Universal Operator Research Articles

Witnessing arbitrary bipartite entanglement in a measurement-device-independent way

Experimental detection of entanglement of an arbitrary state of a given bipartite system is crucial for exploring many areas of quantum information processing. But such a detection should be made in a device-independent way if the preparation process of the state is considered to be faithful, in order to avoid detection of a separable state as an entangled one. The recently developed scheme of detecting bipartite entanglement in a measurement-device-independent way [Phys. Rev. Lett. 110, 060405 (2013)] does require information about the state. Here, by using Auguisiak et al.'s universal entanglement witness scheme for two-qubit states [Phys. Rev. A 77, 030301 (2008)], we provide a universal entanglement detection scheme for two-qubit states in a measurement-device-independent way. We also provide a set of universal witness operators for detecting NPT-ness (negative under partial transpose) of two-qudit states in a measurement-device-independent way. We conjecture that no such universal entanglement witness scheme exists for PPT (positive under partial transpose) entangled states. We also analyze the robustness of some of the experimental schemes---for detecting entanglement in a measurement-device-independent way---under the influence of noise in the inputs (from the referee) as well as in the measurement operator.

Non-Asplund Banach spaces and operators

Let W and Z be Banach spaces such that Z is separable and let R:W⟶Z be a (continuous, linear) operator. We study consequences of the adjoint operator R⁎ having non-separable range. From our main technical result we obtain applications to the theory of basic sequences and the existence of universal operators for various classes of operators between Banach spaces. We also obtain an operator-theoretic characterisation of separable Banach spaces with non-separable dual.

Friedrichs systems in a Hilbert space framework: Solvability and multiplicity

The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide sufficient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples.

Algebraic sets of universal algebras and algebraic closure operator

The paper is a brief survey of the author’s results connected with the lattices of algebraic sets of universal algebras and with the operator of algebraic closure on the subsets of direct powers of basic sets of algebras.

Some new examples of universal hypercyclic operators in the sense of Glasner and Weiss

— A bounded operator on a real or complex separable infinite-dimensional Banach space Z is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation T of a standard Lebesgue probability space (X,B, μ), there exists an A-invariant probability measure ν on H with full support such that the two dynamical systems (X,B, μ;T ) and (H,BH , ν;A) are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or bilateral weighted shifts on `p or c0, show the existence of universal operators on a large class of Banach spaces, and give a criterion for universality in terms of unimodular eigenvectors. We also obtain similar results for operators which are universal for all ergodic systems (not only for invertible ones), and study necessary conditions for an operator on a Hilbert space to be universal.

Universal and homogeneous structures on the Urysohn and Gurarij spaces

Using Fraisse theoretic methods we enrich the Urysohn universal space by universal and homogeneous closed relations, retractions, closed subsets of the product of the Urysohn space itself and some fixed compact metric space, L-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm L on the Lispchitz-free space of the Urysohn space.

Some Muirhead Mean Operators for Intuitionistic Fuzzy Numbers and Their Applications to Group Decision Making.

Muirhead mean (MM) is a well-known aggregation operator which can consider interrelationships among any number of arguments assigned by a variable vector. Besides, it is a universal operator since it can contain other general operators by assigning some special parameter values. However, the MM can only process the crisp numbers. Inspired by the MM’ advantages, the aim of this paper is to extend MM to process the intuitionistic fuzzy numbers (IFNs) and then to solve the multi-attribute group decision making (MAGDM) problems. Firstly, we develop some intuitionistic fuzzy Muirhead mean (IFMM) operators by extending MM to intuitionistic fuzzy information. Then, we prove some properties and discuss some special cases with respect to the parameter vector. Moreover, we present two new methods to deal with MAGDM problems with the intuitionistic fuzzy information based on the proposed MM operators. Finally, we verify the validity and reliability of our methods by using an application example, and analyze the advantages of our methods by comparing with other existing methods.

A cooperative neural network approach for enhancing data traffic prediction

This paper addresses the problem of learning a regression model for the prediction of data traffic in a cellular network. We proposed a cooperative learning strategy that involves two Jordan recurrent neural networks (JNNs) trained using the firefly algorithm (FFA) and resilient backpropagation algorithm (Rprop), respectively. While the cooperative capability of the learning process ensures the effectiveness of the regression model, the recurrent nature of the neural networks allows the model to handle temporally evolving data. Experiments were carried out to evaluate the proposed approach using high-speed downlink packet access data demand and throughput measurements collected from different cell sites of a universal mobile telecommunications system-based cellular operator. The proposed model produced significantly superior results compared to the results obtained on the same problems from the traditional method of separately training a JNN with FFA and Rprop.

A Kirchberg-type tensor theorem for operator systems

We construct operator systems $\mathfrak C_I$ that are universal in the sense that all operator systems can be realized as their quotients. They satisfy the operator system lifting property. Without relying on the theorem by Kirchberg, we prove the Kirchberg type tensor theorem $$\mathfrak C_I \otimes_{\min} B(H) = \mathfrak C_I \otimes_{\max} B(H).$$ Combining this with a result of Kavruk, we give a new operator system theoretic proof of Kirchberg's theorem and show that Kirchberg's conjecture is equivalent to its operator system analogue $$\mathfrak C_I \otimes_{\min} \mathfrak C_I =\mathfrak C_I \otimes_{\rm c} \mathfrak C_I.$$ It is natural to ask whether the universal operator systems $\mathfrak C_I$ are projective objects in the category of operator systems. We show that an operator system from which all unital completely positive maps into operator system quotients can be lifted is necessarily one-dimensional. Moreover, a finite dimensional operator system satisfying a perturbed lifting property can be represented as the direct sum of matrix algebras. We give an operator system theoretic approach to the Effros-Haagerup lifting theorem.

A separable Fréchet space of almost universal disposition

The Gurariĭ space is the unique separable Banach space G which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every ε>0, for all finite-dimensional normed spaces E⊆F, for every isometric embedding e:E→G there exists an ε-isometric embedding f:F→G such that f↾E=e. We show that GN with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Fréchet spaces. The construction relies heavily on the universal operator on the Gurariĭ space, recently constructed by Garbulińska-Węgrzyn and the third author. In addition, we consider a non-graded sequence of semi-norms on GN with which the space GN is of almost universal disposition for finite-dimensional Fréchet spaces with a fixed sequence of semi-norms. In both cases, this yields in particular that GN is universal in the class of all separable Fréchet spaces.

Free pluriharmonic functions on noncommutative polyballs

In this paper, we study free k-pluriharmonic functions on noncommutative regular polyballs. These regular polyballs have universal operator models consisting of left creation operators acting on tensor products of full Fock spaces. We introduce and determine the class of kmulti- Toeplitz operators acting on these tensor products and show that the bounded free k-pluriharmonic functions on regular polyballs are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmonic function has continuous extension to the closed polyball if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in a certain class, which we determine. We provide a Naimark type dilation theorem for direct products of unital free semigroups, and use it to obtain a structure theorem which characterizes the positive free k-pluriharmonic functions on the regular polyball, with operator-valued coefficients. We define the noncommutative the Berezin (resp. Poisson) transform of a completely bounded linear map on the C*-algebra generated by the universal operator model and give necessary an sufficient conditions for a function to be the Poisson transform of a completely bounded (resp. completely positive) map. In the last section of the paper, we obtain Herglotz-Riesz representation theorems for free holomorphic functions on regular polyballs with positive real parts, extending the classical result as well as Korannyi-Pukanszky version in scalar polydisks.

Rota's universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

An introduction to Rota’s universal operators: properties, old and new examples and future issues

AbstractThe Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been an important tool for studying such important problem. In this survey, we focus on Rota’s universal operators, pointing out their main properties and exhibiting some old and recent examples.

The Third State: Toward a Quantum Information Theory of Consciousness

The question of how our perceived reality is constructed and subsequently how our mind has evolved such that we are able to both perceive and subsequently alter our own causality or even our own evolution within this reality has been a long running open question. Referred to as “The Hard Problem”. There have been many theoretical interpretations on the nature of causal self-observance – hereafter referred to as 'consciousness'. The current paper introduces the reader to the indeterminable operator - “The Third State”. The Third State is a term used to describe space itself in relation to the position of all things. As the paper shall show, The Third State is a required omnipresent and universal operator in the otherwise binary realm of data → information. The Third State augments the accepted binary operators to produce the required 'tristate' condition that facilitates the required probabilistic nature of the conscious manifestation. Secondly the Unity Magnitude [Um] scale, which facilitates the bounding of quantum probabilistic memory in a finite model. The paper further introduces the reader to a model and experimental theory that suggests all things we perceive as physical reality can be fabricated from primitive components of data, bits – matter / antimatter / something or nothing. That facilitated by the Third State, our immediate present reality as we experience it and theretofore consciousness is a simultaneous product of the current physical configuration of the frequency stable systems of which we are comprised , interpreted past reality, self-predicted future provided by cyclic frequency stable systems and the immediate physical and sensory state including recursive imagination systems generated by the output of these perturbed frequency stable systems in the cyclic feedback process - ultimately perturbed by, but as one unified with the stochastic processes in relation to the quantum cosmological domain.

Consequences of universality among Toeplitz operators

The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been one tool for studying the problem. The best known universal operators have been adjoints of analytic Toeplitz operators or unitarily equivalent to them. We present many examples of Toeplitz operators whose adjoints are universal operators and exhibit some of their common properties. Some ways in which the invariant subspaces of these universal operators interact with operators in their commutants are given. Special attention is given to the closed subalgebra, not always the zero algebra, of compact operators in their commutants. Finally, three questions connecting shift invariant subspaces and invariant subspaces of analytic Toeplitz operators are raised. Positive answers for both of the first two imply the existence of non-trivial invariant subspaces for every bounded operator on separable Hilbert spaces of dimension two or more.

Controlled formalized operators

In this report is considered the generalization of the principles, which manage the formalization of controlled qubit operators. The research builds on the standard external data sources, for development of a generalized system for conditional operators. Certain controlled operators are viewed as linear combinations of universal operators. This system provides a generalized scheme for the type of the controlled operators, which could be encountered in a circuit, composed entirely of elementary operators. In this report is presented a formalized generalization of single qubit and controlled operators, based on formalized single qubit operator. This work is part of the developed by the author formalized system for designing of algorithmic models for quantum circuits, based on phase encoding, decod- ing and parameterization of primitive quantum operators. In previous publications of the author (6, 7, 8) were defined sev- eral sets of operators on the n qubit, which generalize certain classical characteristics: identity and logical negation. Moreo- ver, some ways were explored in which can be constructed operators as linear combinations of elements from those sets. Such combinations capture the partial application of an opera- tor together with another operator, which in a broader sense is its logical negation.

The Modified Activity-Based Costing Method in Universal Postal Service Area: Case Study of the Montenegro Post

The 3rd EC Postal Directive proposes a calculation approach to separately determine the net cost of universal service obligation. Also proposes presence of liability to compensate the universal service provider. In this paper, authors have presented the necessity of compliance with Postal Directives through application of the activity-based costing method for costs calculation related to the universal postal operator. The objective of this paper is to research the possibility of modification of the existing, traditional ABC method. Method will improve the quality of cost accounting of the universal postal operators. The aim is to present the implementation of the modified activity-based cost accounting method from the point of view of a universal postal service. Conceptual framework for the modified activity-based costing in the area of universal postal service has been developed and applied in the Montenegro Post. This modification of the traditional activity-based costing, called the modified activity-based costing, for solving specific accounting problems in a universal postal service area, is the main contribution of the paper. As long as the modified ABC method is not created and implemented, the Montenegro Post has been unable to adequately respond to the established legal requirements. The presented method can be a guide for other postal companies with matters concerning the calculation of costs and revenues.DOI: http://dx.doi.org/10.5755/j01.ee.26.2.2818

ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing

Medical imaging involves important computational geometric problems, such as image segmentation and analysis, shape approximation, three-dimensional (3D) modeling, and registration of volumetric data. In the last few years, Conformal Geometric Algebra (CGA), based on five-dimensional (5D) Clifford Algebra, is emerging as a new paradigm that offers simple and universal operators for the representation and solution of complex geometric problems. However, the widespread use of CGA has been so far hindered by its high dimensionality and computational complexity. This paper proposes a simplified formulation of the conformal geometric operations (reflections, rotations, translations, and uniform scaling) aimed at a parallel hardware implementation. A specialized coprocessing architecture (ConformalALU) that offers direct hardware support to the new CGA operators, is also presented. The ConformalALU has been prototyped as a complete System-on-Programmable-Chip (SoPC) on the Xilinx ML507 FPGA board, containing a Virtex-5 FPGA device. Experimental results show average speedups of one order of magnitude for CGA rotations, translations, and dilations with respect to the geometric algebra software library Gaigen running on the general-purpose PowerPC processor embedded in the target FPGA device. A suite of medical imaging applications, including segmentation, 3D modeling and registration of medical data, has been used as testbench to evaluate the coprocessor effectiveness.

Anisotropic Lizorkin-Triebel spaces with mixed norms - traces on smooth boundaries

This article deals with trace operators on anisotropic Lizorkin–Triebel spaces with mixed norms over cylindrical domains with smooth boundary. As a preparation we include a rather self-contained exposition of Lizorkin–Triebel spaces on manifolds and extend these results to mixed-norm Lizorkin–Triebel spaces on cylinders in Euclidean space. In addition Rychkov's universal extension operator for a half space is shown to be bounded with respect to the mixed norms, and a support preserving right-inverse of the trace is given explicitly and proved to be continuous in the scale of mixed-norm Lizorkin–Triebel spaces. As an application, the heat equation is considered in these spaces, and the necessary compatibility conditions on the data are deduced.

On Gács' quantum algorithmic entropy

We define an infinite dimensional modification of lower-semicomputability of density operators by G\'acs with an attempt to fix some problem in the paper. Our attempt is partly achieved by showing the existence of universal operator under some additional assumption. It is left as a future task to eliminate this assumption. We also see some properties and examples which stimulate further research. In particular, we show that universal operator has certain nontrivial form if it exists.