Unbiased Unitary Bases (MUUBs) comprising of unitary operators. We note an obvious isomorphism between the vector spaces and beyond that, we define a relevant monoid structure for $\mathcal{H}_d$ isomorphic to one for the subspace of $M (d,\mathbb{C})$. This provides us not only with the maximal number of such MUUBs, but also a recipe for its construction.

Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

It is purpose of this paper to investigate Sturm-Liouville equation on many-interval with the eigenvalue parameter appearing linearly in the boundary conditions and with two supplementary transmission conditions. The classical Sturmian theory did not cover such type of many-interval boundary value transmission problems. For the classical Sturm-Liouville problems it is guaranteed that the problem is self-adjoint with compact resolvent, the spectrum is disctrete and consist of eigenvalues and the corresponding eigenfunctions form an orthogonal basis in the well-known Hilbert space . But the boundary-value-transmission problems are not self-adjoint and the system of eigenfunctions did not form a basis in the classical Hilbert space in general. Taking in view this fact we suggest a new approach for self-adjoint realization of such type transmission problems. Moreover, we define some new Hilbert spaces to establish positiveness of corresponding operator-pencil. At first we define a concept of generalized eigenfunctions for this kind of spectral problems. In particular it is shown that if the potential is continuous then the generalized eigenfunctions satisfies the considered problem is the classical sense. Then we introduce to the consideration some compact operators such a way that the considered boundary-value-transmission problem can be reduced to the appropriate operator-pencil equation. Finally, we prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. It is important to note that the obtained results extends classical results associated with regular Sturm-Liouville problems.

Internal sizes in μ-abstract elementary classes

Working in the context of μ-abstract elementary classes (μ-AECs)—or, equivalently, accessible categories with all morphisms monomorphisms—we examine the two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic, notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary results on the existence and categoricity spectra of μ-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity defined using cardinality) in μ-AECs.

Mutually unbiased bases: a group and graph theoretical approach

In this contribution the main ideas underlying recent work aiming at the construction of mutually unbiased bases in finite dimensional Hilbert spaces are discussed. This approach relies on a systematic use of group and graph theoretical concepts announced by Charnes and Beth (2005 ERATO Conf. on Quantum Information Science) and extended significantly by Charnes (2018 in preparation) recently. A principal feature of this method is its independence of prime number restrictions thus distinguishing it from almost all previous constructions which have relied on finite fields and related concepts of finite geometry. This group and graph theoretical approach offers the possibility to gain new insight into the intricate relation between quantum theoretical complementarity as encoded in mutually unbiased bases and characteristic geometrical structures of the Hilbert space involved.

Bases of Banach and Hilbert Spaces Transformed by Linear Operators

The construction of bases comprising of elements with special structures in various Banach and Hilbert spaces is a major enterprise in many applications. The current article is targeted to study what we call perturbed bases and frames for Banach and Hilbert spaces. To this end, we consider a pair of bounded linear operators related via an inequality of a special form. Our results are in spirit close to a family of bounded linear operators known in fractal approximation theory. Hence, the proposed theory exhibits, in particular, bases and frames consisting of fractal (self-referential) functions for some standard spaces of functions. In the last part of the article we study the structure of the space of (strictly) relatively bounded operators.

$G$-dual Frames in Hilbert $C^{*}$-module Spaces

In this paper, we introduce the concept of $g$-dual frames for Hilbert $C^{*}$-modules, and then the properties and stability results of $g$-dual frames are given. A characterization of $g$-dual frames, approximately dual frames and dual frames of a given frame is established. We also give some examples to show that the characterization of $g$-dual frames for Riesz bases in Hilbert spaces is not satisfied in general Hilbert $C^*$-modules.