Showing papers on "Unitary state published in 2000"

Frames, bases, and group representations

Abstract: Introduction Basic theory for frames Complementary frames and disjointness Frame vectors for unitary systems Gabor type unitary systems Frame wavelets, super-wavelets and frame sets Frame representations for groups Concluding remarks Bibliography.

Geometric phases for mixed states in interferometry.

Abstract: We provide a physical prescription based on interferometry for introducing the total phase of a mixed state undergoing unitary evolution, which has been an elusive concept in the past. We define the parallel transport condition that provides a connection form for obtaining the geometric phase for mixed states. The expression for the geometric phase for mixed state reduces to well known formulas in the pure state case when a system undergoes noncyclic and unitary quantum evolution.

Unitary root-MUSIC with a real-valued eigendecomposition: a theoretical and experimental performance study

Abstract: A real-valued (unitary) formulation of the popular root-MUSIC direction-of-arrival (DOA) estimation technique is considered. This unitary root-MUSIC algorithm reduces the computational complexity in the eigenanalysis stage of root-MUSIC because it exploits the eigendecomposition of a real-valued covariance matrix. The asymptotic performance of unitary root-MUSIC is analyzed and compared with that of conventional root-MUSIC. The results of this comparison show identical asymptotic properties of both algorithms in the case of uncorrelated sources and a better performance of unitary root-MUSIC in scenarios with partially correlated or fully coherent sources. Additionally, our simulations and the results of sonar and ultrasonic real data processing demonstrate an improved threshold performance of unitary root-MUSIC relative to conventional root-MUSIC. It can be then recommended that, as a rule, the unitary root-MUSIC technique should be preferred by the user to the conventional root-MUSIC algorithm.

Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations

Abstract: Entanglement of two parts of a quantum system is a non-local property unaffected by local manipulations of these parts. It is described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of non-local properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by single-bit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can be found efficiently. Without any of these 18 invariants the set is incomplete. Similarly, non-local, entangling properties of two-qubit unitary gates are invariant under single-bit operations. We present a complete set of 3 real polynomial invariants of unitary gates. Our results are useful for optimization of quantum computations since they provide an effective tool to verify if and how a given two-qubit operation can be performed using exactly one elementary two-qubit gate, implemented by a basic physical manipulation (and arbitrarily many single-bit gates).

Hilbert-Schmidt distance and non-classicality of states in quantum optics

Abstract: The Hilbert—Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states. Unitary transformations of both states with the same unitary operator (e.g. the displacement of both states in the phase plane by the same displacement vector and squeezing of both states) do not change this distance. The nearest distance of a given state to the whole set of coherent states is proposed as a quantitative measure of non-classicality of the state which is identical when considering the coherent states as the most classical ones among pure states and the deviations from them as non-classicality. The connection to other definitions of the non-classicality of states is discussed. The notion of distance can also be used for the definition of a neighbourhood of considered states. Inequalities for the distance of states to Fock states are derived. For given neighbourhoods, they restrict common characteristics of the state as the dispersion of the number operato...

Local symmetry properties of pure three-qubit states

Abstract: Entanglement types of pure states of three spin-½ particles are classified by means of their stabilizers in the group of local unitary transformations. It is shown that the stabilizer is generically discrete, and that a larger stabilizer indicates a stationary value for some local invariant. We describe all the exceptional states with enlarged stabilizers.

Structure of hyperbolic unitary groups i. elementary subgroups

Abstract: This is the first in a series of papers dedicated to the structure of hyperbolic unitary groups over form rings and their subgroups. In this part, we recall foundations of the theory and study the elementary subgroup of the hyperbolic unitary group over a form ring. In particular, using a variant of Suslin’s patching method, we prove standard commutator formulae for relative elementary subgroups.

Hua type integrals over unitary groups and over projective limits of unitary groups

Abstract: We discuss some natural maps from a unitary group U(n) to a smaller group U(n-m) (these maps are versions of the Livshic characteristic function). We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces (values of the integrals are products of Gamma-functions). These integrals generalize Hua Loo Keng integrals. We construct inverse limits of unitary groups equipped with analogues of the Haar measure and evaluate some integrals over these inverse limits.

Local symmetry properties of pure 3-qubit states

Abstract: Entanglement types of pure states of 3 qubits are classified by means of their stabilisers in the group of local unitary operations. It is shown that the stabiliser is generically discrete, and that a larger stabiliser indicates a stationary value for some local invariant. We describe all the exceptional states with enlarged stabilisers.

Algebraic orbifold conformal field theories

Abstract: The unitary rational orbifold conformal field theories in the algebraic quantum field theory and subfactor theory framework are formulated. Under general conditions, it is shown that the orbifold of a given unitary rational conformal field theory generates a unitary modular category. Many new unitary modular categories are obtained. It is also shown that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time.

Intergovernmental relations in South Africa : the challenges of co-operative government

Abstract: The new South African constitution, with its emphasis on ‘co-operative government’, sets out the structure of the state and delineates, in broad terms, the responsibilities of the different levels of the public sector. The constitution, however, merely spells out an enabling framework for the development of a system of inter-governmental relations (IGR). The operationalisation of the policy of co-operative government, consequently, is in formation and manifests the extant tensions between a unitary and a federal model of the state. The shortcomings in the existing system of IGR lead to poor co-ordination within and between the different structures of government and limit its capacity to deliver multi-sectoral social programmes. While the government is anxious to develop a regulatory framework for IGR, it is unlikely that legislation on its own will promote greater intergovernmental co-operation and co-ordination; administrative capacity and the evolution of accepted models of interaction are likely to be of equal or greater import. Copyright © 2000 John Wiley & Sons, Ltd.

An Accurate Measure of the Instantaneous Discharge Probability, with Application to Unitary Joint-Event Analysis

Abstract: We present an estimate for the instantaneous discharge probability of a neurone, based on single-trial spike-train analysis. By detecting points where the neurone abruptly changes its firing rate and treating them specially, the method is able to achieve smooth estimates yet avoid the blurring of significant changes. This estimate of instantaneous discharge probability is then applied to the method of unitary event analysis. We show that the unitary event analysis as originally conceived is highly sensitive to firing-rate nonstationarities and covariations, but that it can be considerably improved if calculations of statistical significance use an instantaneous discharge probability instead of a firing-rate estimate based on averaging across multiple trials.

Minimal Escape velocities

Abstract: We give a new derivation of the minimal velocity estimates for unitary evolutions with some optimal bounds.

Monge Metric on the Sphere and Geometry of Quantum States

Abstract: Topological and geometrical properties of the set of mixed quantum states in the N-dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU(2) coherent states and the generalised Husimi distributions to define the Monge distance between arbitrary two density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state \rho_*, proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized 'chaotic' states are close to \rho_*. This contrasts the geometries induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, where the distance from \rho_* is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states, and more suitable for numerical computation.

Why Unitary Boards are Not Best Practice: A Case for Compound Boards

Abstract: The paper identifies the manifold conflicts of interest inherent in firms governed by a unitary board and how they exacerbate information overload and bounded rationality while paradoxically not providing sufficient information, independent of management, to direct, monitor, control, and change management Compound boards introduce a division of power to mediate conflicts, allow access to superior feedback information from stakeholders, and enhance the cybernetic integrity of how a firm is governed Transaction byte analysis is used to explain bounded rationality and identify how compound boards decompose decision making labour Compound boards are identified as a necessary condition for reducing the cost of finance and developing (i) holonic architecture, (ii) social tensegrity, (iii) self-regulation and self-governance, (v) sustainable employee and/or other stakeholder participation in governance, and (vi) superior performance This makes unitary boards inconsistent with convergence and knowledge intensive or network firms seeking to bond human capital with employee ownership

Complementarity and quantum erasure with entangled-photon states

Abstract: A quantum-erasure experiment was performed on superposition states of two photons in either of two spatiotemporal modes. Symmetrical tapping of energy from the two-mode state creates a four-mode entangled state. A direct measurement of one photon in two of the modes reveals the path of the second one, thereby eliminating the possibility of observing interference between the two remaining modes. It is shown that a unitary rotation of one of the two-mode states erases the path information; as a consequence, the visibility of the other two-mode state can be resurrected.

Competition and Structure

Abstract: The essays in this volume, written by well-known economists and other social scientists from North America, Europe and Australia, share to an unusual degree a common concern with the competitive mechanisms that underlie collective decisions and with the way they are embedded in institutional settings. This gives the book a unitary inspiration whose value is clear from the understanding and insights its chapters provide on important theoretical and practical issues such as the social dimension and impact of trust, the management of information in bureaucratic settings, the role of political parties in constitutional evolution, inter-level rivalry and reassignments of powers in federal and unitary systems of government, the impact of ethnicity and nationalism on federal institutions or arrangements, and the response of governments and overarching institutions of globalization. The essays were written in honour of Albert Breton, a pioneer in this field of investigation.

Howe correspondence for real unitary groups II

Abstract: A previous paper by the author describes the Howe correspondence for dual pairs of the form (U(p, q), U(r, s)) with p+ q = r + s, in terms of Langlands parameters. We extend these results to the case p+q = r+s+1.

A Langlands classification for unitary representations

Abstract: The Langlands classification theorem describes all admissible representations of a reductive group $G$ in terms of the tempered representations of Levi subgroups of $G$. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of $G$ in terms of certain very special unitary representations of Levi subgroups.

Ethnic ‘nationalities’, God & the state: Whither the federal republic of Nigeria?

Abstract: There is a continuing contradiction between the state as the corporate representation of Nigerian society and sectional (ethno‐religious) interests struggling to counter perceived margin‐alisation by revenue ‘sharing’ among patrons and clients (Joseph, 1987) Power elites in command of the centre identify with the state and proclaim its indivisible ‘unitary’ character; poly‐ethnic labour activists believe in the Federal Republic but criticise their exclusion from state power by a mege‐rich elite (Ojewale, 2000) Other (southern) leaders are so dissatisfied with the unitary Nigerian state that they are campaigning for a confederation of ethnic ‘nationalities’ or secession into independent republics As well, certain northern leaders are emphasising the Islamic identity of the Hausa‐Fulani ‘nationality’ by substituting Shari'a law for the criminal code so some people believe core northern states are engaging in covert religio‐legal secession (Soyinka, 2000) The state and ethno‐religious sectionalism thus c

Minimal Square Spectral Factors via Triples

Abstract: We consider the problem of parametrizing the set of square minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis in terms of invariant subspaces and of the minimal unitary left divisors of a certain unitary function. We shall use an approach which involves null-pole triples.

Analytical unitary bounds on quantum dynamics: Design of optimum NMR experiments in two-spin-12 systems

Abstract: We report analytical solutions to the unitary bound problem for coherence/polarization transfer in IS two-spin-12 systems by means of unitary operations. Theoretical upper bounds for the transfer efficiency along with the associated optimum transformation operators are obtained analytically by decomposing the unitary operator as a product of exponentials in the special unitary Lie group SU(4). Addressing NMR spectroscopy as a specific example, the method is demonstrated for the non-Hermitian transfers I−→S− and 2I−Sz→S− being relevant for heteronuclear single-quantum coherence (HSQC) experiments as well as the double- to single-quantum transfer I−S−→I−Sβ+IβS− being representative for coherence-order and spin-state-selective transfer in INADEQUATE CR experiments. Furthermore, using a Lagrangian function approach it is demonstrated how the method enables analytical description of two-dimensional bounds for Iz→Sz cross polarization.

Unitary perturbation theory: a generalization of two-by-two rotations

Abstract: A generalization of the two-by-two rotation technique is proposed, permitting a whole row (column) of the matrix to be treated simultaneously. The method is based on the explicit analytical evaluation of the matrix exponent representing a symmetric combination of the individual rotations. Besides constructing the unitary transformation matrices, a new orthogonalization algorithm is also proposed. The resulting “unitary perturbation theory” and orthogonalization method may be useful in different areas.

Quantum synthesis of arbitrary unitary operators

Abstract: Nature provides us with a restricted set of microscopic interactions. The question is whether we can synthesize out of these fundamental interactions an arbitrary unitary operator. In this paper we present a constructive algorithm for realization of any unitary operator which acts on a (truncated) Hilbert space of a single bosonic mode. The algorithm itself is not unitary because it involves a conditional measurement. However, it does yield a constant probability of the conditional measurement which does not depend on the input state of the bosonic system. We consider a physical implementation of unitary transformations acting on one-dimensional vibrational states of a trapped ion. As an example we present an algorithm which realizes the discrete Fourier transform.

Generalised coherent states and the diagonal representation for operators

Abstract: We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with an unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch-Gordan theory and the reciprocity theorems associated with induced unitary group representation. Applications to several examples involving $SU(2),$ $SU(3),$ and the Heisenberg-Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase-space description of quantum mechanics and quantum state reconstruction problems.

Adapted-operator representations: Selective versus collective properties of quantum networks

Abstract: Based on local unitary operators acting on a n-dimensional Hilbert-space, we investigate selective and collective operator basis sets for N-particle quantum networks. Selective cluster operators are used to derive the properties of general cat-states for any n and N. Collective operators are conveniently used to account for permutation symmetry: The respective Hilbert-space dimension is then only polynomial in N and governed by strong selection rules. These selection rules can be exploited for the design of decoherence-free subspaces as well as for the implementation of efficient routes to entanglement if suspended switching between states of different symmetry classes could be realized.

Method of wireless communication using structured unitary space-time signal constellations

Abstract: A method is disclosed for generating unitary space-time signals for wireless communication using an array of M transmitting antennas. Each unitary space-time signal comprises T symbols. The method can be used to generate arbitrarily large signal constellations. The resulting unitary space-time signals have statistical properties that are favorable for reliable use in communication, and they can lead to very high data transmission rates. The disclosed method involves the mapping of binary strings of data to matrix products of the kind obtained by left-multiplying an initial T×M matrix by an ordered multiplicative sequence of exponentiated T×T unitary matrices, referred to as generator matrices. The result of each such multiplication is a unitary space-time signal matrix that may be transmitted. Each generator matrix corresponds to a single bit or a collection of bits of a binary string that is to be mapped. The value of the exponent of each generator matrix is determined by the pertinent bit or collection of bits.

Beyond the Unitary Conception of the United Kingdom Constitution

Abstract: ion, the British constitution is viewed as having other distinguishing historical and institutional features, including an emphasis upon continuity of development, conventions in lieu of higher law, the absence of a constitutional court, the residual protection of civil liberties, a majority voting system and a bicameral legislature with an unelected upper chamber. It is a peculiar feature of our constitutional selfunderstanding that we tend to make either too much or too little of the coincidence of these central features. It is perhaps even more peculiar that these opposite attitudes are connected to one and the same defining trait of our constitutional tradition, namely its unwritten – or nondocumentary2 character. This unwritten character, it has Editor’s Note: This chapter was first published as an article under the same title in (2000) Public Law: pp.384-404. It is reproduced here without amendment or abridgement by the kind permission of the author and Sweet & Maxwell. In compliance with the terms of the publishers’ permission, references in footnotes are also reproduced in the original citation style, which is different to the citation protocol used for other chapters in this volume. For an explanation of the location of this chapter within the scheme of this volume and its relevance to broader Sri Lankan constitutional debates, see the Editor’s Introduction. * I would like to thank participants at seminars in the Law Faculties of the Universities of Glasgow and Edinburgh and in the Politics Department of the University of Aberdeen, as well as the audience at the conference on Public Law of the Turn of the Century, held on November 25, 1999, for their comments on earlier versions of this article. The usual disclaimer applies. 1 See e.g., S. de Smith and R. Brazier, Constitutional and Administrative Law (8th ed., Penguin, London, 1998), Chap. 1; A.W. Bradley and K.D. Ewing, Constitutional and Administrative Law (12th ed., Longman, London, 1997), Chap. 1; E. Barendt, An Introduction to Constitutional Law, (Oxford University Press, 1998), Chap. 1. 2 C. R. Munro, Studies in Constitutional Law, (2nd ed., Butterworths, London, 1999), p. 3.