Product Hilbert Space Research Articles

Weighted Tensor Product Algorithms for Linear Multivariate Problems

We study the ε-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. Λall consists of all linear functionals while Λstd consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1/ε and d. For Λall we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d=1. ForΛstd we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences.

Bell measurements for teleportation

In this paper we investigate the possibility of making complete Bell measurements on a product Hilbert space of two two-level bosonic systems. We restrict our tools to linear elements, such as beam splitters and phase shifters, delay lines and electronically switched linear elements, photodetectors, and auxiliary bosons. As a result we show that with these tools a never failing Bell measurement is impossible.

Generalized Fredholm property and a priori estimates for linear operators on tensor products of Hilbert spaces

For a linear operator acting in a Hilbert space, the generalized Fredholm property (invertibility modulo a certain ideal) is proved to be equivalent to certaina priori estimates. This result is applied to establish a connection between properties of linear operators on tensor products of Hilbert spaces, such asn- andd-normality, the (generalized and ordinary) Fredholm property, and appropriatea priori estimates.

A chiral spin theory in the framework of an invariant evolution parameter formalism

We present a formulation for the construction of first-order equations which describe particles with spin, in the context of a manifestly covariant relativistic theory governed by an invariant evolution parameter; one obtains a consistent quantized formalism dealing with off-shell particles with spin. Our basic requirement is that the second-order equation in the theory is of the Schrödinger–Stueckelberg type, which exhibits features of both the Klein–Gordon and Schrödinger equations. This requirement restricts the structure of the first-order equation, in particular, to a chiral form. One thus obtains, in a natural way, a theory of chiral form for massive particles, which may contain both left and right chiralities, or just one of them. We observe that by iterating the first-order system, we are able to obtain second-order forms containing the transverse and longitudinal momentum relative to a timelike vector tμtμ=−1 used to maintain covariance of the theory. This timelike vector coincides with the one used by Horwitz, Piron, and Reuse to obtain an invariant positive definite space–time scalar product, which permits the construction of an induced representation for states of a particle with spin. We discuss the currents and continuity equations. The transverse and longitudinal aspects of the particle are complementary, and can be treated in a unified manner using a tensor product Hilbert space. Introducing the electromagnetic field we find an equation which gives rise to the correct gyromagnetic ratio, and is fully Hermitian under the proposed scalar product. Finally, we show that the original structure of Dirac’s equation and its solutions is obtained in the highly constrained limit in which pμ is proportional to tμ on mass shell. The chiral nature of the theory is apparent. We define the discrete symmetries of the theory, and find that they are represented by states which are pure left or right handed.

A C++ library using quantum trajectories to solve quantum master equations

Quantum trajectory methods can be used for a wide range of open quantum systems to solve the master equation by unravelling the density operator evolution into individual stochastic trajectories in Hilbert space. This C++ class library offers a choice of integration algorithms for three important unravellings of the master equation. Different physical systems are modelled by different Hamiltonians and environment operators. The program achieves flexibility and user friendliness, without sacrificing execution speed, through the way it represents operators and states in Hilbert space. Primary operators, implemented in the form of simple routines acting on single degrees of freedom, can be used to build up arbitrarily complex operators in product Hilbert spaces with arbitrary numbers of components. Standard algebraic notation is used to build operators and to perform arithmetic operations on operators and states. States can be represented in a local moving basis, often leading to dramatic savings of computing resources. The state and operator classes are very general and can be used independently of the quantum trajectory algorithms. Only a rudimentary knowledge of C++ is required to use this package. The package illustrates how computational physics can profit from object-oriented programming concepts like inheritance.

SComment on 'Local-realism violations in two-particle interferometry'

In a recent article [Phys. Rev. A 53, R1927 (1996)] Wu et al. give a proof of nonlocality for the maximally entangled state of two particles without using inequalities. Here we point out that in order for their argument to go through it is necessary to consider a minimum total of six dimensions in Hilbert space. Indeed, the above argument is found to fail whenever the dimensionality of the product Hilbert space is reduced to four. The explanation for this failure is provided by the fact according to which no direct contradiction between the quantum-mechanical and locally realistic predictions can arise for any fraction of an ensemble of 2\ifmmode\times\else\texttimes\fi{}2 systems configured in the maximally entangled state.

The Gradient Method for Non–Differentiable Operators in Product Hilbert Spaces and Applications to Elliptic Systems of Quasilinear Differential Equations

The Gradient Method for Non–Differentiable Operators in Product Hilbert Spaces and Applications to Elliptic Systems of Quasilinear Differential Equations

Superposition states through correlations of the second kind

We introduce model systems, embedded in R2, for joint systems consisting of two entangled spin-1/2 entities in superposition states: one in a symmetric superposition state, one in an antisymmetric superposition state. We study these model systems and compare them with a model system for a spin-1 quantum entity. All this leads to a new way of looking at the tensor product of Hilbert spaces within the context of quantum mechanics, and thus also to a new approach for the description of joint systems in quantum mechanics.

On a variational approximation method for a class of elliptic eigenvalue problems in composite structures

We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in M M nonoverlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. We present a general finite element method to obtain a numerical solution of the eigenvalue problem, starting from a nonstandard formally equivalent variational formulation in an abstract setting in product Hilbert spaces. We use standard Lagrange finite element spaces on the subdomains. Moreover, the bilinear forms are approximated by suitable numerical quadrature formulas. We obtain error estimates for both the eigenfunctions and the eigenvalues, allowing for the case of multiple exact eigenvalues, by a pure variational method.

Spectral Components of Selfadjoint Block Operator Matrices with Unbounded Entries

AbstractThis paper is devoted to the study of the spectral components of selfadjoint operator matrices which are generated by symmetric operator matrices of the form in the product Hilbert space H1 × H2 where the entries A, B and C are not necessarily bounded operators in the Hilbert spaces H1, H2 or between them, respectively. Under suitable assumptions a selfadjoint operator L is associated with Lo and the spectral properties of L are studied. The main result concerns the case in which the spectra of the selfadjoint operators A and C are weakly separated. If a is a real number such that max σ(C) ≤ α ≤ min σ(A), descriptions of the spectral subspaces of L corresponding to the intervals [— ∞, α] and ]α,∞[ and of the restrictions of L to these subspaces are given. From this main result half range completeness and basis properties for certain parts of the spectrum of L are deduced. The paper closes with two applications to systems of differential operators from magnetohydrodynamics.

On certain states on free *-algebras and their GNS representations

States on free *-algebras expressed in terms of ordered partitions are studied. To this class belong twisted limit states that are derived from a convolution type q- analog of the quantum central limit theorem (qclt) for twisted free *-bialgebras 𝒞q, studied previously in the case of quantum groups. A special form of the GNS representation in terms of generalized creation and annihilation operators, A(v) and A(v*), respectively, living in the infinite tensor product of Hilbert spaces is given leading to a generalization of the Fock space. It is also shown that another family of states of similar combinatorics can be obtained from the convolution series.

Trace operators, Feynman distributions, and multiparameter white noise

Within the framework of white noise analysis on the probability spaceΩ = ℒ*RdRM, the recent work by Johnson and Kallianpur on the Hu-Meyer formula, traces, and natural extensions is generalized to the multiparameter case:d>1. Besides providing a more general setting for these topics, the paper gives an alternative definition for the traces, a distributional version of the natural extension, and a generalized Kallianpur-Feynman distribution. The development illustrates how traces and natural extensions are intimately related to Wick products and the change of covariance formula from quantum field theory, as well as to the projective tensor product of Hilbert spaces from functional analysis.

Essential normality of some classes of operators in infinite tensor products of Hilbert spaces

Essential normality of some classes of operators in infinite tensor products of Hilbert spaces

Essential normality of some classes of operators in tensor products of Hilbert spaces

We establish general criteria for the domains of definition of spectral integrals to be essential. On the basis of these criteria, we prove that some classes of operators are essentially normal in tensor products of Hilbert spaces.

Finitely Correlated Pure States

We study a w*-dense subset of the translation invariant states on an infinite tensor product algebra ⊗ Z A , where A is a matrix algebra. These "finitely correlated states" are explicitly constructed in terms of a finite dimensional auxiliary algebra B and a completely positive map E : A ⊗ B → B . We show that such a state ω is pure if and only if it is extremal periodic and its entropy density vanishes. In this case the auxiliary objects B and E are uniquely determined by ω, and can be expressed in terms of an isometry between suitable tensor product Hilbert spaces.

Self-memorization Equation in Atmospheric Motion

In view of the fact that the atmospheric motion is anirreversible process, a memory function which can recall observational data in the past is suggested. In terms of defining an inner product in Hilbert space, a new concept of self-memorization in the atmospheric motion is proposed, thus the traditional atmospheric motion equation is generalized to a self-memorization equation in inclusion of multi-time observations. Self-memorization equations of a barotropic nondivergent model and a barotropic primitive equation model are exemplified. It is proved that some existing difference schemes can be derived from the self-memorization equation by giving particular values to memory function. However, it demonstrates thatmulti-time numerical prediction models can be unified into a framework of self-memorization equation. If a stochastic method in solving the memory function is taken, the selfmemorization equation will be transformed to a sort of dynamic-stochastic prediction model.

Self-memorization Equation in Atmospheric Motion

In view of the fact that the atmospheric motion isan irreversible process, a memory function which can recall observational data in the past is suggested. In terms of defining an inner product in Hilbert space, a new concept of self-memorization in the atmospheric motion is proposed, thus the traditional atmospheric motion equation is generalized to a self-memorization equation in inclusion of multi-time observations. Self-memorization equations of a barotropic nondivergent model and a barotropic primitive equation model are exemplified. It is proved that some existing difference schemes can be derived from the self-memorization equation by giving particular values to memory function. However, it demonstrates that multi-time numerical prediction models can be unified into a framework of self-memorization equation. If a stochastic method in solving the memory function is taken, the selfmemorization equation will be transformed to a sort of dynamic-stochastic prediction model.

Derivation of the geometrical phase.

The geometric (Berry) phase is shown to have its origin in the nontrivial geometry of the fiber bundle: Hilbert space\ensuremath{\rightarrow}space of states. The nontrivial geometry comes simply from the scalar product in Hilbert space. A comprehensive treatment of the geometrical phase is presented, bringing together the various ideas in the literature.

Particles, particle labels, and quanta: The toll of unacknowledged metaphysics

The practice of describing multiparticle quantum systems in terms of labeled particles indicates that we think of quantum entities as individuatable. The labels, together with particle indistinguishability, create the need for symmetrization or antisymmetrization (or, in principle, higher-order symmetries), which in turn results in “surplus formal structure” in the formalism, formal structure which corresponds to nothing in the real world. We argue that these facts show quanta to be unindividuatable entities, things in principle incapable of supporting labels, and so things which support no factual difference_if two of them are thought of as being switched. When thinking of the metaphysics of quanta, we should eschew the misleading labels of the tensor product Hilbert space formalism and prefer the ontologically more faithful description of the Fock space formalism. This conception eliminates puzzles about the quantum statistics of bosons.

Complemented Subspaces of Products of Hilbert Spaces

Complemented Subspaces of Products of Hilbert Spaces