Arbitrary Hilbert Space Research Articles

Necessary and sufficient conditions for bipartite entanglement

Necessary and sufficient conditions for bipartite entanglement are derived, which apply to arbitrary Hilbert spaces. Motivated by the concept of witnesses, optimized entanglement inequalities are formulated solely in terms of arbitrary Hermitian operators, which makes them useful for applications in experiments. The needed optimization procedure is based on a separability eigenvalue problem, whose analytical solutions are derived for a special class of projection operators. For general Hermitian operators, a numerical implementation of entanglement tests is proposed. It is also shown how to identify bound entangled states with positive partial transposition.

On the commutant of [formula omitted] in [formula omitted]-crossed products by [formula omitted] and their representations

For the C ∗ -crossed product C ∗ ( Σ ) associated with an arbitrary topological dynamical system Σ = ( X , σ ) , we provide a detailed analysis of the commutant, in C ∗ ( Σ ) , of C ( X ) and the commutant of the image of C ( X ) under an arbitrary Hilbert space representation π ˜ of C ∗ ( Σ ) . In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system Σ, the commutant of C ( X ) has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of C ∗ ( Σ ) . We also show that the corresponding statement holds true for the commutant of π ˜ ( C ( X ) ) under the assumption that a certain family of pure states of π ˜ ( C ∗ ( Σ ) ) is total. Furthermore we establish that, if C ( X ) ⊊ C ( X ) ′ , there exist both a C ∗ -subalgebra properly between C ( X ) and C ( X ) ′ which has the aforementioned intersection property, and such a C ∗ -subalgebra which does not have this property. We also discuss existence of a projection of norm one from C ∗ ( Σ ) onto the commutant of C ( X ) .

Novel Multiclass Classifiers Based on the Minimization of the Within-Class Variance

In this paper, a novel class of multiclass classifiers inspired by the optimization of Fisher discriminant ratio and the support vector machine (SVM) formulation is introduced. The optimization problem of the so-called minimum within-class variance multiclass classifiers (MWCVMC) is formulated and solved in arbitrary Hilbert spaces, defined by Mercer's kernels, in order to find multiclass decision hyperplanes/surfaces. Afterwards, MWCVMCs are solved using indefinite kernels and dissimilarity measures via pseudo-Euclidean embedding. The power of the proposed approach is first demonstrated in the facial expression recognition of the seven basic facial expressions (i.e., anger, disgust, fear, happiness, sadness, and surprise plus the neutral state) problem in the presence of partial facial occlusion by using a pseudo-Euclidean embedding of Hausdorff distances and the MWCVMC. The experiments indicated a recognition accuracy rate achieved up to 99%. The MWCVMC classifiers are also applied to face recognition and other classification problems using Mercer's kernels.

Weakly coprime factorization and state-feedback stabilization of discrete-time systems

The LQ-optimal state feedback of a finite-dimensional linear time-invariant system determines a coprime factorization NM −1 of the transfer function. We show that the same is true also for infinite-dimensional systems over arbitrary Hilbert spaces, in the sense that the factorization is weakly coprime, i.e., Nf, $${Mf \in \mathcal {H}^2 \Longrightarrow f \in \mathcal {H}^2}$$ for every function f. The factorization need not be Bezout coprime. We prove that every proper quotient of two bounded holomorphic operator-valued functions can be presented as the quotient of two bounded holomorphic weakly coprime functions. This result was already known for matrix-valued functions with the classical definition gcd(N, M) = I, which we prove equivalent to our definition. We give necessary and sufficient conditions and further results for weak coprimeness and for Bezout coprimeness. We then establish a variant of the inner–outer factorization with the inner factor being “weakly left-invertible”. Most of our results hold also for continuous-time systems and many are new also in the scalar-valued case.

Local realism, detection efficiencies, and probability polytopes

We present a detailed investigation of minimum detection efficiencies, below which local realism cannot be violated by any quantum system of any dimension in bipartite Bell experiments. Lower bounds on these minimum detection efficiencies are determined with the help of linear programming techniques. Our approach is based on the observation that any possible bipartite quantum correlation originating from a quantum state in an arbitrary dimensional Hilbert space is sandwiched between two probability polytopes, namely the local (Bell) polytope and a corresponding nonlocal no-signaling polytope. Numerical results are presented demonstrating the dependence of these lower bounds on the numbers of inputs and outputs of the bipartite physical system.

MCMC METHODS FOR DIFFUSION BRIDGES

We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

A General Hilbert Space Approach to Framelets

In arbitrary separable Hilbert spaces it is possible to deffine multiscale methods of constructive approximation based on product kernels, restricting their choice in certain ways. These wavelet techniques have already filtering and localization properties and they are applicable in many areas due to their generalized deffinition. But they lack detailed information about their stability and redundancy, which are frame properties. So in this work frame conditions are introduced for approximation methods based on product kernels. In order to provide stability and redundancy the choice of product kernel ansatz function has to be restricted. Taking into account the kernel conditions for multiscale and for frame approximations one is able to deffine wavelet frames (= framelets), inheriting the approximation properties of both techniques and providing a more precise tool for multiscale analysis than the normal wavelets.

On the Improvement of Support Vector Techniques for Clustering by Means of Whitening Transform

In this letter, we suggest a novel method for clustering, based on finding the smallest enclosing hyperellipse in arbitrary Hilbert spaces. In particular, we show that the one class support vector method that finds the minimum bounding hypersphere, under the whitening transform, becomes a method for finding the minimum bounding hyperellipse. Afterwards, we generalize the method in order to find the minimum bounding hyperellipse in arbitrary Hilbert spaces. We illustrate the power of the proposed methods in clustering applications.

Markovian extensions of a stochastic process

The problem on existence of a minimal Markovian (in the wide sense) process which contains a given stochastic process as a component is considered and partially solved in Rozanov [1977. On Markovian extensions of random process. Theory Probab. Appl. 22 (1), 194–199]. In this paper the problem (to be formulated precisely below) of finding all Markovian (in the wide sense) extensions for a given stochastic process is solved. In the first part of this paper we give some definitions and known results related to splitting subspaces with respect to arbitrary Hilbert spaces. In the second part we apply these results to the problem of finding all Markovian (in the wide sense) extensions of an arbitrary stochastic process. The approach adopted in this paper is that of Lindquist and Picci [1985. On the stochastic realization problem. SIAM J. Control Optim. 17 (3), 365–389] and Rozanov [1977. On Markovian extensions of random process. Theory Probab. Appl. 22 (1), 194–199].

Minimum Class Variance Support Vector Machines

In this paper, a modified class of support vector machines (SVMs) inspired from the optimization of Fisher's discriminant ratio is presented, the so-called minimum class variance SVMs (MCVSVMs). The MCVSVMs optimization problem is solved in cases in which the training set contains less samples that the dimensionality of the training vectors using dimensionality reduction through principal component analysis (PCA). Afterward, the MCVSVMs are extended in order to find nonlinear decision surfaces by solving the optimization problem in arbitrary Hilbert spaces defined by Mercer's kernels. In that case, it is shown that, under kernel PCA, the nonlinear optimization problem is transformed into an equivalent linear MCVSVMs problem. The effectiveness of the proposed approach is demonstrated by comparing it with the standard SVMs and other classifiers, like kernel Fisher discriminant analysis in facial image characterization problems like gender determination, eyeglass, and neutral facial expression detection.

Characterizations of normal, hyponormal and EP operators

In this paper normal and hyponormal operators with closed ranges, as well as EP operators, are characterized in arbitrary Hilbert spaces. All characterizations involve generalized inverses. Thus, recent results of S. Cheng and Y. Tian [S. Cheng, Y. Tian, Two sets of new characterizations for normal and EP matrices, Linear Algebra Appl. 375 (2003) 181–195] are extended to infinite-dimensional settings.

Quantum error correcting codes from the compression formalism

We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. By solving an algebraic compression problem, we construct qubit codes for such channels on arbitrary dimension Hilbert space, and identify correctable codes for Pauli-error models not obtained by the stabilizer formalism. This is accomplished through an application of a new tool for error correction in quantum computing called the ``higher-rank numerical range''. We describe its basic properties and discuss possible further applications.

Characterization of Oblique Dual Frame Pairs

Given a frame for a subspace W of a Hilbert space H, we consider all possible families of oblique dual frame vectors on an appropriately chosen subspace V. In place of the standard description, which involves computing the pseudoinverse of the frame operator, we develop an alternative characterization which in some cases can be computationally more efficient. We first treat the case of a general frame on an arbitrary Hilbert space, and then specialize the results to shift-invariant frames with multiple generators. In particular, we present explicit versions of our general conditions for the case of shift-invariant spaces with a single generator. The theory is also adapted to the standard frame setting in which the original and dual frames are defined on the same space.

Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity

Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of quantum theory are shown to be compatible with the principle of functional relativity. By accepting the principle as a hypothesis, we then analyze the origin of physical dimensions, provide a geometric interpretation of Planck's constant, and find a simple interpretation of the two-slit experiment and the process of measurement.

Linear algebra and differential geometry on abstract Hilbert space

Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n‐dimensional vector spaces. However, while n‐dimensional spaces in applications are always realized as the Euclidean space Rn, Hilbert spaces admit various useful realizations as spaces of functions. In the paper this simple observation is used to construct a fruitful formalism of local coordinates on Hilbert manifolds. Images of charts on manifolds in the formalism are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations then describe families of functional equations on various spaces of functions. The formalism itself and its applications in linear algebra, differential equations, and differential geometry are carefully analyzed.

A skew normal dilation on the numerical range of an operator

Simple facts about the Poincare-Neumann double layer potential are used in the construction of a normal dilation, on the numerical range of an arbitrary Hilbert space operator. Recent and old ideas in the theory of the numerical range are unified by this framework. A couple of mapping results for the numerical range are derived.

Computer-aided perturbation theory by cumulants: Dimerized and frustrated spin-12chain

This paper demonstrates that a computer aided perturbation theory can easily be realized by use of a cumulant approach. In contrast to a recent alternative formulation on the basis of Wegner's flow equation method the present approach can be applied to systems with arbitrary Hilbert space. In particular an equidistant spectrum of the unperturbed part of the Hamiltonian is not needed. The method is illustrated in detail for dimerized and frustrated spin 1/2 chains for which the ground state energy is calculated up to seventh order perturbation theory.

Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces

The purpose of this paper is to study the separation for the general second order elliptic differential operator G=G 0+V(x),x∈R n, in the weighted Hilbert space H= L 2, k ( R n , H 1), where G 0=−∑ i, j=1 n a ij ( x) D i j is the differential operator with the real positive coefficients a ij ( x)∈ C 2( R n ) and D i j= ∂ 2 ∂x i ∂x j ,i,j=1,…,n . The operator potential V( x)∈ C 1( R n , L( H 1)), where L( H 1) is the space of all bounded linear operators on the arbitrary Hilbert space H 1. Moreover, we study the existence and uniqueness of the solution of the second order differential equation −∑ i, j=1 n a ij ( x) D i j u( x)+ V( x) u( x)= f( x), where f( x)∈ H, in the weighted Hilbert space H= L 2, k ( R n , H 1).

The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

Efficient distillation beyond qubits

We provide generalizations of known two-qubit entanglement distillation protocols for arbitrary Hilbert space dimensions. The protocols, which are analogs of the hashing and breeding procedures, are adapted to bipartite quantum states that are diagonal in a basis of maximally entangled states. We show that the obtained rates are optimal, and thus equal to the distillable entanglement, for a $(d\ensuremath{-}1)$-parameter family of rank deficient states. Methods to improve the rates for other states are discussed. In particular, for isotropic states it is shown that the rate can be improved such that it approaches the relative entropy of entanglement in the limit of large dimensions.