on group theory for quantum gates and quantum coherence finite group extensions offer a natural language to quantum computing in a nutshell one roughly describes the action of a quantum computer as consisting of two finite groups of gates error gates from the general pauli group p and stabilizing gates within an extension group c in this paper one explores the nice adequacy between group theoretical concepts such as commutators normal subgroups group of automorphisms short exact sequences wreath products and the coherent quantum computational primitives the structure of the single qubit and two qubit clifford groups is analyzed in detail as a byproduct one discovers that m20 the smallest perfect group for which the commutator subgroup departs from the set of commutators underlies quantum coherence of the two qubit system one recovers similar results by looking at the automorphisms of a complete set of mutually unbiased bases

qudits of composite dimension mutually unbiased bases and projective ring geometry the d2 pauli operators attached to a composite qudit in dimension d may be mapped to the vectors of the symplectic module being the modular ring as a result perpendicular vectors correspond to commuting operators a free cyclic submodule to a maximal commuting set and disjoint such sets to mutually unbiased bases for dimensions d 6 10 15 12 and 18 the fine structure and the incidence between maximal commuting sets are found to reproduce the projective line over the rings and respectively

analytic representations with theta functions for systems on ℤ d and on an analytic representation with theta functions on a torus for systems with variables in d is considered another analytic representation with theta functions on a strip for systems with positions in a circle x1d54a and momenta in is also considered the reproducing kernel formalism for these two systems is studied wigner and weyl functions in this language are also studied

z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

on the polar decomposition of the quantum su 2 algebra polar decomposition of the generators of the su 2 q algebra is performed and its contraction limit to the deformed oscillator algebra is considered using the phase operator occurring in the decomposition and its conjugate the authors write a realisation of the su 2j 1 algebra into which they embed the deformed su 2 realisation finally they discuss the classical limit of the polar decomposition and the case of the quantum su 2 dynamics

multiterminal source coding with high resolution we consider separate encoding and joint decoding of correlated continuous information sources subject to a difference distortion measure we first derive a multiterminal extension of the shannon lower bound for the rate region then we show that this shannon outer bound is asymptotically tight for small distortions these results imply that the loss in the sum of the coding rates due to the separation of the encoders vanishes in the limit of high resolution furthermore lattice quantizers followed by slepian wolf lossless encoding are asymptotically optimal we also investigate the high resolution rate region in the remote coding case where the encoders observe only noisy versions of the sources for the quadratic gaussian case we establish a separation result to the effect that multiterminal coding aimed at reconstructing the noisy sources subject to the rate constraints followed by estimation of the remote sources from these reconstructions is optimal under certain regularity conditions on the structure of the coding scheme

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

an extended weyl wigner transformation for special finite spaces abstract we extend the weyl wigner transformation to those particular degrees of freedom described by a finite number of states using a technique of constructing operator bases developed by schwinger discrete transformation kernels are presented instead of continuous coordinate momentum pair system and systems such as the one dimensional canonical continuous coordinate momentum pair system and the two dimensional rotation system are described by special limits expressions are explicitly given for the spin one half case

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

the complete heyting algebra of subsystems and contextuality the finite set of subsystems of a finite quantum system with variables in z n is studied as a heyting algebra the physical meaning of the logical connectives is discussed it is shown that disjunction of subsystems is more general concept than superposition consequently the quantum probabilities related to commuting projectors in the subsystems are incompatible with associativity of the join in the heyting algebra unless if the variables belong to the same chain this leads to contextuality which in the present formalism has as contexts the chains in the heyting algebra logical bell inequalities which contain heyting factors are discussed the formalism is also applied to the infinite set of all finite quantum systems which is appropriately enlarged in order to become a complete heyting algebra

quantum mechanics on profinite groups and partial order inverse limits and profinite groups are used in a quantum mechanical context two cases are considered a quantum system with positions in the profinite group and momenta in the group and a quantum system with positions in the profinite group and momenta in the group the corresponding schwatz bruhat spaces of wavefunctions and the heisenberg weyl groups are discussed the sets of subsystems of these systems are studied from the point of view of partial order theory it is shown that they are directed complete partial orders it is also shown that they are topological spaces with t0 topologies and this is used to define continuity of various physical quantities the physical meaning of profinite groups non archimedean metrics partial orders and t0 topologies in a quantum mechanical context is discussed

asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization two results are given first using a result of csiszar 1973 the asymptotic i e high resolution low distortion performance for entropy constrained tessellating vector quantization heuristically derived by gersho 1979 is proven for all sources with finite differential entropy this implies using gersho s conjecture and zador s formula that tessellating vector quantizers are asymptotically optimal for this broad class of sources and generalizes a rigorous result of gish and pierce 1968 from the scalar to the vector case second the asymptotic performance is established for zamir and feder s 1992 randomized lattice quantization with the only assumption that the source has finite differential entropy it is proven that the low distortion performance of the zamir feder universal vector quantizer is asympotically the same as that of the deterministic lattice quantizer

asymptotic design of quantizers for decentralized mmse estimation conceptual and practical encoding decoding aimed at accurately reproducing remotely collected observations has been heavily investigated since the pioneering works by shannon about source coding however when the goal is not to reproduce the observables but making inference about an embedded parameter and the scenario consists of many unconnected remote nodes the landscape is less certain we consider a multiterminal system designed for efficiently estimating a random parameter according to the minimum mean square error mmse criterion the analysis is limited to scalar quantizers followed by a joint entropy encoder and it is performed in the high resolution regime where the problem can be more easily mathematically tackled focus is made on the peculiarities deriving from the estimation task as opposed to that of reconstruction as well as on the multiterminal as opposite to centralized character of the inference the general form of the optimal nonuniform quantizer is derived and examples are given

quantum mechanics on p adic numbers a quantum system with positions in and momenta in is studied displacement operators in the phase space of this system and the corresponding heisenberg weyl group are studied it is shown that such a system can be constructed from a semi infinite chain of spins which are coupled in a particular way

on the sum rate of gaussian multiterminal source coding new proofs and results we show that the lower bound on the sum rate of the direct and indirect gaussian multiterminal source coding problems can be derived in a unified manner by exploiting the semidefinite partial order of the distortion covariance matrices associated with the minimum mean squared error mmse estimation and the so called reduced optimal linear estimation through which an intimate connection between the lower bound and the berger tung upper bound is revealed we give a new proof of the minimum sum rate of the indirect gaussian multiterminal source coding problem i e the gaussian ceo problem for the direct gaussian multiterminal source coding problem we derive a general lower bound on the sum rate and establish a set of sufficient conditions under which the lower bound coincides with the berger tung upper bound we show that the sufficient conditions are satisfied for a class of sources and distortion constraints in particular they hold for arbitrary positive definite source covariance matrices in the high resolution regime in contrast with the existing proofs the new method does not rely on shannon s entropy power inequality

the non regular ceo problem we consider the ceo problem for non regular source distributions such as uniform or truncated gaussian a group of agents observe independently corrupted versions of data and transmit coded versions over rate limited links to a ceo the ceo then estimates the underlying data based on the received coded observations agents are not allowed to convene before transmitting their observations this formulation is motivated by the practical problem of a firm s ceo estimating non regular beliefs about a sequence of events before acting on them agents observations are modeled as jointly distributed with the underlying data through a given conditional probability density function we study the asymptotic behavior of the minimum achievable mean squared error distortion at the ceo in the limit when the number of agents l and the sum rate r tend to infinity we establish a 1 r 2 convergence of the distortion an intermediate regime of performance between the exponential behavior in discrete ceo problems berger zhang and viswanathan 1996 and the 1 r behavior in gaussian ceo problems viswanathan and berger 1997 achievability is proved by a layered architecture with scalar quantization distributed entropy coding and midrange estimation the converse is proved using the bayesian chazan zakai ziv bound

on sic povms in prime dimensions the generalized pauli group and its normalizer the clifford group have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete povms sic povms to date almost every known sic povm fiducial vector is an eigenstate of a canonical unitary in the clifford group i show that every canonical unitary in prime dimensions p 3 lies in the same conjugacy class of the clifford group and give a class representative for all such dimensions it follows that if even one such sic povm fiducial vector is an eigenvector of such a unitary then all of them are for a given such dimension i also conjecture that in all dimensions d the number of conjugacy classes is bounded above by 3 and depends only on dmod9 and i support this claim with computer computations in all dimensions 48

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

on the sum rate loss of quadratic gaussian multiterminal source coding this work studies the sum rate loss of quadratic gaussian multiterminal source coding i e the difference between the minimum sum rates of distributed encoding and joint encoding both with joint decoding of correlated gaussian sources subject to mse distortion constraints on individual sources it is shown that under the nondegraded assumption i e all target distortions are simultaneously achievable by a gaussian berger tung scheme the supremum of the sum rate loss of distributed encoding over joint encoding of l jointly gaussian sources increases almost linearly in the number of sources l with an asymptotic slope of 0 1083 bit per sample per source as l goes to infinity this result is obtained even though we currently do not have the full knowledge of the minimum sum rate for the distributed encoding case the main idea is to upper bound the minimum sum rate of multiterminal source coding by that achieved by parallel gaussian test channels while lower bounding the minimum sum rate of joint encoding by a reverse water filling solution to a relaxed joint encoding problem of the same set of gaussian sources with a sum distortion constraint that equals the sum of the individual target distortions we show that under the nondegraded assumption the supremum difference between the upper bound for distributed encoding and the lower bound for joint encoding is achieved in the bi eigen equal variance with equal distortion case in which both bounds are known to be tight

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

universal quantization for separate encodings and joint decoding of correlated sources we consider the multi user lossy source coding problem for continuous alphabet sources in a previous work ziv proposed a single user universal coding scheme which uses uniform quantization with dither followed by a lossless source encoder entropy coder in this paper we generalize ziv s scheme to the multi user setting for this generalized universal scheme upper bounds are derived on the redundancies defined as the differences between the actual rates and the closest corresponding rates on the boundary of the rate region for the mean square error distortion measure it is shown that this scheme can achieve redundancies of no more than 0 754 b per sample for each user these bounds are obtained without the knowledge of the multi user rate region which is an open problem in general as a direct consequence of these results the inner and outer bounds on the rate distortion achievable region are obtained

on the structure of real time encoding and decoding functions in a multiterminal communication system a real time communication system with two encoders communicating with a single receiver over separate noisy channels is considered the two encoders make distinct partial observations of a markov source each encoder must encode its observations into a sequence of discrete symbols the symbols are transmitted over noisy channels to a finite memory receiver that attempts to reconstruct some function of the state of the markov source encoding and decoding must be done in real time that is the distortion measure does not tolerate delays under the assumption that the encoders observations are conditionally independent markov chains given an unobserved time invariant random variable results on the structure of optimal real time encoding and decoding functions are obtained it is shown that there exist finite dimensional sufficient statistics for the encoders the problem with noiseless channels and perfect memory at the receiver is then considered a new methodology to find the structure of optimal real time encoders is employed a sufficient statistic with a time invariant domain is found for this problem this methodology exploits the presence of common information between the encoders and the receiver when communication is over noiseless channels

rate region of the quadratic gaussian two encoder source coding problem we determine the rate region of the quadratic gaussian two encoder source coding problem this rate region is achieved by a simple architecture that separates the analog and digital aspects of the compression furthermore this architecture requires higher rates to send a gaussian source than it does to send any other source with the same covariance our techniques can also be used to determine the sum rate of some generalizations of this classical problem our approach involves coupling the problem to a quadratic gaussian ldquoceo problem rdquo

an efficient numerical method for compressible flows of a real gas using arithmetic averaging an efficient numerical method is presented for the solution of the euler equations governing the compressible flow of a real gas the scheme is based on the approximate solution of a specially constructed set of linearised riemann problems an average of the flow variables across the interface between cells is required and this is chosen to be the arithmetic mean for computational efficiency which is in contrast to the usual square root averaging the scheme is applied to a test problem for five different equations of state

finite dimensional hilbert space and frame quantization the quantum observables used in the case of quantum systems with finite dimensional hilbert space are defined either algebraically in terms of an orthonormal basis and discrete fourier transformation or by using a continuous system of coherent states we present an alternative approach to these important quantum systems based on the finite frame quantization finite systems of coherent states usually called finite tight frames can be defined in a natural way in the case of finite quantum systems novel examples of such tight frames are presented the quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system they are obtained by using a finite frame and a klauder berezin toeplitz type quantization semi classical aspects of tight frames are studied through lower symbols of basic classical observables

some results on the parametrization of complex hadamard matrices in this paper we provide an analytical procedure which leads to a system of n 2 2 polynomial equations whose solutions give the parametrization of the complex n n hadamard matrices it is shown that in general the hadamard matrices depend on a number of arbitrary phases and a lower bound for this number is given the moduli equations define interesting geometrical objects whose study will shed light on the parametrization of the hadamard matrices as well as on some interesting geometrical varieties defined by them

mutually unbiased bases and hadamard matrices of order six we report on a search for mutually unbiased bases mubs in six dimensions we find only triplets of mubs and thus do not come close to the theoretical upper bound 7 however we point out that the natural habitat for sets of mubs is the set of all complex hadamard matrices of the given order and we introduce a natural notion of distance between bases in hilbert space this allows us to draw a detailed map of where in the landscape the mub triplets are situated we use available tools such as the theory of the discrete fourier transform to organize our results finally we present some evidence for the conjecture that there exists a four dimensional family of complex hadamard matrices of order 6 if this conjecture is true the landscape in which one may search for mubs is much larger than previously thought

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

discrete canonical transforms that are hadamard matrices the group of symplectic linear canonical transformations has an integral kernel which has quadratic and linear phases and which is realized by the geometric paraxial optical model the discrete counterpart of this model is a finite hamiltonian system that acts on n point signals through n n matrices whose elements also have a constant absolute value although they do not form a representation of that group those matrices that are also unitary are hadamard matrices we investigate the manifolds of these n n matrices under the equivalence imposed by the model and find them to be on two sided cosets by means of an algorithm we determine representatives that lead to collections of mutually unbiased bases

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

multiterminal source coding achievable rates and reliability some open problems concerning classical multiterminal configuration in which two correlated sources x and y are encoded separately and decoded by common decoder with respect to a fidelity criterion for messages of both sources are resolved the results of berger and yeung 1989 are expanded in two directions first we determine the rate distortion region r spl delta of achievable rates r sub x r sub y for a given spl delta spl delta sub x spl delta sub y with spl delta sub x the permissible distortion level for reproduction of x and spl delta sub y for y thus we give a full solution of the problem noted by kaspi and berger 1982 they constructed the inner bound for r spl delta and solved by berger and yeung for the case of one distortion second we introduce the notion of rate reliability distortion region for the multiterminal source encoding problem more precisely the region of admissible rates of codes ensuring exponential decrease with a given exponent e 0 of the probability of exceeding the distortion levels spl delta sub x or spl delta sub y the inner and outer bounds for this region r e spl delta are constructed the analytical representation obtained for r spl delta proved to be somewhat simpler than that considered by kaspi and berger and by berger and yeung ours uses only variables pertaining to the problem statement with no auxiliary ones

there is no generalization of known formulas for mutually unbiased bases in a quantum system having a finite number n of orthogonal states two orthonormal bases ai and bj are called mutually unbiased if all inner products ai bj have the same modulus 1 n this concept appears in several quantum information problems the number of pairwise mutually unbiased bases is at most n 1 and various constructions of such n 1 bases have been found when n is a power of a prime number we study families of formulas that generalize these constructions to arbitrary dimensions using finite rings we then prove that there exists a set of n 1 mutually unbiased bases described by such formulas if and only if n is a power of a prime number

phase operators phase states and vector phase states for su3 and su2 1 this paper focuses on phase operators phase states and vector phase states for the sl3 lie algebra we introduce a one parameter generalized oscillator algebra a 2 which provides a unified scheme for dealing with su3 for 0 and h4 h4 for 0 symmetries finite and infinite dimensional representations of a 2 are constructed for 0 and 0 respectively phase operators associated with a 2 are defined and temporally stable phase states as well as vector phase states are constructed as eigenstates of these operators finally we discuss a relation between quantized phase states and a quadratic discrete fourier transform and show how to use these states for constructing mutually unbiased bases

the metaplectic group within the heisenberg weyl ring the heisenberg weyl ring contains the metaplectic group of canonical transforms acting unitarily on l 2 r these ring elements are characterized through i the integral transform kernels ii coset distributions and iii classical functions under any quantization scheme the isomorphism under group composition leads to several new relations involving twisted products and quantization of gaussian classical functions the wigner inversion operator is a special central group element it is shown that the only quantization scheme invariant under metaplectic transformations is the weyl scheme the structure studied here appears to be relevant to the study of wave optics with aberration

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

feynman s path integral and mutually unbiased bases our previous work on quantum mechanics in hilbert spaces of finite dimension n is applied to elucidate the deep meaning of feynman s path integral pointed out by g svetlichny he speculated that the secret of the feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases we confirm the corresponding property of the short time propagator by using a specially devised n n approximation of quantum mechanics in applied to our finite dimensional analogue of a free quantum particle

multiterminal source encoding with one distortion criterion the authors unify earlier investigations concerning the encoding of two correlated sources x sub k y sub k by means of separate encoders decoding is done by a single decoder which receives the outputs from both encoders the reconstruction of x sub k is required to be perfect in the usual shannon sense the authors determine the admissible rate region r d where d is the distortion of the reconstruction of y sub k the binary hamming case is investigated explicitly

lossy broadcasting with complementary side information a pair of strings x y put out by a memoryless source needs to be reliably communicated over a memoryless broadcast channel receiver 1 has y as side information and must reconstruct x to within some distortion receiver 2 has x and must reconstruct y to within some distortion the problem is motivated by the broadcast phase downlink of the two way relay channel we characterize reliable communication for gaussian sources with quadratic distortion functions conditionally independent sources deterministic distortion functions and small distortions with hamming distortion functions the last result is obtained by solving a new version of the broadcast problem with steinberg s common reconstruction decoding constraint

partial order and a t0 topology in a set of finite quantum systems a whole part theory is developed for a set of finite quantum systems n with variables in z n the partial order subsystem is defined by embedding various attributes of the system m quantum states density matrices etc into their counterparts in the supersystem n for m n the compatibility of these embeddings is studied the concept of ubiquity is introduced for quantities which fit with this structure it is shown that various entropic quantities are ubiquitous the sets of various quantities become t0 topological spaces with the divisor topology which encapsulates fundamental physical properties these sets can be converted into directed complete partial orders by adding top elements the continuity of various maps among these sets is studied

an improved outer bound for multiterminal source coding we prove a new outer bound on the rate distortion region for the multiterminal source coding problem this bound subsumes the best outer bound in the literature and improves upon it strictly in some cases the improved bound enables us to obtain a new conclusive result for the binary erasure version of the ceo problem the bound recovers many of the converse results that have been established for special cases of the problem including the recent one for the gaussian two encoder problem

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

on sampling and coding for distributed acoustic sensing the issue of how to efficiently represent the data collected by a network of microphones recording spatio temporal acoustic wave fields is addressed each sensor node in the network samples the sound field quantizes the samples and transmits the encoded samples to some central unit which computes an estimate of the original sound field based on the information received from all the microphones our analysis is based on the spectral properties of the sound field which are induced by the physics of wave propagation and have a significant impact on the efficiency of the chosen sampling lattice and coding scheme as field acquisition by a sensor network typically implies spatio temporal sampling of the field a multidimensional sampling theorem for homogeneous random fields with compactly supported spectral measures is proved to assess the loss of information implied by source coding rate distortion functions for various coding schemes and sampling lattices are determined in particular centralized coding independent coding and some multiterminal schemes are compared under the assumption of spectral whiteness of the sound field it is shown that sampling with a quincunx lattice followed by independent coding is optimal as it achieves the lower bound given by centralized coding

the rate distortion function for the quadratic gaussian ceo problem a new multiterminal source coding problem called the ceo problem was presented and investigated by berger zhang and viswanathan recently viswanathan and berger have investigated an extension of the ceo problem to gaussian sources and call it the quadratic gaussian ceo problem they considered this problem from a statistical viewpoint deriving some interesting results in this paper we consider the quadratic gaussian ceo problem from a standpoint of multiterminal rate distortion theory we regard the ceo problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given this viewpoint leads us to a complete solution to the problem we determine the tradeoff between the total amount of rate and squared distortion deriving an explicit formula of the rate distortion function the derived function has the form of a sum of two nonnegative functions one is a classical rate distortion function for single gaussian source and the other is a new rate distortion function which dominates the performance of the system for a relatively small distortion it follows immediately from our result that the conjecture of viswanathan and berger on the asymptotic behavior of minimum squared distortion for large rates is true

weak mutually unbiased bases quantum systems with variables in are considered the properties of lines in the phase space of these systems are studied weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d 1 2 or alternatively to one of the d 1 2i 0 where di is a divisor of d apart from d 1 they are designed for the geometry of the phase space in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin in the special case of prime d there are no divisors of d apart from 1 d and the weak mutually unbiased bases are mutually unbiased bases

selfish distributed compression over networks correlation induces anarchy we consider the min cost multicast problem under network coding with multiple correlated sources where each terminal wants to losslessly reconstruct all the sources we study the inefficiency brought forth by the selfish behavior of the terminals in this scenario by modeling it as a noncooperative game among the terminals the degradation in performance due to the lack of regulation is measured by the price of anarchy poa which is defined as the ratio between the cost of the worst possible wardrop equilibrium and the socially optimum cost our main result is that in contrast with the case of independent sources the presence of source correlations can significantly increase the price of anarchy toward establishing this result we first characterize the socially optimal flow and rate allocation in terms of four intuitive conditions next we show that the wardrop equilibrium is a socially optimal solution for a different set of related cost functions using this we construct explicit examples that demonstrate that the poa 1 and determine near tight upper bounds on the poa as well the main techniques in our analysis are lagrangian duality theory and the usage of the supermodularity of conditional entropy

an efficient numerical scheme for the euler equations a finite difference scheme based on flux difference splitting is presented for the solution of the euler equations for the compressible flow of an ideal gas a linearised riemann problem is defined and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem an average of the flow variables across the interface between cells is required and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging this is in contrast to the usual square root averages found in this type of riemann solver where the computational expense can be prohibitive the method of upwind differencing is used for the resulting scalar problems together with a flux limiter for obtaining a second order scheme which avoids nonphysical spurious oscillations the scheme is applied to a shock tube problem and a blast wave problem each approximate solution compares well with those given by other schemes and for the shock tube problem is in agreement with the exact solution

similarity solutions for multi component flows a one dimensional shock reflection test problem in the case of slab cylindrical or spherical symmetry is discussed for multi component flows the differential equations for a similarity solution are derived and then solved numerically in conjunction with the rankine hugoniot shock relations

a wavelet based two stage near lossless coder in this paper we present a two stage near lossless compression scheme it belongs to the class of lossy plus residual coding and consists of a wavelet based lossy layer followed by arithmetic coding of the quantized residual to guarantee a given linfin error bound in the pixel domain we focus on the selection of the optimum bit rate for the lossy layer to achieve the minimum total bit rate unlike other similar lossy plus lossless approaches using a wavelet based lossy layer the proposed method does not require iteration of decoding and inverse discrete wavelet transform in succession to locate the optimum bit rate we propose a simple method to estimate the optimal bit rate with a theoretical justification based on the critical rate argument from the rate distortion theory and the independence of the residual error

information rates of densely sampled data distributed vector quantization and scalar quantization with transforms for gaussian sources this paper establishes rates attainable by several lossy schemes for coding a continuous parameter source to a specified mean squared error distortion based on sampling at asymptotically large rates first a densely sampled spatiotemporal stationary gaussian source is distributively encoded the berger tung bound to the distributed rate distortion function and three convergence theorems are used to obtain an upper bound expressed in terms of the source spectral density to the smallest attainable rate at asymptotically large sampling rates the bound is tighter than that recently obtained by kashyap both indicate that with ideal distributed lossy coding dense sensor networks can efficiently sense and convey a field in contrast to the negative result obtained by marco for encoders based on scalar quantization and slepian wolf distributed lossless coding the second scheme is transform coding with scalar coefficient quantization a new generalized transform coding analysis as well as the aforementioned convergence theorems is used to find the smallest attainable rate at asymptotically large sampling rates in terms of the source spectral density and the operational rate distortion function of the family of quantizers which in contrast to previous analyses need not be convex the result shows that when a transform is used scalar quantization need not cause the poor performance found by marco as a corollary the final result pursues an approach originally proposed by berger to show that the inverse water pouring formula for the rate distortion function can be attained at high sampling rates by transform coding with ideal vector quantization to encode the coefficients also established in the paper are relations between operational rate distortion and distortion rate functions for a continuous parameter source and those for the discrete parameter source that results from sampling

the distributed karhunen loeve transform the karhunen loeve transform klt is a key element of many signal processing and communication tasks many recent applications involve distributed signal processing where it is not generally possible to apply the klt to the entire signal rather the klt must be approximated in a distributed fashion this paper investigates such distributed approaches to the klt where several distributed terminals observe disjoint subsets of a random vector we introduce several versions of the distributed klt first a local klt is introduced which is the optimal solution for a given terminal assuming all else is fixed this local klt is different and in general improves upon the marginal klt which simply ignores other terminals both optimal approximation and compression using this local klt are derived two important special cases are studied in detail namely the partial observation klt which has access to a subset of variables but aims at reconstructing them all and the conditional klt which has access to side information at the decoder we focus on the jointly gaussian case with known correlation structure and on approximation and compression problems then the distributed klt is addressed by considering local klts in turn at the various terminals leading to an iterative algorithm which is locally convergent sometimes reaching a global optimum depending on the overall correlation structure for compression it is shown that the classical distributed source coding techniques admit a natural transform coding interpretation the transform being the distributed klt examples throughout illustrate the performance of the proposed distributed klt this distributed transform has potential applications in sensor networks distributed image databases hyper spectral imagery and data fusion

linear estimation of correlated data in wireless sensor networks with optimum power allocation and analog modulation in this paper we study the energy efficient distributed estimation problem for a wireless sensor network where a physical phenomena that produces correlated data is sensed by a set of spatially distributed sensor nodes and the resulting noisy observations are transmitted to a fusion center via noise corrupted channels we assume a gaussian network model where i the data samples being sensed at different sensors have a correlated gaussian distribution and the correlation matrix is known at the fusion center ii the links between the local sensors and the fusion center are subject to fading and additive white gaussian noise awgn and the fading gains are known at the fusion center and iii the central node uses the squared error distortion metric we consider two different distortion criteria i individual distortion constraints at each node and ii average mean square error distortion constraint across the network we determine the achievable power distortion regions under each distortion constraint taking the delay constraint into account we investigate the performance of an uncoded transmission strategy where the noisy observations are only scaled and transmitted to the fusion center at the fusion center two different estimators are considered i the best linear unbiased estimator blue that does not require knowledge of the correlation matrix and ii the minimum mean square error mmse estimator that exploits the correlations for each estimation method we determine the optimal power allocation that results in a minimum total transmission power while satisfying some distortion level for the estimate under both distortion criteria the numerical comparisons between the two schemes indicate that the mmse estimator requires less power to attain the same distortion provided by the blue and this performance gap becomes more dramatic as correlations between the observations increase furthermore comparisons between power distortion region achieved by the theoretically optimum system and that achieved by the uncoded system indicate that the performance gap between the two systems becomes small for low levels of correlation between the sensor observations if observations at all sensor nodes are uncorrelated the uncoded system with mmse estimator attains the theoretically optimum system performance

the rate loss in the wyner ziv problem the rate distortion function for source coding with side information at the decoder the wyner ziv problem is given in terms of an auxiliary random variable which forms a markov chain with the source and the side information this markov chain structure typical to the solution of multiterminal source coding problems corresponds to a loss in coding rate with respect to the conditional rate distortion function i e to the case where the encoder is fully informed we show that for difference or balanced distortion measures this loss is bounded by a universal constant which is the minimax capacity of a suitable additive noise channel furthermore in the worst case this loss is equal to the maximin redundancy over the rate distortion function of the additive noise test channel for example the loss in the wyner ziv problem is less than 0 5 bit sample in the squared error distortion case and it is less than 0 22 bit for a binary source with hamming distance these results have implications also in universal quantization with side information and in more general multiterminal source coding problems

on source coding for distributed temperature sensing with shift invariant geometries we study the source coding problem in sensor networks deployed to monitor the evolution of spatio temporal temperature distributions the sensors sample the temperature field quantize the samples and transmit the encoded samples through digital channels to some central unit which computes an estimate of the original temperature field our analysis is based on the heat kernel s spectral properties which are induced by the physics of heat diffusion we determine rate distortion functions for various source coding schemes in particular we compare centralized coding independent coding berger tung coding and predictive quantization

rate distortion for correlated sources with partially separated encoders we resolve some open problems concerning the encoding of a pair of correlated sources with respect to a fidelity criterion two encoders are used one encoder observes only the output of the first source while the other is supplied both with the second source output and with partial information about the first source at some predetermined rate a general coding theorem is proved which established that cal s ast a certain region defined in terms of single letter information theoretic quantities is an inner bound to the region of all attainable vectors of rates and distortions for certain cases of interest the converse is proved too thereby establishing the rate distortion region for these cases

distributed functional scalar quantization simplified distributed functional scalar quantization dfsq theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired we address two limitations of previous works prohibitively expensive decoder design and a restriction to source distributions with bounded support we show that a much simpler decoder has equivalent asymptotic performance to the conditional expectation estimator studied previously thus reducing decoder design complexity the simpler decoder features decoupled communication and computation blocks moreover we extend the dfsq framework with the simpler decoder to source distributions with unbounded support finally through simulation results we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis and we give new insight on the rate of convergence

about mutually unbiased bases in even and odd prime power dimensions mutually unbiased bases generalize the x y and z qubit bases they possess numerous applications in quantum information science it is well known that in prime power dimensions n pm with p prime and m a positive integer there exists a maximal set of n 1 mutually unbiased bases in the present paper we derive an explicit expression for those bases in terms of the operations of the associated finite field galois division ring of n elements this expression is shown to be equivalent to the expressions previously obtained by ivanovic 1981 j phys a math gen 14 3241 in odd prime dimensions and wootters and fields 1989 ann phys 191 363 in odd prime power dimensions in even prime power dimensions we derive a new explicit expression for the mutually unbiased bases the new ingredients of our approach are basically the following we provide a simple expression of the generalized pauli group in terms of the additive characters of the field and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized pauli group renormalized by a well chosen phase factor

mutually unbiased bases generalized spin matrices and separability abstract a collection of orthonormal bases for a complex d dimensional hilbert space is called mutually unbiased mub if for any two vectors v and w from different bases the square of the inner product equals 1 d v w 2 1 d the mub problem is to prove or disprove the existence of a maximal set of d 1 bases it has been shown in ann phys 191 1989 363 that such a collection exists if d is a power of a prime number p we revisit this problem and use d d generalizations of the pauli spin matrices to give a constructive proof of this result specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the mub problem additionally we give formulas from which the orthogonal bases can be readily computed we show how the techniques developed here provide a natural way to analyze the separability of the bases the techniques used require properties of algebraic field extensions and the relevant part of that theory is included in the appendix

symplectic transformations and quantum tomography in finite quantum systems quantum systems where the position and momentum are in the ring zd d is an odd integer are considered symplectic transformations are studied and the order of sp 2 zd is calculated quantum tomography is also discussed it is shown that measurements used in the inverse radon transform need to be made on j2 d lines where j2 d is the jordan totient function

a semi continuous version of the berger yeung problem we present a continuous dual of multiterminal source encoding with one distortion criterion the berger yeung 1989 problem a continuous source x is encoded with high resolution d sub x spl rarr 0 with the aid of a helper i e a correlated discrete or continuous source y that is encoded separately subject to some arbitrary distortion criterion d sub y we find the asymptotic form of the set of achievable coding rates r d sub x d sub y of the x and y encoders as d sub x spl rarr 0 two extreme cases of our result provide high resolution interpretations to the classical work of wyner 1975 1978 ahlswede 1975 korner 1975 and ziv 1976 on source coding with side information

projective ring line of an arbitrary single qudit as a continuation of our previous work havlicek and saniga 2007 j phys a math gen 40 f943 52 preprint 0708 4333 an algebraic geometrical study of a single d dimensional qudit is made with d being any positive integer the study is based on an intricate relation between the symplectic module of the generalized pauli group of the qudit and the fine structure of the projective line over the modular ring explicit formulae are given for both the number of generalized pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of we find remarkably that a perp set is not a set theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes the operators are also seen to be structured into disjoint layers according to the degree of their representing vectors a brief comparison with some multiple qudit cases is made

distributed scalar quantization for computing high resolution analysis and extensions communication of quantized information is frequently followed by a computation we consider situations of distributed functional scalar quantization distributed scalar quantization of possibly correlated sources followed by centralized computation of a function under smoothness conditions on the sources and function companding scalar quantizer designs are developed to minimize mean squared error mse of the computed function as the quantizer resolution is allowed to grow striking improvements over quantizers designed without consideration of the function are possible and are larger in the entropy constrained setting than in the fixed rate setting as extensions to the basic analysis we characterize a large class of functions for which regular quantization suffices consider certain functions for which asymptotic optimality is achieved without arbitrarily fine quantization and allow limited collaboration between source encoders in the entropy constrained setting a single bit per sample communicated between encoders can have an arbitrarily large effect on functional distortion in contrast such communication has very little effect in the fixed rate setting

on multiterminal source code design multiterminal mt source coding refers to separate lossy encoding and joint decoding of multiple correlated sources recently the rate region of both direct and indirect mt source coding in the quadratic gaussian setup with two encoders was determined we are thus motivated to design practical mt source codes that can potentially achieve the entire rate region in this paper we present two practical mt coding schemes under the framework of slepian wolf coded quantization swcq for both direct and indirect mt problems the first asymmetric swcq scheme relies on quantization and wyner ziv coding and it is implemented via source splitting to achieve any point on the sum rate bound in the second conceptually simpler scheme symmetric swcq the two quantized sources are compressed using symmetric slepian wolf coding via a channel code partitioning technique that is capable of achieving any point on the slepian wolf sum rate bound our practical designs employ trellis coded quantization and turbo low density parity check ldpc codes for both asymmetric and symmetric slepian wolf coding simulation results show a gap of only 0 139 0 194 bit per sample away from the sum rate bound for both direct and indirect mt coding problems

the isotropic lines of bb z _ d 2 we show that the isotropic lines in the lattice are the lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice the set of isotropic lines decomposes into orbits under the action of we give an explicit description of those orbits as well as their number and respective cardinalities we also develop two group actions on the group related to the topic

group theoretical construction of mutually unbiased bases in hilbert spaces of prime dimensions mutually unbiased bases in hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity an alternative proof of existence of a maximal collection of n 1 mutually unbiased bases in hilbert spaces of prime dimension n is given by exploiting the finite heisenberg group also called the pauli group and the action of on finite phase space implemented by unitary operators in the hilbert space crucial for the proof is that for prime is also a finite field

on quasi orthogonal signatures for cdma systems sum capacity optimal signatures in synchronous code division multiple access cdma systems are functions of the codebook length as well as the number of active users a new signature set must be assigned every time the number of active users changes this correspondence considers signature sets that are less sensitive to changes in the number of active users equiangular signature sequences are proven to solve a certain max min signal to interference plus noise problem which results from their interference invariance unions of orthonormal bases have subsets that come close to satisfying the welch bound bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided connections are made between these signature design problems grassmannian line packing frame theory and algebraic geometry

partial ordering of weak mutually unbiased bases a quantum system with variables in where with prime numbers is considered the non near linear geometry of the phase space is studied the lines through the origin are factorized in terms of prime factor lines in weak mutually unbiased bases wmub which are products of the mutually unbiased bases in the prime factor hilbert spaces are also considered the factorization of both lines and wmub is analogous to the factorization of integers in terms of prime numbers the duality between lines and wmub is discussed it is shown that there is a partial order in the set of subgeometries of isomorphic to the partial order in the set of subsystems of

phase operators temporally stable phase states mutually unbiased bases and exactly solvable quantum systems we introduce a one parameter generalized oscillator algebra that covers the case of the harmonic oscillator algebra and discuss its finite and infinite dimensional representations according to the sign of the parameter we define an hamiltonian operator associated with and examine the degeneracies of its spectrum for the finite when 0 and the infinite when 0 representations of we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators to overcome the difficulties related to the phase operator in the infinite dimensional case and to avoid the degeneracy problem for the finite dimensional case we introduce a truncation procedure which generalizes the one used by pegg and barnett for the harmonic oscillator this yields a truncated generalized oscillator algebra where s denotes the truncation order we construct two types of temporally stable states for as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of two applications are considered in this paper the first concerns physical realizations of and in the context of one dimensional quantum systems with finite morse system or infinite poschl teller system discrete spectra the second deals with mutually unbiased bases used in quantum information

hamiltonians of quantum systems with positions and momenta in gf pl a quantum system with positions and momenta in gf pl is considered such a system can be constructed from l smaller systems in which the positions and momenta take values in zp if the hamiltonian of this l partite system is compatible with gf pl the concept of compatibility of a hamiltonian with gf pl allows the quantum formalism in the l partite system to be expressed in terms of galois arithmetic transformations of the basis in gf pl produce unitary transformations of the quantum states which form a representation of gl l zp they are used to define which subset of the general set of hamiltonians in the l partite system is compatible with gf pl

the ceo problem multiterminal source coding we consider a new problem in multiterminal source coding motivated by the following decentralized communication estimation task a firm s chief executive officer ceo is interested in the data sequence x t sub t 1 sup spl infin which cannot be observed directly perhaps because it represents tactical decisions by a competing firm the ceo deploys a team of l agents who observe independently corrupted versions of x t sub t 1 sup spl infin because x t is only one among many pressing matters to which the ceo must attend the combined data rate at which the agents may communicate information about their observations to the ceo is limited to say r bits per second if the agents were permitted to confer and pool their data then in the limit as l spl rarr spl infin they usually would be able to smooth out their independent observation noises entirely then they could use their r bits per second to provide the ceo with a representation of x t with fidelity d r where d spl middot is the distortion rate function of x t in particular with such data pooling d can be made arbitrarily small if r exceeds the entropy rate h of x t suppose however that the agents are not permitted to convene agent i having to send data based solely on his own noisy observations y sub i t we show that then there does not exist a finite value of r for which even infinitely many agents can make d arbitrarily small furthermore in this isolated agents case we determine the asymptotic behavior of the minimal error frequency in the limit as l and then r tend to infinity

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina