on symmetric invariants of centralisers in reductive lie algebras let g be a finite dimensional simple lie algebra of rank l over an algebraically closed field of characteristic 0 let e be a nilpotent element of g and let ge be the centraliser of e in g in this paper we study the algebra s ge ge of symmetric invariants of ge we prove that if g is of type a or c then s ge ge is always a graded polynomial algebra in l variables and we show that this continues to hold for some nilpotent elements in the lie algebras of other types in type a we prove that the invariant algebra s ge ge is freely generated by a regular sequence in s ge and describe the tangent cone at e to the nilpotent variety of g

strong consistency and asymptotic efficiency for adaptive quantum estimation problems it is shown that for an adaptive quantum estimation scheme based on locally unbiased measurements the sequence of maximum likelihood estimators is strongly consistent and asymptotically efficient

a fibre bundle over manifolds of quantum channels and its application to quantum statistics a fibre bundle structure is introduced over manifolds of quantum channels this structure has a close connection with the problem of estimating an unknown quantum channel specified by a parameter it is shown that the quantum fisher information of the family of output states maximized over all input states which quantifies the ultimate statistical distinguishability of the parameter is expressed in terms of a geometrical quantity on the fibre bundle using this formula a criterion for the maximum quantum fisher information of the nth extended channel id n to be o n is derived this criterion further proves that for almost all quantum channels the maximum quantum fisher information increases in the order of o n

optimal estimation of one parameter quantum channels we explore the task of optimal quantum channel identification and in particular the estimation of a general one parameter quantum process we derive new characterizations of optimality and apply the results to several examples including the qubit depolarizing channel and the harmonic oscillator damping channel we also discuss the geometry of the problem and illustrate the usefulness of using entanglement in process estimation

complete reducibility and separability let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 a subgroup of g is said to be separable in g if its global and infinitesimal centralizers have the same dimension we study the interaction between the notion of separability and serre s concept of g complete reducibility for subgroups of g the separability hypothesis appears in many general theorems concerning g complete reducibility we demonstrate that many of these results fail without this hypothesis on the other hand we prove that if g is a connected reductive group and p is very good for g then any subgroup of g is separable we deduce that under these hypotheses on g a subgroup h of g is g completely reducible provided the lie algebra of g is semisimple as an h module recently guralnick has proved that if h is a reductive subgroup of g and c is a conjugacy class of g then the intersection of c and g is a finite union of h conjugacy classes for generic p when certain extra hypotheses hold including separability this follows from a well known tangent space argument due to richardson but in general it rests on lusztig s deep result that a connected reductive group has only finitely many unipotent conjugacy classes we show that the analogue of guralnick s result is false if one considers conjugacy classes of n tuples of elements from h for n 1

semi direct products of lie algebras and covariants abstract let q be a connected algebraic group with lie algebra q symmetric invariants of q i e the q invariants in the symmetric algebra s q of q is a first approximation to the understanding of the coadjoint action q q and coadjoint q orbits in this article we study a class of non reductive lie algebras where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant theoretic properties if g is a semisimple group with lie algebra g and v is g module then we define q to be the semi direct product of g and v then we are interested in the case where the generic isotropy group for the g action on v is reductive and commutative it turns out that in this case symmetric invariants of q can be constructed via certain g equivariant maps from g to v covariants

reductive pairs arising from representations let g be a reductive algebraic group and v a g module we consider the question of when gl v g is a reductive pair of algebraic groups where is the representation afforded by v we first make some observations about general g and v then specialise to the group sl2 k with k algebraically closed of positive characteristic p for this group we provide complete answers for the classes of simple and weyl modules the behaviour being determined by the base p expansion of the highest weight of the module we conclude by illustrating some of the results from the first section with examples for the group sl3 k

a clebsch gordan decomposition in positive characteristic abstract let k be an algebraically closed field and g sl 2 k let e be the natural module and s r e the rth symmetric power we consider here for r s 0 the tensor product of s r e and the dual of s s e in characteristic zero this tensor product decomposes according to the clebsch gordan formula we consider here the situation when k is a field of positive characteristic we show that each indecomposable component occurs with multiplicity one and identify which modules occur for given r and s

on the coadjoint representation of z2 contractions of reductive lie algebras abstract we study the coadjoint representation of contractions of reductive lie algebras associated with symmetric decompositions let g g 0 g 1 be a symmetric decomposition of a reductive lie algebra g then the semi direct product of g 0 and the g 0 module g 1 is a contraction of g we conjecture that these contractions have many properties in common with reductive lie algebras in particular it is proved that in many cases the algebra of invariants is polynomial we also discuss the so called codim 2 property for coadjoint representations and its relationship with the structure of algebra of invariants

quantum parameter estimation of a generalized pauli channel we present a quantum parameter estimation theory for a generalized pauli channel d d where the parameter is regarded as a coordinate system of the probability simplex d2 1 we show that for each degree n of extension id n d d n d d n the sld fisher information matrix for the output states takes the maximum when the input state is an n tensor product of a maximally entangled state me d d we further prove that for the corresponding quantum cramer rao inequality there is an efficient estimator if and only if the parameter is m affine in d2 1 these results rely on the fact that the family id me of output states can be identified with d2 1 in the sense of quantum information geometry this fact further allows us to investigate submodels of generalized pauli channels in a unified manner

geometry of optimal estimation scheme for su d channels the problem of estimating an unknown su d channel u u u is studied based on the quantum cramer rao inequality it is shown that the minimum estimation error is of o 1 n2 where n is the degree of extension of the channel the mechanism behind this asymptotic behaviour is investigated from a differential geometrical point of view

a geometric approach to complete reducibility let g be a connected reductive linear algebraic group we use geometric methods to investigate g completely reducible subgroups of g giving new criteria for g complete reducibility we show that a subgroup of g is g completely reducible if and only if it is strongly reductive in g this allows us to use ideas of r w richardson and hilbert mumford kempf from geometric invariant theory we deduce that a normal subgroup of a g completely reducible subgroup of g is again g completely reducible thereby providing an affirmative answer to a question posed by j p serre and conversely we prove that the normalizer of a g completely reducible subgroup of g is again g completely reducible some rationality questions and applications to the spherical building of g are considered many of our results extend to the case of non connected g

quantum statistical holonomy a faithful density matrix can be represented by a hilbert schmidt operator up to composition with a unitary operator this ambiguity can be described in terms of a principal fibre bundle which is a modification of a construction by uhlmann 1986 the hs norm provides a real metric which determines a natural connection via the kaluza klein mechanism the authors compute the curvature and show that at the boundary its holonomy reproduces the berry phase for pure non degenerate states and the non abelian holonomy of wilczek and zee 1984 for k fold degenerate states

the polynomiality of the poisson center and semi center of a lie algebra and dixmier s fourth problem abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we provide necessary and also some sufficient conditions in order for its poisson center and semi center to be polynomial algebras over k this occurs for instance if g is quadratic of index 2 with g g g and also if g is nilpotent of index at most 2 the converse holds for filiform lie algebras of type l n q n r n and w n we show how dixmier s fourth problem for an algebraic lie algebra g can be reduced to that of its canonical truncation g moreover dixmier s statement holds for all lie algebras of dimension at most eight the nonsolvable ones among them possess a polynomial poisson center and semi center

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

higher extensions between modules for abstract we calculate ext sl 2 k ext sl 2 k l ext sl 2 k l ext sl 2 k l l where is the weyl module of highest weight l is the simple sl 2 k module of highest weight and our field k is algebraically closed of positive characteristic we also get analogous results for the dipper donkin quantisation to do thus we construct the lyndon hochschild serre spectral sequence in a new way and find a new condition for the e 2 page of any spectral sequence to be the same as the e page

maximal quantum fisher information for mixed states we study quantum metrology for unitary dynamics analytic solutions are given for both the optimal unitary state preparation starting from an arbitrary mixed state and the corresponding optimal measurement precision this represents a rigorous generalization of known results for optimal initial states and upper bounds on measurement precision which can only be saturated if pure states are available in particular we provide a generalization to mixed states of an upper bound on measurement precision for time dependent hamiltonians that can be saturated with optimal hamiltonian control these results make precise and reveal the full potential of mixed states for quantum metrology

computing invariants and semi invariants by means of frobenius lie algebras abstract let u be the enveloping algebra of a finite dimensional lie algebra over a field k of characteristic zero z u its center and sz u its semi center a sufficient condition is given in order for sz u to be a polynomial algebra over k surprisingly this condition holds for many lie algebras especially among those for which the radical is nilpotent in which case sz u z u in particular it allows the explicit description of z u for more than half of all complex indecomposable nilpotent lie algebras of dimension at most 7

comparison between the cramer rao and the mini max approaches in quantum channel estimation in a unified viewpoint in quantum channel estimation we compare the cramer rao and the mini max approaches which gives the bayesian bound in the group covariant model for this purpose we introduce the local asymptotic mini max bound whose maximum is shown to be equal to the asymptotic limit of the mini max bound it is shown that the local asymptotic mini max bound is strictly larger than the cramer rao bound in the phase estimation case while both bounds coincide when the minimum mean square error decreases with the order o frac 1 n we also derive a sufficient condition so that the minimum mean square error decreases with the order o frac 1 n

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

the poisson center and polynomial maximal poisson commutative subalgebras especially for nilpotent lie algebras of dimension at most seven abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we collect some general results on the poisson center of s g including some simple criteria regarding its polynomiality and also on certain poisson commutative subalgebras of s g these facts are then used to complete our previous work on the subject ooms 2009 o5 5 i e to give an explicit description for the poisson center of all indecomposable nilpotent lie algebras of dimension at most seven among other things we also provide a polynomial maximal poisson commutative subalgebra of s g enjoying additional properties as a by product we show that a conjecture by milovanov is valid in this situation these results easily carry over to the enveloping algebra u g

a degree inequality for lie algebras with a regular poisson semi center for lie algebras whose poisson semi center is a polynomial ring we give a bound for the sum of the degrees of the generating semi invariants this bound was previously known in many special cases

an estimation theoretical characterization of coherent states we introduce a class of quantum pure state models called the coherent models a coherent model is an even dimensional manifold of pure states whose tangent space is characterized by a symplectic structure in a rigorous framework of noncommutative statistics it is shown that a coherent model inherits and expands the original spirit of the minimum uncertainty property of coherent states

analysis of a convenient information bound for general quantum channels open questions from sarovar and milburn 2006 j phys a math gen 39 8487 are answered sarovar and milburn derived a convenient upper bound for the fisher information of a one parameter quantum channel they showed that for quasi classical models their bound is achievable and they gave a necessary and sufficient condition for positive operator valued measures povms attaining this bound they asked i whether their bound is attainable more generally ii whether explicit expressions for optimal povms can be derived from the attainability condition we show that the symmetric logarithmic derivative sld quantum information is less than or equal to the sm bound i e h c and we find conditions for equality as the fisher information is less than or equal to the sld quantum information i e fm h we can deduce when equality holds in fm c equality does not hold for all channels as a consequence the attainability condition cannot be used to test for optimal povms for all channels these results are extended to multi parameter channels