hamburger and stieltjes moment problems in several variables in this paper we give solutions to the hamburger and stieltjes moment problems in several variables in algebraic terms via extended sequences some characterizations of the uniqueness of the solutions are also presented

cubic column relations in truncated moment problems for the truncated moment problem associated to a complex sequence 2n ij i j z i j 2n to have a representing measure it is necessary for the moment matrix m n to be positive semidefinite and for the algebraic variety v to satisfy rankm n cardv as well as a consistency condition the riesz functional vanishes on every polynomial of degree at most 2n that vanishes on v in previous work with l fialkow and h m moller the first named author proved that for the extremal case rankm n cardv positivity and consistency are sufficient for the existence of a representing measure in this paper we solve the truncated moment problem for cubic column relations in m 3 of the form z3 itz uz u t r we do this by checking consistency for u t in the open cone determined by 0 u t 2 u we first prove that the algebraic variety has exactly 7 points and rankm 3 7 we then apply the above mentioned result to obtain a concrete computable necessary and sufficient condition for the existence of a representing measure

non extremal sextic moment problems abstract positive semidefiniteness recursiveness and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems also positive semidefiniteness combined with another necessary condition consistency is a sufficient condition in the case of extremal moment problems i e when the rank of the moment matrix denoted by r and the cardinality of the associated algebraic variety denoted by v are equal however these conditions are not sufficient for non extremal sextic or higher order truncated moment problems in this paper we settle three key instances of the non extremal i e r v sextic moment problem as follows when r 7 positive semidefiniteness consistency and the variety condition guarantee the existence of a 7 atomic representing measure when r 8 we construct two determining algorithms corresponding to the cases v 9 and v to accomplish this we generalize the rank reduction technique developed in previous work where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure

random quantum circuits are approximate 2 designs given a universal gate set on two qubits it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately haar distributed unitary however this requires exponential time we show that random circuits of only polynomial length will approximate the first and second moments of the haar distribution thus forming approximate 1 and 2 designs previous constructions required longer circuits and worked only for specific gate sets as a corollary of our main result we also improve previous bounds on the convergence rate of random walks on the clifford group

evenly distributed unitaries on the structure of unitary designs we clarify the mathematical structure underlying unitary t designs these are sets of unitary matrices evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group we present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design lower bounds for the number of elements of 2 designs are derived we show how to turn mutually unbiased bases into approximate 2 designs whose cardinality is optimal in leading order designs of higher order are discussed and an example of a unitary 5 design is presented we comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups further we sketch connections to problems in linear optics and questions regarding typical entanglement

kerdock codes determine unitary 2 designs the non linear binary kerdock codes are known to be gray images of certain extended cyclic codes of length n 2 m over mathbb z 4 we show that exponentiating these mathbb z 4 valued codewords by i triangleq sqrt 1 produces stabilizer states that are quantum states obtained using only clifford unitaries these states are also the common eigenvectors of commuting hermitian matrices forming maximal commutative subgroups mcs of the pauli group we use this quantum description to simplify the derivation of the classical weight distribution of kerdock codes next we organize the stabilizer states to form n 1 mutually unbiased bases and prove that automorphisms of the kerdock code permute their corresponding mcs thereby forming a subgroup of the clifford group when represented as symplectic matrices this subgroup is isomorphic to the projective special linear group psl 2 n we show that this automorphism group acts transitively on the pauli matrices which implies that the ensemble is pauli mixing and hence forms a unitary 2 design the kerdock design described here was originally discovered by cleve et al 2016 but the connection to classical codes is new which simplifies its description and translation to circuits significantly sampling from the design is straightforward the translation to circuits uses only clifford gates and the process does not require ancillary qubits finally we also develop algorithms for optimizing the synthesis of unitary 2 designs on encoded qubits i e to construct logical unitary 2 designs software implementations are available at https github com nrenga symplectic arxiv18a which we use to provide empirical gate complexities for up to 16 qubits

quantum data hiding we expand on our work on quantum data hiding hiding classical data among parties who are restricted to performing only local quantum operations and classical communication locc we review our scheme that hides one bit between two parties using bell 1964 states and we derive upper and lower bounds on the secrecy of the hiding scheme we provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme we give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement a candidate data hiding scheme that does not use entanglement is presented we show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme

dimensional stability in truncated moment problems abstract an approach to truncated moment problems is developed via the riesz functional and an assumed dimensional stability of its associated hilbert spaces although equivalent to a concept of flatness introduced by r curto and l fialkow the dimensional stability discussed in this paper has a different geometric aspect and leads to statements parallel to those of the quoted authors as well as to some newer ones obtained by simpler arguments a stability equation giving a local characterization of the dimensional stability is also presented

supplement to a table of elliptic integrals of the second kind c c double precision function rf x y z errtol ierr c c this function subroutine computes the incomplete elliptic c integral of the first kind c rf x y z integral from zero to infinity of c c 1 2 1 2 1 2 c 1 2 t x t y t z dt c c where x y and z are nonnegative and at most one of them c is zero if one of them is zero the integral is complete c the duplication theorem is iterated until the variables are c nearly equal and the function is then expanded in taylor c series to fifth order c references b c carlson and e m notis algorithms for c incomplete elliptic integrals acm transactions on mathemac tical software 7 1981 398 403 b c carlson computing c elliptic integrals by duplication numer math 33 1979 c 1 16 c authors b c carlson and elaine m notis ames laboratoryc doe iowa state university ames ia 50011 and r l pexton c lawrence livermore national laboratory livermore ca 94550 c aug 1 1979 revised jan 15 1987 c c check value rf 0 1 2 1 31102 87771 46059 90523 24198 c check by addition theorem rf xx z x w rf y y z y w c rf o z w where x y z w are positive and x y z w c integer ierr printr double precision c1 c2 c3 e2 e3 epslon errtol lamda double precision lolim mu s uplimx xn xndev xnroot double precision yyn yndev ynroot z zn zndev znroot c intrinsic functions used dabs dmax1 dmin1 dsqrt c c printr is the unit number of the printer data printr 6 c

unitary 2 designs from random x and z diagonal unitaries unitary 2 designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries often referred to as haar random unitaries they are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many body systems here we show that unitary 2 designs can be approximately implemented by alternately repeating random unitaries diagonal in the pauli z basis and pauli x basis we also provide a converse about the number of repetitions needed to achieve unitary 2 designs these results imply that the process after l repetitions achieves a d l approximate unitary 2 design based on the construction we further provide quantum circuits that efficiently implement approximate unitary 2 designs although a more efficient implementation of unitary 2 designs is known our quantum circuit has its own merit that it is divided into a constant number of commuting parts which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time we finally interpret the result in terms of the dynamics generated by time dependent hamiltonians and provide for the first time a random disordered time dependent hamiltonian that generates a unitary 2 design after switching interactions only a few times unitary 2 designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries often referred to as haar random unitaries they are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many body systems here we show that unitary 2 designs can be approximately implemented by alternately repeating random unitaries diagonal in the pauli z basis and pauli x basis we also provide a converse about the number of repetitions needed to achieve unitary 2 designs these results imply that the process after l repetitions achieves a d l approximate unitary 2 design based on the construction we further provide quantum circuits that efficiently implement approximate unitary 2 designs although a more efficient implementation of unitary 2 designs is known our quantum circuit has its own merit that it is divided into a constant number of commuting parts which

positivity of riesz functionals and solutions of quadratic and quartic moment problems abstract we employ positivity of riesz functionals to establish representing measures or approximate representing measures for truncated multivariate moment sequences for a truncated moment sequence y we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set k r n if and only if the associated riesz functional l y is k positive for a determining set k we prove that if l y is strictly k positive then y admits a representing measure supported in k as a consequence we are able to solve the truncated k moment problem of degree k in the cases i n k 2 4 and k r 2 ii n 1 k 2 and k is defined by one quadratic equality or inequality in particular these results solve the truncated moment problem in the remaining open cases of hilbert s theorem on sums of squares

precise and fast computation of a general incomplete elliptic integral of second kind by half and double argument transformations this is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind fukushima 2010 2011 21 23 we developed a method to compute an associate incomplete elliptic integral of the third kind j f n m p f n m f f m n by the half argument formulas of the sine and cosine amplitude functions and the double argument transformation of the integral the relative errors of j f n m computed by the new method are sufficiently small as less than 20 machine epsilons meanwhile the simplicity of the adopted algorithm makes the new method run 1 5 to 3 7 times faster than carlson s duplication method the combination of the new method and that to compute simultaneously two other associate incomplete elliptic integrals of the second kind b f m e f m 1 m f f m m and d f m f f m e f m m which we established recently 23 enables a precise and fast computation of arbitrary linear combination of legendre s incomplete elliptic integrals of all three kinds f f m e f m and p f n m these new procedures share the same device the half argument transformations while the double argument transformation of j f n m includes those of b f m and d f m as its sub component as a result the simultaneous computation of the three associate integrals is significantly faster than computing them separately in fact our combined procedure is 2 7 to 5 9 times faster than the combination of carlson s duplication method to compute r d and r j

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

the quantum reverse shannon theorem based on one shot information theory the quantum reverse shannon theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel s entanglement assisted classical capacity in this paper we provide a new proof of this theorem which has previously been proved by bennett devetak harrow shor and winter our proof has a clear structure being based on two recent information theoretic results one shot quantum state merging and the post selection technique for quantum channels

comment on random quantum circuits are approximate 2 designs by a w harrow and r a low commun math phys 291 257 302 2009 in a w harrow and r a low commun math phys 291 1 257 302 2009 it was shown that a quantum circuit composed of random 2 qubit gates converges to an approximate quantum 2 design in polynomial time we point out and correct a flaw in one of the paper s main arguments our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence

on square positive extensions and cubature formulas we consider linear functionals ld defined over pd p r x1 xn and study how to extend ld to a square positive l p r i e to a linear functional l with l p2 0 for all p p we use the connection of square positive functionals to real ideals to orthogonal polynomials and to moment matrices finally we report how such extension techniques are used to construct and analyse cubature formulas

a fully quantum asymptotic equipartition property the classical asymptotic equipartition property is the statement that in the limit of a large number of identical repetitions of a random experiment the output sequence is virtually certain to come from the typical set each member of which is almost equally likely in this paper a fully quantum generalization of this property is shown where both the output of the experiment and side information are quantum an explicit bound on the convergence is given which is independent of the dimensionality of the side information this naturally leads to a family of reacutenyi like quantum conditional entropies for which the von neumann entropy emerges as a special case

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

one shot decoupling if a quantum system a which is initially correlated to another system e undergoes an evolution separated from e then the correlation to e generally decreases here we study the conditions under which the correlation disappears almost completely resulting in a decoupling of a from e we give a criterion for decoupling in terms of two smooth entropies one quantifying the amount of initial correlation between a and e and the other characterizing the mapping that describes the evolution of a the criterion applies to arbitrary such mappings in the general one shot setting furthermore the criterion is tight for mappings that satisfy certain natural conditions one shot decoupling has a number of applications both in physics and information theory e g as a building block for quantum information processing protocols as an example we give a one shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption production

min and max relative entropies and a new entanglement monotone two new relative entropy quantities called the min and max relative entropies are introduced and their properties are investigated the well known min and max entropies introduced by renner are obtained from these we define a new entanglement monotone which we refer to as the max relative entropy of entanglement and which is an upper bound to the relative entropy of entanglement we also generalize the min and max relative entropies to obtain smooth min and max relative entropies these act as parent quantities for the smooth renyi entropies eth zurich ph d dissertation 2005 and allow us to define the analogues of the mutual information in the smooth renyi entropy framework further the spectral divergence rates of the information spectrum approach are shown to be obtained from the smooth min and max relative entropies in the asymptotic limit

numerical inversion of a general incomplete elliptic integral we present a numerical method to invert a general incomplete elliptic integral with respect to its argument and or amplitude the method obtains a solution by bisection accelerated by half argument formulas and addition theorems to evaluate the incomplete elliptic integrals and jacobian elliptic functions required in the process if faster execution is desirable at the cost of complexity of the algorithm the sequence of bisection is switched to allow an improvement by using newton s method halley s method or higher order schroder methods in the improvement process the elliptic integrals and functions are computed by using maclaurin series expansion and addition theorems based on the values obtained at the end of the bisection also the derivatives of the elliptic integrals and functions are recursively evaluated from their values by adopting 0 2 as the critical value of the length of the solution interval to shift to the improvement process we suppress the expected number of bisections to be as low as four on average the typical number of applications of update formulas in the double precision environment is three for newton s method and two for halley s method or higher order schroder methods whether the improvement process is added or not our method requires none of the procedures to compute the incomplete elliptic integrals and jacobian elliptic functions but only those to evaluate the complete elliptic integrals once at the beginning as a result it runs fairly quickly in general for example when using the improvement process it is around 2 5 times faster than newton s method using boyd s starter boyd 2012 25 in inverting e f m legendre s incomplete elliptic integral of the second kind

a duality proof of tchakaloff s theorem abstract tchakaloff s theorem establishes the existence of a quadrature rule of prescribed degree relative to a positive compactly supported measure that is absolutely continuous with respect to lebesgue measure on r d subsequent extensions were obtained by mysovskikh and by putinar we provide new proofs and partial extensions of these results based on duality techniques utilized by stochel we also obtain new uniqueness criteria in the truncated complex moment problem