crossed products of nuclear c algebras and their traces abstract this work addresses the blackadar kirchberg question bkq as it pertains to group and reduced crossed product c algebras the bkq asks if every stably finite c algebra admits the matricial field mf property we characterize the mf property for a class of reduced crossed product c algebras and for their traces we prove that every ordered abelian group is in a sense mf and show that there is no k theoretic obstruction to an affirmative answer to the bkq using classification techniques and induced k theoretic dynamics we give an affirmative answer to the bkq for reduced crossed products of ah algebras of real rank zero by free groups combining our results with recent progress in elliott s classification program shows that the reduced crossed product of a separable simple unital nuclear uct and monotracial c algebra by the free group is always mf we also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties by appealing to a result of ozawa rordam and sato we show that semidirect products of discrete amenable groups by free groups admit mf reduced group c algebras

the monge metric on the sphere and geometry of quantum states 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 generalized husimi distributions to define the monge distance between two arbitrary 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 proportional to the identity matrix admits the largest value for the coherent states while the delocalized chaotic states are close to this contrasts the geometry induced by the standard trace hilbert schmidt or bures metrics for which the distance from 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

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

majorization entropic uncertainty relations entropic uncertainty relations in a finite dimensional hilbert space are investigated making use of the majorization technique we derive explicit lower bounds for the sum of r enyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables obtained bounds are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix numerical simulations show that for a generic unitary matrix of size n 5 our bound is stronger than the well known result of maassen and uffink mu with a probability larger than 98 we also show that the bounds investigated are invariant under the dephasing and permutation operations finally we derive a classical analogue of the mu uncertainty relation which is formulated for stochastic transition matrices

unified products for leibniz algebras applications abstract let g be a leibniz algebra and e a vector space containing g as a subspace all leibniz algebra structures on e containing g as a subalgebra are explicitly described and classified by two non abelian cohomological type objects hl g 2 v g provides the classification up to an isomorphism that stabilizes g and hl 2 v g will classify all such structures from the view point of the extension problem here v is a complement of g in e a general product called the unified product is introduced as a tool for our approach the crossed resp bicrossed products between two leibniz algebras are introduced as special cases of the unified product the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem the description and the classification of all complements of a given extension g e of leibniz algebras are given as a converse of the factorization problem they are classified by another cohomological object denoted by ha 2 h g where is the canonical matched pair associated to a given complement h several examples are worked out in details

a lie grading which is not a semigroup grading this note is devoted to the construction of a graded lie algebra whose grading is not given by a semigroup thus providing a counterexample to an assertion by patera and zassenhaus

further properties of lp dimension abstract this article establishes more properties of the l p dimension introduced in a previous article given an amenable group acting by translation on l p a number satisfying dimension like properties is associated to the usually infinite dimensional subspaces y of l p which are invariant under the action of this may be interpreted as a von neumann dimension for l p spaces as a consequence for p 1 2 if y is a closed non trivial invariant subspace of l p and let y n be an increasing sequence of closed invariant subspace such that y n l p v then there exist a k such that y k y 0

n dimensional filiform leibniz algebras of length n 1 and their derivations abstract in this work n dimensional filiform leibniz algebras admitting a gradation of length n 1 are classified derivations of such algebras are also described

preparation uncertainty implies measurement uncertainty in a class of generalized probabilistic theories in quantum theory it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values and that there is no joint measurement of them they are called preparation uncertainty and measurement uncertainty respectively and research has unveiled that they are not independent from but related with each other in a quantitative way this study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories gpts in particular a certain class of gpts is considered which can be characterized by transitivity and self duality and regarded as extensions of quantum theory it is proved that there are close connections expressed quantitatively between two types of uncertainty on a pair observables also in those theories if preparation uncertainty exists then measurement uncertainty also exists and they are described by similar inequalities our results manifest that their correspondences are not specific to quantum theory but more universal ones

entropic bounds on semiclassical measures for quantized one dimensional maps quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic hamiltonian systems are equidistributed in phase space this however does not prohibit existence of exceptional sequences which might converge to different non liouville classical invariant measures it has been recently shown by n anantharaman and s nonnenmacher in 20 21 with h koch that for anosov geodesic flows the metric entropy of any semiclassical measure must satisfy a certain bound this remarkable result seems to be optimal for manifolds of constant negative curvature but not in the general case where it might become even trivial if the negative curvature of the riemannian manifold varies a lot it has been conjectured by the same authors that in fact a stronger bound valid in the general case should hold

on the structure of graded leibniz triple systems abstract we study the structure of a leibniz triple system e graded by an arbitrary abelian group g which is considered of arbitrary dimension and over an arbitrary base field k we show that e is of the form e u j 1 i j with u a linear subspace of the 1 homogeneous component e 1 and any ideal i j of e satisfying i j e i k i j i k e e i j i k 0 if j k where the relation in 1 g g 1 l g 0 defined by g h if and only if g is connected to h

lie algebras with a set grading abstract consider g a lie algebra graded by an arbitrary set i set grading we show that g decomposes as the sum of well described graded ideals plus maybe a certain linear subspace under mild conditions the simplicity of g is characterized and it is shown that the above decomposition is actually the direct sum of the family of its minimal graded ideals each one being a simple set graded lie algebra

on extended graded poisson algebras abstract we introduce the class of extended graded poisson algebras as a generalization of the one of graded poisson algebras and study its structure if p is an extended graded poisson algebra we show that p is of the form p u i i i with u a linear subspace of p 0 and any i i a well described ideal of p satisfying i i i j i i i j 0 if i qj it is also shown that under certain conditions p is the direct sum of the family of its simple ideals

equivalence theorem of uncertainty relations we present an equivalence theorem to unify the two classes of uncertainty relations i e the variance based ones and the entropic forms showing that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators this means that an uncertainty relation in the language of entropy may be mapped onto a variance based one and vice versa employing the equivalence theorem alternative formulations of entropic uncertainty relations are obtained for the qubit system that are stronger than the existing ones in the literature and variance based uncertainty relations for spin systems are reached from the corresponding entropic uncertainty relations

higher entropic uncertainty relations for anti commuting observables uncertainty relations provide one of the most powerful formulations of the quantum mechanical principle of complementarity yet very little is known about such uncertainty relations for more than two measurements here we show that sufficient unbiasedness for a set of binary observables in the sense of mutual anticommutation is good enough to obtain maximally strong uncertainty relations in terms of the shannon entropy we also prove nearly optimal relations for the collision entropy this is the first systematic and explicit approach to finding an arbitrary number of measurements for which we obtain maximally strong uncertainty relations our results have immediate applications to quantum cryptography

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

maximally informative ensembles for sic povms in dimension 3 in order to find out for which initial states of the system the uncertainty of the measurement outcomes will be minimal one can look for the minimizers of the shannon entropy of the measurement in case of group covariant measurements this question becomes closely related to the problem of how informative the measurement is in the sense of its informational power namely the orbit under group action of the entropy minimizer corresponds to a maximally informative ensemble of equiprobable elements we give a characterization of such ensembles for three dimensional group covariant weyl heisenberg symmetric informationally complete positive operator valued measures sic povms in both geometric and algebraic terms it turns out that a maximally informative ensemble arises from the input state orthogonal to a subspace spanned by three linearly dependent vectors defining a sic povm geometrically or from an eigenstate of a certain weyl matrix algebraically

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

general entropy like uncertainty relations in finite dimensions we revisit entropic formulations of the uncertainty principle up for an arbitrary pair of positive operator valued measures povm a and b acting on finite dimensional hilbert space salicr generalized entropies including r nyi and tsallis ones among others are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables we obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals which is valid for both pure and mixed states the bound depends on the overlap triplet with ca respectively cb being the overlap between the elements of the povm a respectively b and the overlap between the pair of povm our approach is inspired by that of de vicente and s nchez ruiz 2008 phys rev a 77 042110 and consists in a minimization of the entropy sum subject to the landau pollak inequality that links the maximum probabilities of both observables we solve the constrained optimization problem in a geometrical way and furthermore when dealing with r nyi or tsallis entropic formulations of the up we overcome the h lder conjugacy constraint imposed on the entropic indices by the riesz thorin theorem in the case of nondegenerate observables we show that for given the bound obtained is optimal and that for r nyi entropies our bound improves deutsch one but maassen uffink bound prevails when finally we illustrate by comparing our bound with known previous results in particular cases of r nyi and tsallis entropies

on groups with quasidiagonal c algebras abstract we examine the question of quasidiagonality for c algebras of discrete amenable groups from a variety of angles we give a quantitative version of rosenberg s theorem via paradoxical decompositions and a characterization of quasidiagonality for group c algebras in terms of embeddability of the groups we consider several notable examples of groups such as topological full groups associated with cantor minimal systems and abels celebrated example of a finitely presented solvable group that is not residually finite and show that they have quasidiagonal c algebras finally we study strong quasidiagonality for group c algebras exhibiting classes of amenable groups with and without strongly quasidiagonal c algebras

asymptotic entropic uncertainty relations we analyze entropic uncertainty relations for two orthogonal measurements on a n dimensional hilbert space performed in two generic bases it is assumed that the unitary matrix u relating both bases is distributed according to the haar measure on the unitary group we provide lower bounds on the average shannon entropy of probability distributions related to both measurements the bounds are stronger than those obtained with use of the entropic uncertainty relation by maassen and uffink and they are optimal up to additive constants we also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations which hold with high probability with respect to the random choice of bases the lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by wehner and winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity as a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the haar measure which are of independent interest

on sofic actions and equivalence relations abstract the notion of sofic equivalence relation was introduced by gabor elek and gabor lippner their technics employ some graph theory here we define this notion in a more operator algebraic context starting from connes embedding problem and prove the equivalence of these two definitions we introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic we also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic

unified entropy trade off relations for a single quantum channel many important properties of quantum channels are quantified by means of entropic functionals characteristics of such a kind are closely related to different representations of a quantum channel in the jamiolkowski choi representation the given quantum channel is described by the so called dynamical matrix entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence within the so called natural representation the quantum channel is formally posed by another matrix obtained as the reshuffling of the dynamical matrix the corresponding entropies characterize an amount with which the receiver a priori knows the channel output as was previously shown the map and receiver entropies are mutually complementary characteristics indeed there exists a non trivial lower bound on their sum first we extend the concept of receiver entropy to the family of unified entropies developing the previous results we further derive non trivial lower bounds on the sum of the map and receiver q s entropies the derivation is essentially based on some inequalities with the schatten norms and anti norms

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

randomizing quantum states constructions and applications the construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver we show that if only near perfect security is required the size of the key can be reduced by a factor of two more specifically we show that there exists a set of roughly d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing as compared to the d2 operators required to randomize perfectly aside from the private quantum channel variations of this construction can be applied to many other tasks in quantum information processing we show for instance that it can be used to construct locc data hiding schemes for bits and qubits that are much more efficient than any others known allowing roughly log d qubits to be hidden in 2 log d qubits the method can also be used to exhibit the existence of quantum states with locked classical correlations an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key our construction also provides the basic building block for a method of remotely preparing arbitrary d dimensional pure quantum states using approximately log d bits of communication and log d ebits of entanglement

wehrl s entropy of spin states and lieb s conjecture wehrl s entropy is the entropy of the probability distribution in the phase space corresponding to the q representation antinormal ordering of operators of a quantum state in terms of coherent states the hilbert space of a system of n spins or n two level atoms is of 2n dimensions the subspace with maximum total spin n 2 is of n 1 dimensions all discussions are restricted within this subspace the probability amplitude for an arbitrary pure state in this subspace is a polynomial of degree n which can be factorised into the product of n linear factors and each root can be identified with one point on the unit sphere hence an arbitrary state in the subspace can be represented geometrically by n unit vectors the general expression for the wehrl entropy is obtained as a finite series expansion in terms of some symmetric functions of sin2 omega ij 2 where omega ij is the angle between a pair of unit vectors the special cases of coherent and almost coherent spin states are then considered

more non semigroup lie gradings this note is devoted to the construction of two very easy examples of respective dimensions 4 and 6 of graded lie algebras whose grading is not given by a semigroup the latter one being a semisimple algebra it is shown that 4 is the minimal possible dimension for such graded lie algebras

on the structure of graded commutative algebras abstract consider a a commutative algebra graded by means of an abelian group g we show that under the hypothesis a 0 g a g a g where g g a g 0 the algebra a is of the form a j i j with any i j a well described graded ideal of a satisfying i j i k 0 if j k under certain conditions the simplicity of a is characterized and it is shown that a is the direct sum of the family of its minimal graded ideals each one being a simple graded commutative algebra

a transform of complementary aspects with applications to entropic uncertainty relations even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood here we construct special sets of up to 2n 1 mutually unbiased bases mubs in dimension d 2 n which have particularly beautiful symmetry properties derived from the clifford algebra more precisely we show that there exists a unitary transformation that cyclically permutes such bases this unitary can be understood as a generalization of the fourier transform which exchanges two mubs to multiple complementary aspects we proceed to prove a lower bound for min entropic entropic uncertainty relations for any set of mubs and show that symmetry plays a central role in obtaining tight bounds for example we obtain for the first time a tight bound for four mubs in dimension d 4 which is attained by an eigenstate of our complementarity transform finally we discuss the relation to other symmetries obtained by transformations in discrete phase space and note that the extrema of discrete wigner functions are directly related to min entropic uncertainty relations for mubs

renyi wehrl entropies as measures of localization in phase space we generalize the concept of the wehrl entropy of quantum states which gives a basis independent measure of their localization in phase space we discuss the minimal values and the typical values of these r enyi wehrl entropies for pure states for spin systems according to lieb s conjecture the minimal values are provided by the spin coherent states though lieb s conjecture remains unproven we give new proofs of partial results that may be generalized for other systems we also investigate random pure states and calculate the mean r enyi wehrl entropies averaged over the natural measure in the space of pure quantum states

entropy and the variational principle for actions of sofic groups recently lewis bowen introduced a notion of entropy for measure preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy by applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants and we establish the variational principle in this context in the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group c algebra

entropic measures of joint uncertainty effects of lack of majorization we compute renyi entropies for the statistics of a noisy simultaneous observation of two complementary observables in two dimensional quantum systems the relative amount of uncertainty between two states depends on the uncertainty measure used these results are not reproduced by a more standard duality relation we show that these behaviors are consistent with the lack of majorization relation between the corresponding statistics

a lower bound for the wehrl entropy of quantum spin with sharp high spin asymptotics a lower bound for the wehrl entropy of a single quantum spin is derived the high spin asymptotics of this bound coincides with lieb s conjecture up to but not including terms of first and higher order in the inverse spin quantum number the result presented here may be seen as complementary to the verification of the conjecture in cases of lowest spin by schupp commun math phys 207 481 1999 the present result for the wehrl entropy is obtained from interpolating a sharp norm bound that also implies a sharp lower bound for the so called renyi wehrl entropy with certain indices that are evenly spaced by half of the inverse spin quantum number

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

quasi filiform lie algebras of maximum length abstract we prove the existence of adequate homogeneous bases in the connected integer gradation with the greatest number of non trivial subspaces on a nilpotent lie algebra in the quasi filiform case the existence of those bases allows the description of the structure of the family in each dimension finally we obtain the classification of the graded quasi filiform lie algebras considered

on the entropic formulation of uncertainty for quantum measurements the formulation of the uncertainty principle based on entropy is given with the noncommuting observables described via positive operator valued measures

sofic equivalence relations we introduce the notion of sofic measurable equivalence relations using them we prove that connes embedding conjecture as well as the measurable determinant conjecture of luck sauer and wegner hold for treeable equivalence relations

uncertainty in quantum measurements the object of this letter is to show that except in the case of canonically conjugate observables the generalized heisenberg inequality does not properly express the quantum uncertainty principle it is in general too weak an inequality is obtained which does express the principle

decompositions of linear spaces induced by bilinear maps abstract let v be an arbitrary linear space and f v v v a bilinear map we show that for any choice of basis b of v the bilinear map f induces on v a decomposition v j j v j as a direct sum of linear subspaces which is f orthogonal in the sense f v j v k 0 when j k and in such a way that any v j is strongly f invariant in the sense f v j v f v v j v j we also characterize the f simplicity of any v j finally an application to the structure theory of arbitrary algebras is also provided

superspace geometry the exact uncertainty relationship between complementary aspects after reviewing the standard uncertainty relations due to heisenberg 1927 robertson 1929 and schrodinger 1930 as well as the relations of deutsch and of maassen and uffink including the so called entropic relations the author presents a complete account of the uncertainty relationship between complementary aspects in terms of superspace geometry an approach not hitherto employed two incompatible properties a sigma alpha aalpha mod alpha alpha mod and n sigma nnn mod n n mod belong to a pair of complementary aspects defined by two orthonormal bases mod alpha and mod n in the hilbert space h if the state is mod psi then p alpha mod alpha mod psi mod 2 is the probability of obtaining the value aalpha in a measurement of a and p n mod n mod psi mod 2 is the probability of obtaining the value nn in a measurement of n the two aspects are characterised relative to mod psi by the numbers so called purities pi sigma alpha p alpha 2 and pi sigma n p n 2 both or 1 he gives a complete characterisation of the uncertainty relationship between a and n more precisely between their aspects in terms of the range of joint values of pi pi for arbitrary initial states pure as well as mixed a theorem of lenard is given an alternative proof employing only elementary superspace geometry the results depend on two angles phi m minimal angle and phi m maximal angle between the two aspects exact expressions for phi m and phi m are obtained in terms of the overlap matrix lambda lambda alpha n mod alpha mod n mod 2 as a corollary he finds the uncertainty relation for a pure state mod psi pi pi or 1 1 g 1 1 g cos phi m where g dim h and a sharper one for mixed states pi pi 2 is obtainable if and only if the intersection of the aspects holds a pure state if phi m pi 2 maximal incompatibility then pi pi or 1 1 g is a special case of a stronger relation sigma mu g o pi mu 2 which one obtains for g 1 maximally incompatible aspects by means of a thereom of ivanovic

entropic uncertainty relations for the infinite well we calculate the entropic and standard uncertainty relations for the position and momentum of the infinite potential well as functions of its quantum states we show that the entropic uncertainty relations express more adequately the uncertainty principle than the standard ones which use the dispersions as measures of the uncertainties for two non commuting observables

on split leibniz superalgebras abstract we study the structure of arbitrary split leibniz superalgebras we show that any of such superalgebras l is of the form l u j i j with u a subspace of an abelian graded subalgebra h and any i j a well described graded ideal of l satisfying i j i k 0 if j k in the case of l being of maximal length the simplicity of l is also characterized in terms of connections of roots

entropy power uncertainty relations towards a tight inequality for all gaussian pure states we show that a proper expression of the uncertainty relation for a pair of canonically conjugate continuous variables relies on entropy power a standard notion in shannon information theory for real valued signals the resulting entropy power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to bialynicki birula and mycielski but can be further extended to rotated variables hence based on a reasonable assumption we give a partial proof of a tighter form of the entropy power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity interestingly it implies the generalized rotation invariant schrodinger robertson uncertainty relation exactly as the original entropy power uncertainty relation implies heisenberg relation it is saturated for all gaussian pure states in contrast with hitherto known entropic formulations of the uncertainty principle

graded differential geometry of graded matrix algebras we study the graded derivation based noncommutative differential geometry of the z2 graded algebra m n m of complex n m n m matrices with the usual block matrix grading for n m beside the infinite dimensional algebra of graded forms the graded cartan calculus graded symplectic structure graded vector bundles graded connections and curvature are introduced and investigated in particular we prove the universality of the graded derivation based first order differential calculus and show that m n m is a noncommutative graded manifold in a stricter sense there is a natural body map and the cohomologies of m n m and its body coincide as in the case of ordinary graded manifolds

phase space localization of antisymmetric functions upper and lower bounds are written down for the minimum information entropy in phase space of an antisymmetric wave function in any number of dimensions similar bounds are given when the wave function is restricted to belong to any of the proper subspaces of the fourier transform operator

universal uncertainty relations uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined the modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring noncommuting observables however there is no fundamental reason for using entropies as quantifiers any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation starting from a very reasonable assumption of invariance under mere relabeling of the measurement outcomes we show that schur concave functions are the most general uncertainty quantifiers we then discover a fine grained uncertainty relation that is given in terms of the majorization order between two probability vectors significantly extending a majorization based uncertainty relation first introduced in m h partovi phys rev a 84 052117 2011 such a vector type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers our relation is therefore universal and captures the essence of uncertainty in quantum theory

extended quantum conditional entropy and quantum uncertainty inequalities quantum states can be subjected to classical measurements whose incompatibility or uncertainty can be quantified by a comparison of certain entropies there is a long history of such entropy inequalities between position and momentum recently these inequalities have been generalized to the tensor product of several hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized our proofs utilize the technique of the original derivation of strong subadditivity of the von neumann entropy

the norm of polynomials in large random and deterministic matrices let bf x n x 1 n ldots x p n be a family of n n independent normalized random matrices from the gaussian unitary ensemble we state sufficient conditions on matrices bf y n y 1 n ldots y q n possibly random but independent of xn for which the operator norm of p bf x n bf y n bf y n converges almost surely for all polynomials p limits are described by operator norms of objects from free probability theory taking advantage of the choice of the matrices yn and of the polynomials p we get for a large class of matrices the no eigenvalues outside a neighborhood of the limiting spectrum phenomena we give examples of diagonal matrices yn for which the convergence holds convergence of the operator norm is shown to hold for block matrices even with rectangular gaussian blocks a situation including non white wishart matrices and some matrices encountered in mimo systems

gradings on simple jordan and lie algebras abstract in this paper we describe all group gradings by a finite abelian group g of several types of simple jordan and lie algebras over an algebraically closed field f of characteristic zero

on lie gradings iii gradings of the real forms of classical lie algebras abstract maximal abelian subgroups of diagonalizable automorphisms of lie algebra so called mad groups play a crucial role in the construction of fine gradings of lie algebra our aim is to give a description of mad groups for real forms of classical lie algebras we introduce four types of matrix subgroups of g l n c called out groups ad groups out groups and ad groups for each type of these subgroups we define a relation of equivalence the problem of classifying of all non conjugate mad groups on real forms of sl n c o n c or sp n c is transformed to the problem of classifying these equivalence classes the classification of these equivalence classes is presented here

an lp version of von neumann dimension for banach space representations of sofic groups abstract in 5 a gournay defined a notion of l p dimension for invariant subspaces of l q n with amenable the number dim l q l p v is dim v when p q and is preserved by a certain class of equivariant bounded linear isomorphisms we develop a notion of dim l p y where y is a banach space with a uniformly bounded action of a sofic group and is a sofic approximation in particular our definition makes sense for a large class of non amenable groups we also develop a notion of dim s p y with an r embeddable group and s p the space of finite dimensional schatten p class operators these numbers are invariant under bounded equivariant linear isomorphisms and under the natural translation action of dim l p l p v dim v and dim s p l p v dim v for 1 p 2 in particular this shows that l p v is not isomorphic to l p w as a representation of if dim v dim w and is r embeddable we discuss other concrete computations in a follow up paper including proving that our dimension agrees with von neumann dimension for representations contained in a multiple of the left regular representation

renyi formulation of the entropic uncertainty principle for povms uncertainty relations for a pair of arbitrary measurements are posed in the form of inequalities using the renyi entropies the formulation deals with povm measurements on finite dimensional hilbert space both the entropic uncertainty relations with state dependent and state independent bounds are presented the results are illustrated on the example of measurements for state discrimination

entropic uncertainty relations and the stabilizer formalism entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy central questions include the derivation of lower bounds on the total uncertainty for given observables the characterization of observables that allow strong uncertainty relations and the construction of such relations for the case of several observables we demonstrate how the stabilizer formalism can be applied to these questions we show that the maassen uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases we compare the relative strengths of variance based and various entropic uncertainty relations for dichotomic anticommuting observables

differential entropy and dynamics of uncertainty we analyze the functioning of gibbs type entropy functionals in the time domain with emphasis on shannon and kullback leibler entropies of time dependent continuous probability distributions the shannon entropy validity is extended to probability distributions inferred from l 2 r n quantum wave packets in contrast to the von neumann entropy which simply vanishes on pure states the differential entropy quantifies the degree of probability de localization and its time development the associated dynamics of the fisher information functional quantifies nontrivial power transfer processes in the mean both in dissipative and quantum mechanical cases

entropic energy time uncertainty relation energy time uncertainty plays an important role in quantum foundations and technologies and it was even discussed by the founders of quantum mechanics however standard approaches e g robertson s uncertainty relation do not apply to energy time uncertainty because in general there is no hermitian operator associated with time following previous approaches we quantify time uncertainty by how well one can read off the time from a quantum clock we then use entropy to quantify the information theoretic distinguishability of the various time states of the clock our main result is an entropic energy time uncertainty relation for general time independent hamiltonians stated for both the discrete time and continuous time cases our uncertainty relation is strong in the sense that it allows for a quantum memory to help reduce the uncertainty and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry because of the operational relevance of entropy we anticipate that our uncertainty relation will have information processing applications

on the structure of graded lie algebras we study the structure of arbitrary graded lie algebras we show that any of such algebras l with a symmetric g support is of the form l u jij with u a subspace of l1 and any ij a well described graded ideal of l satisfying ij ik 0 if j k under certain conditions the gr simplicity of l is characterized and it is shown that l is the direct sum of the family of its minimal graded ideals each one being a gr simple graded lie algebra

uncertainty relations for joint localizability and joint measurability in finite dimensional systems two quantities quantifying uncertainty relations are examined busch and pearson j math phys 48 082103 2007 investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall width and error bar width in this paper we show a simple relationship between these quantities for finite dimensional systems our result indicates that if there is a bound on joint localizability it is possible to obtain a similar bound on joint measurability for finite dimensional systems uncertainty relations for a pair of general projection valued measures are obtained as by products

stronger uncertainty relations for all incompatible observables the heisenberg robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator notably it does not capture the concept of incompatible observables because it can be trivial i e the lower bound can be null even for two noncompatible observables here we give two stronger uncertainty relations relating to the sum of variances whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system

models of the lie algebra f4 abstract we describe the models of the exceptional lie algebra f 4 which are based on its semisimple subalgebras of rank 4 the underlying fact is that any reductive subalgebra of maximal rank of a simple lie algebra induces a grading on this algebra by means of an abelian group in such a way that the nontrivial components of the grading are irreducible modules

some extensions of the uncertainty principle abstract we study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities extending results recently derived by bialynicki birula i bialynicki birula formulation of the uncertainty relations in terms of the renyi entropies physical review a 74 5 2006 052101 and zozor et al s zozor c vignat on classes of non gaussian asymptotic minimizers in entropic uncertainty principles physica a 375 2 2007 499 517 those inequalities can be considered as generalizations of the heisenberg uncertainty principle since they measure the mutual uncertainty of a wave function and its fourier transform through their associated renyi entropies with conjugated indices we consider here the general case where the entropic indices are not conjugated in both cases where the state space is discrete and continuous we discuss the existence of an uncertainty inequality depending on the location of the entropic indices and in the plane our results explain and extend a recent study by luis a luis quantum properties of exponential states physical review a 75 2007 052115 where states with quantum fluctuations below the gaussian case are discussed at the single point 2 2

quantifying the incompatibility of quantum measurements relative to a basis motivated by quantum resource theories we introduce a notion of incompatibility for quantum measurements relative to a reference basis the notion arises by considering states diagonal in that basis and investigating whether probability distributions associated with different quantum measurements can be converted into one another by probabilistic postprocessing the induced preorder over quantum measurements is directly related to multivariate majorization and gives rise to families of monotones i e scalar quantifiers that preserve the ordering for the case of orthogonal measurement we establish a quantitative connection between incompatibility quantum coherence and entropic uncertainty relations we generalize the construction to include arbitrary positive operator valued measurements and report complete families of monotones

on the structure of split lie color algebras abstract the class of lie color algebras contains the one of lie superalgebras and so the one of lie algebras in order to begin an approach to the structure of arbitrary lie color algebras with no restrictions neither on the dimension nor on the base field we introduce the class of split lie color algebras as the natural extension of the classes of split lie algebras and split lie superalgebras by developing techniques of connections of roots for this kind of algebra we show that any such algebra l is of the form l u j i j with u a subspace of the abelian graded subalgebra h and eachx i j a well described graded ideal of l satisfying i j i k 0 if j k under certain conditions the simplicity of l is characterized and it is shown that l is the direct sum of the family of its minimal graded ideals each one being a simple split lie color algebra

on split leibniz algebras abstract in order to begin an approach to the structure of arbitrary leibniz algebras with no restrictions neither on the dimension nor on the base field we introduce the class of split leibniz algebras as the natural extension of the class of split lie algebras by developing techniques of connections of roots for this kind of algebras we show that any of such algebras l is of the form l u j i j with u a subspace of the abelian subalgebra h in the sense h h 0 and any i j a well described ideal of l satisfying i j i k 0 if j k in the case of l being of maximal length we characterize the simplicity of the algebra in terms of connections of roots

lie gradings on associative algebras abstract in this paper we apply the method of functional identities to the study of group gradings by an abelian group g on simple lie algebras under very mild restrictions on the grading group or the base field of coefficients