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

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

an extreme form of superactivation for quantum zero error capacities the zero error capacity of a channel is the rate at which it can send information perfectly with zero probability of error and has long been studied in classical information theory we show that the zero error capacity of quantum channels exhibits an extreme form of nonadditivity one which is not possible for classical channels or even for the usual capacities of quantum channels by combining probabilistic arguments with algebraic geometry we prove that there exist channels and with no zero error classical capacity whatsoever but whose joint zero error quantum capacity is positive this striking effect is an extreme form of the superactivation phenomenon as it implies that both the classical and quantum zero error capacities of these channels can be superactivated simultaneously while being a strictly stronger property of capacities superactivation of the quantum zero error capacity was not previously known

additivity for unital qubit channels additivity of the holevo capacity is proved for product channels under the condition that one of the channels is a unital qubit channel with the other completely arbitrary as a byproduct this proves that the holevo bound is the classical information capacity of such qubit channels and provides an explicit formula for this classical capacity additivity of minimal entropy and multiplicativity of p norms are also proved under the same assumptions the proof relies on a new bound for the p norm of an output state from the half noisy phase damping channel

isometries rigidity and universal covers the goal of this paper is to describe all closed aspherical riemannian manifolds m whose universal covers m have have a nontrivial amount of symmetry by this we mean that isom m is not discrete by the well known theorem of myers steenrod ms this condition is equivalent to isom m 1 m also note that if any cover of m has a nondiscrete isometry group then so does its universal cover m our description of such m is given in theorem 1 2 below the proof of this theorem uses methods from lie theory harmonic maps large scale geometry and the homological theory of transformation groups the condition that m have nondiscrete isometry group appears in a wide variety of problems in geometry since theorem 1 2 provides a taxonomy of such m it can be used to reduce many general problems to verifications of specific examples actually it is not always theorem 1 2 which is applied directly but the main subresults from its proof after explaining in 1 1 the statement of theorem 1 2 we give in 1 2 a number of such applications these range from new characterizations of locally symmetric manifolds to the classification of contractible manifolds covering both compact and finite volume manifolds to a new proof of the nadel frankel theorem in complex geometry

rtni a symbolic integrator for haar random tensor networks we provide a computer algebra package called random tensor network integrator rtni it allows to compute averages of tensor networks containing multiple haar distributed random unitary matrices and deterministic symbolic tensors such tensor networks are represented as multigraphs with vertices corresponding to tensors or random unitaries and edges corresponding to tensor contractions input and output spaces of random unitaries may be subdivided into arbitrary tensor factors with dimensions treated symbolically the algorithm implements the graphical weingarten calculus and produces a weighted sum of tensor networks representing the average over the unitary group we illustrate the use of this algorithmic tool on some examples from quantum information theory including entropy calculations for random tensor network states as considered in toy models for holographic duality mathematica and python implementations are supplied

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

measure homology and singular homology are isometrically isomorphic measure homology is a variation of singular homology designed by thurston in his discussion of simplicial volume zastrow and hansen showed independently that singular homology with real coefficients and measure homology coincide algebraically on the category of cw complexes it is the aim of this paper to prove that this isomorphism is isometric with respect to the l1 seminorm on singular homology and the seminorm on measure homology induced by the total variation this in particular implies that one can calculate the simplicial volume via measure homology as already claimed by thurston for example measure homology can be used to prove gromov s proportionality principle of simplicial volume

an infinite sequence of additive channels the classical capacity of cloning channels we introduce an infinite sequence of quantum channels for which the holevo capacity is additive the channel series is closely related to the quantum channels arising from universal quantum cloning machines the additivity proof is motivated by a special property the studied channels enjoy the property of conjugate degradability as a consequence of the announced proof we also provide an easy way of proving the additivity of the holevo capacity for the original unruh channel for which the quantum capacity is already known consequently we present not only an infinite series of finite dimensional channels but also a nontrivial example of an infinite dimensional channel for which the classical and quantum channel capacities are easily calculable

fast imaging of mean axial and radial diffusion kurtosis diffusion kurtosis imaging dki is being increasingly reported to provide sensitive biomarkers of subtle changes in tissue microstructure however dki also imposes larger data requirements than diffusion tensor imaging dti hence the widespread adaptation and exploration of dki would benefit from more efficient acquisition and computational methods to meet this demand we recently developed a method capable of estimating mean kurtosis with only 13 diffusion weighted images this approach was later shown to provide very accurate mean kurtosis estimates and to be more efficient in terms of contrast to noise per unit time however insofar the computation of two other critical dki parameters radial and axial kurtosis has required the estimation of all 22 variables parameterizing the full dki signal expression here we present two strategies for estimating all of dki s principal parameters mean kurtosis radial kurtosis and axial kurtosis using only 19 diffusion weighted images compared to the current state of the art acquisitions typically requiring about 60 images the first approach is based on axially symmetric diffusion and kurtosis tensors presented here for the first time and referred to as axially symmetric dki the second approach is applicable in tissues with a priori known principal diffusion direction and does not require fitting of any kind the approaches are evaluated in human brain in vivo as well as in fixed rat spinal cord and are demonstrated to provide metrics in good agreement with their full dki counterparts estimated with nonlinear least squares for small data sets and in white matter axially symmetric dki provides more accurate and robust estimates than unconstrained dki the significant acceleration achieved further paves the way to routine application of the technique

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

revisiting additivity violation of quantum channels we prove additivity violation of minimum output entropy of quantum channels by straightforward application of epsilon net argument and levy s lemma the additivity conjecture was disproved initially by hastings later a proof via asymptotic geometric analysis was presented by aubrun szarek and werner which uses dudley s bound on gaussian process or dvoretzky s theorem with schechtman s improvement in this paper we develop another proof along dvoretzky s theorem in milman s view showing additivity violation in broader regimes than the existing proofs importantly dvoretzky s theorem works well with norms to give strong statements but these techniques can be extended to functions which have norm like structures positive homogeneity and triangle inequality then a connection between hastings method and ours is also discussed in addition we make some comments on relations between regularized minimum output entropy and classical capacity of quantum 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 dimension of subspaces with bounded schmidt rank we consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states this is motivated in part by results of hayden et al e print arxiv quant ph 0407049 commun math phys 265 95 2006 which show that in large d d dimensional systems there exist random subspaces of dimension almost d2 all of whose states have entropy of entanglement at least logd o 1 it is also a generalization of results on the dimension of completely entangled subspaces which have connections with the construction of unextendible product bases here we take as entanglement measure the schmidt rank and determine for every pair of local dimensions da and db and every r the largest dimension of a subspace consisting only of entangled states of schmidt rank r or larger this exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in hayden et al we also determine the converse the l

generic local distinguishability and completely entangled subspaces a subspace of a multipartite hilbert space is completely entangled if it contains no product states such subspaces can be large with a known maximum size smax approaching the full dimension of the system d we show that almost all subspaces with dimension s smax are completely entangled and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n d smax this condition holds for almost all sets of states of all multipartite systems and reveals something surprising the criterion is identical for separable and nonseparable states entanglement makes no difference

perturbations of certain crossed product algebras by free groups given two von neumann algebras m and n acting on the same hilbert space d m n d m n is defined to be the hausdorff distance between their unit balls the kadison kastler problem asks whether two sufficiently close von neumann algebras are spatially isomorphic in this article we show that if p 0 p 0 is an injective von neumann algebra with a cyclic tracial vector g is a free group is a free action of g on p 0 p 0 and n is a von neumann algebra such that d n p 0 g 7 d n p 0 g 1 7 10 7 then n and p 0 g p 0 g are spatially isomorphic suitable choices of the actions give the first examples of infinite noninjective factors for which this problem has a positive solution

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

the intrinsic asymmetry and inhomogeneity of teichmuller space let s sg n be a connected orientable surface of genus g 0 with n 0 punctures let teich s denote the corresponding teichmuller space and let mod s denote the mapping class group of s understanding the analogy of teich s with symmetric spaces is a wellknown theme recall that a complete riemannian manifold x is symmetric if it is symmetric at each point x x the map t 7 t which flips geodesics about x is an isometry symmetric spaces x are homogeneous the isometry group isom x acts transitively on x in his famous paper ro royden studied the possible symmetry and homogeneity of teich s endowed with the teichmuller metric

proper actions of lattices on contractible manifolds every lattice in a connected semi simple lie group g acts properly discontinuously by isometries on the contractible manifold g k k a maximal compact subgroup of g we prove that if acts on a contractible manifold w and if either 1 the action is properly discontinuous or 2 w is equipped with a complete riemannian metric the action is by isometries and with unbounded orbits g is simple with finite center and rank 1 then dimw dimg k

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

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

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