comments on hastings additivity counterexamples hastings 12 recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy in this paper we present an expanded version of hastings proof in addition to a careful elucidation of the details of the proof we also present bounds for the minimal dimensions needed to obtain a counterexample

a proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional eckart young approach the proper generalized decomposition pgd is a methodology initially proposed for the solution of partial differential equations pde defined in tensor product spaces it consists in constructing a separated representation of the solution of a given pde in this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting in particular we introduce a generalization of eckart and young theorem which allows to prove the convergence of the so called progressive pgd for a large class of linear problems defined in tensor product hilbert spaces

boundary regularity of rotating vortex patches we show that the boundary of a rotating vortex patch or v state in the terminology of deem and zabusky is c provided the patch is close to the bifurcation circle in the lipschitz norm the rotating patch is also convex if it is close to the bifurcation circle in the c2 norm our proof is based on burbea s approach to v states

generalized surface quasi geostrophic equations with singular velocities this paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals the first family is a generalized surface quasigeostrophic sqg equation with the velocity field u related to the scalar by where and is the zygmund operator the borderline case 1 corresponds to the sqg equation and the situation is more singular for 1 we obtain the local existence and uniqueness of classical solutions the global existence of weak solutions and the local existence of patch type solutions the second family is a dissipative active scalar equation with which is at least logarithmically more singular than the velocity in the first family we prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data this result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by k ohkitani 2012 wiley periodicals inc

a quantum computing primer for operator theorists this is an exposition of some of the aspects of quantum computation and quantum information that have connections with operator theory after a brief introduction we discuss quantum algorithms we outline basic properties of quantum channels or equivalently completely positive trace preserving maps the main theorems for quantum error detection and correction are presented and we conclude with a description of a particular passive method of quantum error correction

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

additive bounds of minimum output entropies for unital channels and an exact qubit formula we find an upper bound for the minimum output entropy of a unital quantum channel and obtain an exact formula for general qubit channels our techniques incorporate the renyi entropies particularly with renyi parameter alpha 2 moreover since our upper bound is additive under tensor product we get as a corollary an upper bound for the classical capacity of unital quantum channels interestingly our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless hermitian operators and is tight in the sense that it gives the precise quantity of classical capacity of the werner holevo channel as an example we study quantum channels with operator sum representation that is made of the discrete weyl operators generalized pauli operators and explain how our formula works in this case finally we find new examples for which the minimum output renyi 2 entropy is additive

reduced order modeling of parameterized pdes using time space parameter principal component analysis this paper presents a methodology for constructing low order surrogate models of finite element finite volume discrete solutions of parameterized steady state partial differential equations the construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high dimensional discrete solutions using only a few coefficients an incremental greedy approach is developed for efficiently tackling problems with high dimensional parameter spaces for numerical experiments and validation several non linear steady state convection diffusion reaction problems are considered first in one spatial dimension with two parameters and then in two spatial dimensions with two and five parameters in the two dimensional spatial case with two parameters it is shown that a 7 7 coefficient matrix is sufficient to accurately reproduce the expected solution while in the five parameters problem a 13 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy the proposed methodology is expected to find applications to parameter variation studies uncertainty analysis inverse problems and optimal design

the geometric measure of multipartite entanglement and the singular values of a hypermatrix it is shown that the geometric measure of entanglement of a pure multipartite state satisfies a polynomial equation generalizing the singular value equation of the matrix of coefficients of a bipartite state the equation is solved for a class of three qubit states

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

relative equilibria with holes for the surface quasi geostrophic equations abstract we study the existence of doubly connected rotating patches for the inviscid surface quasi geostrophic equation left open by de la hoz hassainia and hmidi in 2016 in 10 by using the approach proposed in 2016 by castro cordoba and gomez serrano in 4 we also prove that close to the annulus the boundaries are actually analytic curves

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

quantum channels irreducibly covariant with respect to the finite group generated by the weyl operators in matrix algebras we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the weyl operators in particular we analyze the irreducibly covariant quantum channels that is the completely positive and trace preserving linear maps interestingly imposing additional symmetries leads to the so called generalized pauli channels which were recently considered in the context of the non markovian quantum evolution finally we provide examples of irreducibly covariant positive but not necessarily completely positive maps

remarks on additivity of the holevo channel capacity and of the entanglement of formation the purpose of this article is to discuss the relation between the additivity questions regarding the quantities holevo capacityofaquantumchannelt and entanglementofformationofabipartitestate in particular using the stinespring dilation theorem we give a formula for the channel capacity involving entanglement of formation this can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels we demonstrate this connection for some families of channels allowing us to calculate the entanglement cost for many states including some where a strictly smaller upper bound on the distillable entanglement is known group symmetry is used for more sophisticated analysis giving formulas valid for a class of channels this is presented in a general framework extending recent findings of vidal dur and cirac we also discuss the property of superadditivity of the entanglement of formation which would imply both the general additivity of this function under tensor products and of the holevo capacity with or without linear cost constraints

the capacity of the quantum depolarizing channel the information carrying capacity of the d dimensional depolarizing channel is computed it is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space and using measurements which are products of projections onto this same orthonormal basis in other words neither entangled signal states nor entangled measurements give any advantage for information capacity the result follows from an additivity theorem for the product channel spl delta spl ominus spl psi where spl delta is the depolarizing channel and spl psi is a completely arbitrary channel we establish the amosov holevo werner see probl inform transm vol 36 p 305 313 2000 p norm conjecture for this product channel for all p spl ges 1 and deduce from this the additivity of the minimal entropy and of the holevo quantity spl chi sup

a strong converse for classical channel coding using entangled inputs a fully general strong converse for channel coding states that when the rate of sending classical information exceeds the capacity of a quantum channel the probability of correctly decoding goes to zero exponentially in the number of channel uses even when we allow code states which are entangled across several uses of the channel such a statement was previously only known for classical channels and the quantum identity channel by relating the problem to the additivity of minimum output entropies we show that a strong converse holds for a large class of channels including all unital qubit channels the d dimensional depolarizing channel and the werner holevo channel this further justifies the interpretation of the classical capacity as a sharp threshold for information transmission

existence for the α patch model and the qg sharp front in sobolev spaces abstract we consider a family of contour dynamics equations depending on a parameter with 0 1 the vortex patch problem of the 2 d euler equation is obtained taking 0 and the case 1 corresponds to a sharp front of the qg equation we prove local in time existence for the family of equations in sobolev spaces

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

reduced dimensional gaussian process emulators of parametrized partial differential equations based on isomap in this paper isomap and kernel isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a gaussian process emulator of parametrized partial differential equations the output space consists of spatial or spatio temporal fields that are functions of multiple input variables for such problems standard multi output gaussian process emulation strategies are computationally impractical and or make restrictive assumptions regarding the correlation structure the method we develop can be applied without modification to any problem involving vector valued targets and vector valued inputs it also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques

on the existence of stationary patches abstract in this paper we show the existence of the first nontrivial family of analytic stationary patch solutions of the sqg and gsqg equations this answers an open problem in f de la hoz et al 2016 13

inequalities for trace norms of 2 2 block matrices this paper derives an inequality relating the p norm of a positive 2 2 block matrix to the p norm of the 2 2 matrix obtained by replacing each block by its p norm the inequality had been known for integer values of p so the main contribution here is the extension to all values p 1 in a special case the result reproduces hanner s inequality a weaker inequality which applies also to non positive matrices is presented as an application in quantum information theory the inequality is used to obtain some results concerning maximal p norms of product channels

nonadditivity of renyi entropy and dvoretzky s theorem the goal of this note is to show that the analysis of the minimum output p renyi entropy of a typical quantum channel essentially amounts to applying milman s version of dvoretzky s theorem about almost euclidean sections of high dimensional convex bodies this conceptually simplifies the nonconstructive argument by hayden winter disproving the additivity conjecture for the minimal output p renyi entropy for p 1

constructive counterexamples to the additivity of the minimum output renyi entropy of quantum channels for all p 2 we present a constructive example of the violation of the additivity of minimum output renyi entropy for each p 2 the example is provided by an antisymmetric subspace of a suitable dimension we discuss the possibility of extension of the result to go beyond p 2 and obtain additivity for p 0 for a class of entanglement breaking channels

converse bounds for quantum and private communication over holevo werner channels werner states have a host of interesting properties which often serve to illuminate the unusual properties of quantum information starting from these states one may define a family of quantum channels known as the holevo werner channels which themselves afford several unusual properties in this paper we use the teleportation covariance of these channels to upper bound their two way assisted quantum and secret key capacities this bound may be expressed in terms of relative entropy distances such as the relative entropy of entanglement and also in terms of the squashed entanglement most interestingly we show that the relative entropy bounds are strictly sub additive for a sub class of the holevo werner channels so that their regularisation provides a tighter performance these information theoretic results are first found for point to point communication and then extended to repeater chains and quantum networks under different types of routing strategies

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

a least squares approximation of partial differential equations with high dimensional random inputs uncertainty quantification schemes based on stochastic galerkin projections with global or local basis functions and also stochastic collocation methods in their conventional form suffer from the so called curse of dimensionality the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem in this paper to overcome the curse of dimensionality a low rank separated approximation of the solution of a stochastic partial differential spde with high dimensional random input data is obtained using an alternating least squares als scheme it will be shown that in theory the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system for the case of an elliptic spde an a priori error analysis of the algorithm is derived finally different aspects of the proposed methodology are explored through its application to some numerical experiments

galerkin reduced order modeling scheme for time dependent randomly parametrized linear partial differential equations summary in this paper we consider the problem of constructing reduced order models of a class of time dependent randomly parametrized linear partial differential equations our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates parameter space and time the proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions the unknown basis functions in the decomposition are estimated using an alternating iterative galerkin projection scheme numerical studies on the time dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos based galerkin projection schemes comparison studies are also made against nouy s generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages copyright 2012 john wiley sons ltd

anchored analysis of variance petrov galerkin projection schemes for linear stochastic structural dynamics in this paper we propose anchored functional analysis of variance petrovgalerkin aapg projection schemes originally developed in the context of parabolic stochastic partial differential equatio

maximization of capacity and lp norms for some product channels it is conjectured that the holevo capacity of a product channel is achieved when product states are used as input amosov holevo and werner have also conjectured that the maximal lp norm of a product channel is achieved with product input states in this article we establish both of these conjectures in the case that is arbitrary and is a cq or qc channel as defined by holevo we also establish the amosov holevo and werner conjecture when is arbitrary and either is a qubit channel and p 2 or is a unital qubit channel and p is integer our proofs involve a new conjecture for the norm of an output state of the half noisy channel i when is a qubit channel we show that this conjecture in some cases also implies additivity of the holevo capacity

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

doubly connected v states for the generalized surface quasi geostrophic equations in this paper we prove the existence of doubly connected v states for the generalized sqg equations with 0 1 they can be described by countable branches bifurcating from the annulus at some explicit eigenvalues related to bessel functions of the first kind contrary to euler equations hmidi et al doubly connected v states for the planar euler equations arxiv 1409 7096 2015 we find v states rotating with positive and negative angular velocities at the end of the paper we discuss some numerical experiments concerning the limiting v states

an algorithm to explore entanglement in small systems a quantum state s entanglement across a bipartite cut can be quantified with entanglement entropy or more generally schmidt norms using only schmidt decompositions we present a simple iterative algorithm to maximize schmidt norms depending on the choice of norm the optimizing states maximize or minimize entanglement possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace recognizing that convergence but not success is certain we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels

counterexample to an additivity conjecture for output purity of quantum channels a conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel as measured by the p norm should be multiplicative with respect to the tensor product of channels we disprove this conjecture for p 4 79 the same example with p also disproves a conjecture for the multiplicativity of the injective norm of hilbert space tensor products

counterexamples to additivity of minimum output p renyi entropy for p close to 0 complementing recent progress on the additivity conjecture of quantum information theory showing that the minimum output p renyi entropies of channels are not generally additive for p 1 we demonstrate here by a careful random selection argument that also at p 0 and consequently for sufficiently small p there exist counterexamples

on the v states for the generalized quasi geostrophic equations we prove the existence of the v states for the generalized inviscid sqg equations with alpha in 0 1 these structures are special rotating simply connected patches with m fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity this produces inter alia an infinite family of non stationary global solutions with uniqueness

an application of decomposable maps in proving multiplicativity of low dimensional maps in this paper we present a class of maps for which the multiplicativity of the maximal output p norm holds for p 2 and p 4 the class includes all positive trace preserving maps from b c3 to b c2 in this sense the result is a generalization of the corresponding result in the work of king and koldan new multiplicativity results for qubit maps j math phys 47 042106 2006 where the multiplicativity was proved for all positive trace preserving maps from b c2 to b c2 with p 2 and p 4 interestingly by contrast the multiplicativity of p norm was investigated in the context of quantum information theory and shown not to hold in general for high dimensional quantum channels hayden p and winter a counterexamples to the maximal p norm multiplicativity conjecture for all p 1 commun math phys 284 263 2008 moreover the werner holevo channel which is a map from b c3 to b c3 is a counterexample for p 4 79 werner and holevo j math phys 43 4353 2002

almost one bit violation for the additivity of the minimum output entropy in a previous paper we proved that in the appropriate asymptotic regime the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic compact set kk t we also showed that the set kk t is obtained up to an intersection as the unit ball of the dual of a free compression norm in this paper we identify the maximum of ell p norms on the set kk t and prove that the maximum is attained on a vector of shape a b b where a b in particular we compute the precise limit value of the minimum output entropy of a single random quantum channel as a corollary we show that for any varepsilon 0 it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183 and that for appropriate choice of parameters the violation can be as large as log 2 varepsilon conversely our result implies that with probability one in the limit one does not obtain a violation of additivity using conjugate random quantum channels and the bell state in dimension 182 and less

weak multiplicativity for random quantum channels it is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p norms for any p 1 in this work we show that a weaker variant of multiplicativity nevertheless holds for these channels for any constant p 1 given a random quantum channel mathcal n i e a channel whose stinespring representation corresponds to a random subspace s we show that with high probability the maximum output p norm of mathcal n otimes n decays exponentially with n the proof is based on relaxing the maximum output norm of mathcal n to the operator norm of the partial transpose of the projector onto s then calculating upper bounds on this quantity using ideas from random matrix theory

multiplicativity of the maximal output 2 norm for depolarized werner holevo channels we study the multiplicativity of the output 2 norm for depolarized werner holevo channels and show that multiplicativity holds for a product of two identical channels in this class moreover it is shown that the depolarized werner holevo channels do not satisfy the entrywise positivity condition introduced by king and ruskai e print quant ph 0909181v2 e print quant ph 0981026v1 which suggests that the main result is nontrivial

existence and regularity of rotating global solutions for the generalized surface quasi geostrophic equations motivated by the recent work of hassainia and hmidi z hassainia t hmidi on the vstates for the generalized quasi geostrophic equations arxiv preprint arxiv 1405 0858 we close the question of the existence of convex global rotating solutions for the generalized surface quasigeostrophic equation for 2 1 2 we also show c 1 regularity of their boundary for all 2 0 2

on the minimum output entropy of random orthogonal quantum channels we consider the sequences of random quantum channels defined by using the stinespring formula with haar distributed random orthogonal matrices for any fixed sequence of input states we study the asymptotic eigenvalue distribution of the outputs through the tensor powers of random channels we show that the input states achieving minimum output entropy are tensor products of maximally entangled states bell states when the tensor power is even this phenomenon is completely different from the one for random quantum channels constructed from haar distributed random unitary matrices which leads us to formulate some conjectures about the regularized minimum output entropy

additivity of the classical capacity of entanglement breaking quantum channels we show that for the tensor product of an entanglement breaking quantum channel with an arbitrary quantum channel both the minimum entropy of an output of the channel and the holevo schumacher westmoreland capacity are additive in addition for the tensor product of two arbitrary quantum channels we give a bound involving entanglement of formation for the amount of subadditivity for minimum entropy output or superadditivity for classical capacity that can occur

decomposability of linear maps under tensor powers both completely positive and completely copositive maps stay decomposable under tensor powers i e under tensoring the linear map with itself but are there other examples of maps with this property we show that this is not the case any decomposable map that is neither completely positive nor completely copositive will lose decomposability eventually after taking enough tensor powers moreover we establish explicit bounds to quantify when this happens to prove these results we use a symmetrization technique from the theory of entanglement distillation and analyze when certain symmetric maps become non decomposable after taking tensor powers finally we apply our results to construct new examples of non decomposable positive maps and establish a connection to the positive partial transpose squared conjecture

counterexamples to the maximal p norm multiplicativity conjecture for all p 1 for all p 1 we demonstrate the existence of quantum channels with non multiplicative maximal output p norms equivalently for all p 1 the minimum output renyi entropy of order p of a quantum channel is not additive the violations found are large in all cases the minimum output renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors since p 1 corresponds to the von neumann entropy these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true it cannot be proved as a consequence of any channel independent guarantee of maximal p norm multiplicativity we also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p 2

new multiplicativity results for qubit maps let be a trace preserving positivity preserving but not necessarily completely positive linear map on the algebra of complex 2 2 matrices and let be any finite dimensional completely positive map for p 2 and p 4 we prove that the maximal p norm of the product map is the product of the maximal p norms of and restricting to the class of completely positive maps this settles the multiplicativity question for all qubit channels in the range of values p 4