Terms Of Unitary Operators Research Articles

Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) H of entire functions associated with operator valued kernel functions. de Branges operators E=(E−,E+) analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on X, where X is a complex separable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS B(E) based on the de Branges operator E=(E−,E+) has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces B(E) has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces B(E) has been given.

A dynamical proof of the van der Corput inequality

We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on the Furstenberg correspondence principle. This is done by reducing the inequality to the mean ergodic theorem for contractions on Hilbert spaces. The key difficulty therein is that the Furstenberg correspondence principle is, a priori, limited to scalar-valued sequences. We, therefore, discuss how interpreting the Furstenberg correspondence principle via the Gelfand–Naimark–Segal construction for -algebras allows to study not just scalar but general Hilbert space-valued sequences in terms of unitary operators. This yields a proof of the van der Corput inequality in the spirit of the Furstenberg correspondence principle and the flexibility of this method is discussed via new proofs for different variants of the inequality.

New Maximally Entangled States for Pattern-Association Through Evolutionary Processes in a Two-Qubit System

New set of maximally entangled states (Singh-Rajput MES), constituting orthonormal eigen bases, has been revisited and its superiority and suitability in pattern-association (Quantum Associative Memory, QuAM) have been demonstrated. Using these MES as memory states in the evolutionary process of pattern storage in a two-qubit system, it has been shown that the first two states of Singh-Rajput MES are useful for storing the pattern |11> and the last two of these MES are useful in storing the pattern |10> Recall operations of quantum associate memory (QuAM) have been conducted through evolutionary process in terms of unitary operators by separately choosing Singh-Rajput MES and Bell’s MES as memory states and it has been shown that Singh-Rajput MES as valid memory states for recalling the patterns in a two-qubit system are much more suitable than Bell’s MES.

Phase-space-region operators and the Wigner function: Geometric constructions and tomography

Quasiprobability measures on a canonical phase space give rise through the action of Weyl's quantization map to operator-valued measures and, in particular, to region operators. Spectral properties, transformations, and general construction methods of such operators are investigated. Geometric trace-increasing maps of density operators are introduced for the construction of region operators associated with one-dimensional domains, as well as with two-dimensional shapes (segments, canonical polygons, lattices, etc.). Operational methods are developed that implement such maps in terms of unitary operations by introducing extensions of the original quantum system with ancillary spaces (qubits). Tomographic methods of reconstruction of the Wigner function based on the radon transform technique are derived by the construction methods for region operators. A Hamiltonian realization of the region operator associated with the radon transform is provided, together with physical interpretations.

Fermionic zero modes and spontaneous symmetry breaking on the light front

Spontaneous symmetry breaking is studied in a simple version of the light-front $O(2)$ sigma model with fermions. Its vacuum structure is derived by an implementation of global symmetries in terms of unitary operators in a finite volume with a periodic Fermi field. Because of the dynamical fermion zero mode, the vector and axial $U(1)$ charges do not annihilate the light-front vacuum. The latter is transformed into a continuous set of degenerate vacuum states, leading to the spontaneous breakdown of the axial symmetry. The existence of associated massless Goldstone boson is demonstrated.

Conjugated operators in quantum algorithms

This paper addresses the question of understanding quantum algorithms in terms of unitary operators. Many quantum algorithms can be expressed as applications of operators formed by conjugating so-called classical operators. The operators that are used for conjugation are determined by the problem and any additional structure possessed by the Hilbert space that is acted upon. We prove many new commutative laws between these different operators, and we use those to phrase and analyze old and new problems and algorithms. As an example, we review the Abelian subgroup problem. We then introduce the problem of determining a group homomorphism, and we give classical and quantum algorithms for it. We also generalize Deutsch''s problem and improve the previous best algorithms for earlier generalizations of it.

Projective group representations in quaternionic Hilbert space

We extend the discussion of projective group representations in quaternionic Hilbert space that was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi-centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed.

Superscattering operators, lie- admissible generalizations of quantum mechanics and dynamical semigroups

We analyse the notion of superscattering operators (operators which can describe transitions from an initial to a final density operator via an irreversible process) within the frameworks provided by Lie-admissible generalizations of quantum mechanics and by the theory of dynamical semigroups. Mignani has shown that the non-factorizability of the superscattering operator is a straightforward consequence of the non-potential scattering theory in the framework of the Lie-admissible formalism. A very naive reading of this formalism leads us to the somewhat dual statement that superscattering operators always factorize, though in general not in terms of unitary operators. It is furthermore found that superscattering operators can be hosted in the theory of semigroups at the cost of a serious restriction of the notion and a suitable enlargement of the definition of semigroup.