darboux transformations and supersymmetry for the generalized schrodinger equations in 1 1 dimensions we study intertwining relations supersymmetry and darboux transformations for generalized schrodinger equations in 1 1 dimensions we obtain intertwiners in an explicit form by means of which we construct arbitrary order darboux transformations for our class of equations we develop a corresponding supersymmetric formulation and prove equivalence of the darboux transformations with the supersymmetry formalism finally we show that our darboux transformations can also be constructed by means of point transformations avoiding the use of intertwiners

subalgebras with converging star products in deformation quantization an algebraic construction for cpn based on a closed formula for a star product of wick type on cpn which has been discovered in an earlier article of the authors we explicitly construct a subalgebra of the formal star algebra with coefficients contained in the uniformly dense subspace of representative functions with respect to the canonical action of the unitary group that consists of converging power series in the formal parameter thereby giving an elementary algebraic proof of a convergence result already obtained by cahen gutt and rawnsley in this subalgebra the formal parameter can be substituted by a real number the resulting associative algebras are infinite dimensional except for the case 1 k k a positive integer where they turn out to be isomorphic to the finite dimensional algebra of linear operators in the kth energy eigen space of an isotropic harmonic oscillator with n 1 degrees of freedom other examples like the 2n torus and the poincare disk are discussed

semi classical properties of berezin toeplitz operators with c k symbol we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators

semiclassical spectral analysis of toeplitz operators on symplectic manifolds the case of discrete wells we consider toeplitz operators associated with the renormalized bochner laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold we study the asymptotic behavior in the semiclassical limit of low lying eigenvalues and the corresponding eigensections of a self adjoint toeplitz operator under assumption that its principal symbol has a non degenerate minimum with discrete wells as an application we prove upper bounds for low lying eigenvalues of the bochner laplacian in the semiclassical limit

quantized nambu poisson manifolds and n lie algebras we investigate the geometric interpretation of quantized nambu poisson structures in terms of noncommutative geometries we describe an extension of the usual axioms of quantization in which classical nambu poisson structures are translated to n lie algebras at quantum level we demonstrate that this generalized procedure matches an extension of berezin toeplitz quantization yielding quantized spheres hyperboloids and superspheres the extended berezin quantization of spheres is closely related to a deformation quantization of n lie algebras as well as the approach based on harmonic analysis we find an interpretation of nambu heisenberg n lie algebras in terms of foliations of rn by fuzzy spheres fuzzy hyperboloids and noncommutative hyperplanes some applications to the quantum geometry of branes in m theory are also briefly discussed

frechet algebraic deformation quantization of the poincare disk starting from formal deformation quantization we use an explicit formula for a star product on the poincare disk n to introduce a frechet topology making the star product continuous to this end a general construction of locally convex topologies on algebras with countable vector space basis is introduced and applied several examples of independent interest are investigated as e g group algebras over finitely generated groups and infinite matrices in the case of the star product on n the resulting frechet algebra is shown to have many nice features it is a strongly nuclear kothe space the symmetry group su 1 n acts smoothly by continuous automorphisms with an inner infinitesimal action and evaluation functionals at all points of n are positive functionals

gl and geometric quantization an axiomatic approach to the approximation of infinite dimensional algebras is presented examples illustrating the need for a rigorous treatment of this subject geometric quantization is employed to construct systematicallysu n approximations of diffeomorphism algebras which first appeared in the theory of relativistic membranes

monotonicity under local operations linear entropic formulas all correlation measures classical and quantum must be monotonic under local operations in this paper we characterize monotonic formulas that are linear combinations of the von neumann entropies associated with the quantum state of a physical system that has n parts we show that these formulas form a polyhedral convex cone which we call the monotonicity cone and enumerate its facets we illustrate its structure and prove that it is equivalent to the cone of monotonic formulas implied by strong subadditivity we explicitly compute its extremal rays for n leq5 we also consider the symmetric monotonicity cone in which the formulas are required to be invariant under subsystem permutations we describe this cone fully for all n

non perturbative deformation quantization of cartan domains abstract we construct families of non commuting c algebras of quantized functions for bounded irreducible hermitian symmetric spaces for this procedure we use algebras of toeplitz operators defined with respect to a perturbation of the ordinary bergman metric we prove the deformation quantization conditions for these algebras

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

generalized bergman kernels and geometric quantization in geometric quantization it is well known that if f is an observable and f a polarization on a symplectic manifold m then the condition xf leaves f invariant where xf denotes the hamiltonian vector field associated to f is sufficient to guarantee that one does not have to compute the bks kernel explicitly in order to know the corresponding quantum operator it is shown in this paper that this condition on f can be weakened to xf leaves f f invariant and the corresponding quantum operator is then given implicitly by formula 4 8 in particular when f is a positive kahler polarization all observables can be quantized directly and moreover an explicit formula for the corresponding quantum operator is derived theorem 5 8 applying this to the phase space r2n one obtains a quantization prescription which ressembles the normal ordering of operators in quantum field theory when we translate this prescription to the usual position representation of quantum mechanics the result i

the deformation quantizations of the hyperbolic plane we describe the space of all invariant both formal and non formal deformation quantizations on the hyperbolic plane mathbb d as solutions of the evolution of a second order hyperbolic differential operator the construction is entirely explicit and relies on non commutative harmonic analytical techniques on symplectic symmetric spaces the present work presents a unified method producing every quantization of mathbb d and provides in the 2 dimensional context an exact solution to weinstein s wkb quantization program within geometric terms the construction reveals the existence of a metric of lorentz signature canonically attached or dual to the geometry of the hyperbolic plane through the quantization process

convergence of the wick star product we construct a frechet space as a subspace of c omega mathbb c n where the wick star product converges and is continuous the resulting frechet algebra mathcal a is studied in detail including a representation of mathcal a in the bargmann fock space and a discussion of star exponentials and coherent states

green s functions sum rules and matrix elements for susy partners it is now well known that the solutions in the schrodinger equations for two potentials which are supersymmetric partners are linked by intertwining relations involving first order differential operators in this paper we explore the consequence of this linkage for the green s functions corresponding to susy partners and the relation between the sums over the inverses of the eigenvalues for the two potentials we also establish some relations between the matrix elements of certain operators evaluated between the eigenstates of two partner potentials we show that there can be circumstances where some matrix elements vanish as a consequence of the presence of supersymmetry

a new inequality for the von neumann entropy strong subadditivity of von neumann entropy proved in 1973 by lieb and ruskai is a cornerstone of quantum coding theory all other known inequalities for entropies of quantum systems may be derived from it here we prove a new inequality for the von neumann entropy which we prove is independent of strong subadditivity it is an inequality which is true for any four party quantum state provided that it satisfies three linear relations constraints on the entropies of certain reduced states

a trace formula for dirac green s functions related by darboux transformations we construct green s functions and a trace formula for pairs of stationary dirac equations under sturm liouville boundary conditions where the equations are related to each other by a darboux transformation our findings generalize former results pozdeeva e 2008 j phys a math theor at press

quantization of kahler manifolds ii we use berezin s dequantization procedure to define a formal product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous kahler manifold m we prove that this formal product isconvergent when m isa hermitian symmetric space 1993 american mathematical society

the inequalities of quantum information theory let spl rho denote the density matrix of a quantum state having n parts 1 n for i spl sube n 1 n let spl rho sub i tr sub n spl bsol i spl rho denote the density matrix of the state comprising those parts i such that i spl isin i and let s spl rho sub i denote the von neumann 1927 entropy of the state spl rho sub i the collection of spl nu 2 sup n numbers s spl rho sub i sub i spl sube n may be regarded as a point called the allocation of entropy for spl rho in the vector space r sup spl nu let a sub n denote the set of points in r sup spl nu that are allocations of entropy for n part quantum states we show that a sub n the topological closure of a sub n is a closed convex cone in r sup spl nu this implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies lieb and ruskai 1973 have established a number of inequalities for multipartite quantum states strong subadditivity and weak monotonicity we give a finite set of instances of these inequalities that is complete in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations and independent in the sense that none of them can be deduced from the others by taking positive linear combinations let b sub n denote the polyhedral cone in r sup spl nu determined by these inequalities we show that a sub n b sub n for n spl les 3 the status of this equality is open for n spl ges 4 we also consider a symmetric version of this situation in which s spl rho sub i depends on i only through the number i spl ne i of indexes in i and can thus be denoted s spl rho sub i in this case we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy s spl rho sub i sub 0 spl les i spl les n in r sup n 1

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

on quantum fidelities and channel capacities for the discrete memoryless quantum channel we show the equivalence of two different notions of quantum channel capacity that which uses the entanglement fidelity as its criterion for success in transmission and that which uses the minimum fidelity of pure states in a subspace of the input hilbert space as its criterion as a corollary any source with entropy rate less than the capacity may be transmitted with high entanglement fidelity we also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel this enables us to simplify a known upper bound for the channel capacity it also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity

entanglement assisted capacity of a quantum channel and the reverse shannon theorem the entanglement assisted classical capacity of a noisy quantum channel c sub e is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel assuming that the sender and receiver have access to the resource of shared quantum entanglement which may be used up by the communication protocol we show that the capacity c sub e is given by an expression parallel to that for the capacity of a purely classical channel i e the maximum over channel inputs spl rho of the entropy of the channel input plus the entropy of the channel output minus their joint entropy the latter being defined as the entropy of an entangled purification of spl rho after half of it has passed through the channel we calculate entanglement assisted capacities for two interesting quantum channels the qubit amplitude damping channel and the bosonic channel with amplification attenuation and gaussian noise we discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel alone or in the presence of ancillary resources such as prior entanglement in the classical analog of entanglement assisted communication communication over a discrete memoryless channel dmc between parties who share prior random information we show that one parameter is sufficient i e that in the presence of prior shared random information all dmcs of equal capacity can simulate one another with unit asymptotic efficiency

green s functions and trace formulas for generalized sturm liouville problems related by darboux transformations we study green s functions of the generalized sturm liouville problems that are related to each other by darboux equivalently supersymmetrical transformations we establish an explicit relation between the corresponding green s functions and derive a simple formula for their trace the class of equations considered here includes the conventional schrodinger equation and generalizations such as for position dependent mass and with linearly energy dependent potential as well as the stationary fokker planck equation

quantization towards a comparison between methods in this paper it is shown that the procedure of geometric quantiztion applied to kahler manifolds gives the following result the hilbert space h consists roughly speaking of holomorphic functions on the phase space m and to each classical observable f i e a real function on m is associated an operator f on h as follows first multiply by f 1 4 drf dr being the laplace de rham operator on the kahler manifold m and then take the holomorphic part see g m tuynman j math phys 27 573 1987 this result is correct on compact kahler manifolds and correct modulo a boundary term md on noncompact kahler manifolds in this way these results can be compared with the quantization procedure of berezin math ussr izv 8 1109 1974 9 341 1975 commun math phys 40 153 1975 which is strongly related to quantization by products e g see c moreno and p ortega navarro amn inst h poincare sec a 38 215 1983 lett math phys 7 181 1983 c moreno lett math phys 11 3

weighted bergman kernels and quantization let be a bounded pseudoconvex domain in c n two positive functions on such that log log are plurisubharmonic and z a point at which log is smooth and strictly plurisubharmonic we show that as k the bergman kernels with respect to the weights k have an asymptotic expansion for x y near z where x y is an almost analytic extension of phi x x x and similarly for further if in addition is of finite type behave reasonably at the boundary and log log are strictly plurisubharmonic on we obtain also an analogous asymptotic expansion for the berezin transform and give applications to the berezin quantization finally for smoothly bounded and strictly pseudoconvex and a smooth strictly plurisubharmonic defining function for we also obtain results on the berezin toeplitz quantization

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

robustness of quantum markov chains if the conditional information of a classical probability distribution of three random variables is zero then it obeys a markov chain condition if the conditional information is close to zero then it is known that the distance minimum relative entropy of the distribution to the nearest markov chain distribution is precisely the conditional information we prove here that this simple situation does not obtain for quantum conditional information we show that for tri partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum markov chain state but the distance can be much greater indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor

a non shannon type conditional inequality of information quantities given n discrete random variables spl omega x sub 1 x sub n associated with any subset spl alpha of 1 2 n there is a joint entropy h x sub spl alpha where x sub spl alpha x sub i i spl isin spl alpha this can be viewed as a function defined on 2 sup 1 2 n taking values in 0 spl infin we call this function the entropy function of spl omega the nonnegativity of the joint entropies implies that this function is nonnegative the nonnegativity of the conditional joint entropies implies that this function is nondecreasing and the nonnegativity of the conditional mutual information implies that this function is two alternative these properties are the so called basic information inequalities of shannon s information measures an entropy function can be viewed as a 2 sup n 1 dimensional vector where the coordinates are indexed by the subsets of the ground set 1 2 n as introduced by yeng see ibid vol 43 no 6 p 1923 34 1997 spl gamma sub n stands for the cone in ir 2 sup n 1 consisting of all vectors which have all these properties let spl gamma sub n be the set of all 2 sup n 1 dimensional vectors which correspond to the entropy functions of some sets of n discrete random variables a fundamental information theoretic problem is whether or not spl gamma sub n spl gamma sub n here spl gamma sub n stands for the closure of the set spl gamma sub n we show that spl gamma sub n is a convex cone spl gamma sub 2 spl gamma sub 2 spl gamma sub 3 spl ne spl gamma sub 3 but spl gamma sub 3 spl gamma sub 3 for four random variables we have discovered a conditional inequality which is not implied by the basic information inequalities of the same set of random variables this lends an evidence to the plausible conjecture that spl gamma sub n spl ne spl gamma sub n for n 3

explicit darboux transformations of arbitrary order for generalized time dependent schrodinger equations we construct darboux transformations of arbitrary order for a generalized linear time dependent schrodinger equation special cases of which correspond to time dependent hamiltonians coupled to a magnetic field with position dependent mass and with weighted energy our darboux transformation reduces correctly to these known cases and also to new generalized schrodinger equations furthermore fundamental properties of the conventional darboux transformation are maintained such as factorization of the nth order transformation into first order transformations and existence of a reality condition for the transformed potentials

susy transformation of the green function and a trace formula an integral relation is established between the green functions corresponding to two hamiltonians which are supersymmetric susy partners and in general may possess both discrete and continuous spectra it is shown that when the continuous spectrum is present the trace of the difference of the green functions for susy partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each green function our findings are illustrated by using the free particle example considered both on the whole real line and on a half line

heralded channel holevo superadditivity bounds from entanglement monogamy we introduce a class of channels called heralded channels which generalize the quantum erasure channel composed with an arbitrary other quantum channel we show that monogamy of squashed entanglem

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure