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

hastings s additivity counterexample via dvoretzky s theorem the goal of this note is to show that hastings s counterexample to the additivity of minimal output von neumann entropy can be readily deduced from a sharp version of dvoretzky s theorem

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

persistence in nonequilibrium surface growth persistence probabilities of the interface height in 1 1 and 2 1 dimensional atomistic solid on solid stochastic models of surface growth are studied using kinetic monte carlo simulations with emphasis on models that belong to the molecular beam epitaxy mbe universality class both the initial transient and the long time steady state regimes are investigated we show that for growth models in the mbe universality class the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady state regimes for the mbe universality class the positive and negative persistence exponents in the steady state are found to be 0 66 0 02 and 0 78 0 02 respectively in 1 1 dimensions and 0 76 0 02 and 0 85 0 02 respectively in 2 1 dimensions the noise reduction technique is applied on some of the 1 1 dimensional models in order to obtain accurate values of the persistence exponents we show analytically that a relation between the steady state persistence exponent and the dynamic growth exponent found earlier to be valid for linear models should be satisfied by the smaller of the two steady state persistence exponents in the nonlinear models our numerical results for the persistence exponents are consistent with this prediction we also find that the steady state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state the dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form

equivalence after extension and matricial coupling coincide with schur coupling on separable hilbert spaces it is known that two banach space operators that are schur coupled are also equivalent after extension or equivalently matricially coupled the converse implication that operators which are equivalent after extension or matricially coupled are also schur coupled was only known for fredholm hilbert space operators and fredholm banach space operators with index 0 we prove that this implication also holds for hilbert space operators with closed range generalizing the result for fredholm operators and banach space operators that can be approximated in operator norm by invertible operators the combination of these two results enables us to prove that the implication holds for all operators on separable hilbert spaces

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

entanglement of random subspaces via the hastings bound recently hastings a counterexample to additivity of minimum output entropy nat phys 5 255 2009 e print arxiv 0809 3972v3 proved the existence of random unitary channels which violate the additivity conjecture in this paper we use hastings method to derive new bounds for the entanglement of random subspaces of bipartite systems as an application we use these bounds to prove the existence of nonunital channels which violate additivity of minimal output entropy

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

survival of a diffusing particle in a transverse shear flow a first passage problem with continuously varying persistence exponent we consider a particle diffusing in the y direction dy dt t subject to a transverse shear flow in the x direction dx dt f y where x 0 and x 0 is an absorbing boundary we treat the class of models defined by f y v y where the upper lower sign refers to y 0 y 0 we show that the particle survives with probability q t t with 1 4 independent of if v v if v v however we show that depends on both and the ratio v v and we determine this dependence

schur coupling and related equivalence relations for operators on a hilbert space abstract for operators on hilbert spaces of any dimension we show that equivalence after extension coincides with equivalence after one sided extension thus obtaining a proof of their coincidence with schur coupling we also provide a concrete description of this equivalence relation in several cases in particular for compact operators

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

hypercontractivity for semigroups of unital qubit channels hypercontractivity is proved for products of qubit channels that belong to self adjoint semigroups the hypercontractive bound gives necessary and sufficient conditions for a product of the form e t 1 h 1 otimes cdots otimes e t n h n to be a contraction from l p to l q where l p is the algebra of 2 n dimensional matrices equipped with the normalized schatten norm and each generator h j is a self adjoint positive semidefinite operator on the algebra of 2 dimensional matrices as a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels

persistence of fractional brownian motion with moving boundaries and applications we consider various problems related to the persistence probability of fractional brownian motion fbm which is the probability that the fbm x stays below a certain level until time t recently oshanin et al 2012 arxiv 1209 3313v2 have studied a physical model where persistence properties of fbm are shown to be related to scaling properties of a quantity jn called the steady state current it turns out that for this analysis it is important to determine persistence probabilities of fbm with a moving boundary we show that one can add a boundary of logarithmic order to an fbm without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in oshanin et al 2013 phys rev lett at press arxiv 1209 3313v2 moreover we complement their findings by considering the continuous time version of jt finally we use the results for moving boundaries in order to improve estimates by molchan 1999 commun math phys 205 97 111 concerning the persistence properties of other quantities of interest such as the time when an fbm reaches its maximum on the time interval 0 1 or the last zero in the interval 0 1

the minimum entropy output of a quantum channel is locally additive we show that the minimum von neumann entropy output of a quantum channel is locally additive hastings counterexample for the additivity conjecture makes this result quite surprising in particular it indicates that the nonadditivity of the minimum entropy output is a global effect of quantum channels

persistence probabilities of two sided integrated sums of correlated stationary gaussian sequences we study the persistence probability for some two sided discrete time gaussian sequences that are discrete time analogues of fractional brownian motion and integrated fractional brownian motion respectively our results extend the corresponding ones in continuous time in molchan commun math phys 205 1 97 111 1999 and molchan j stat phys 167 6 1546 1554 2017 to a wide class of discrete time processes

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

minimal entropy of states emerging from noisy quantum channels we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel which is modeled by the action of a completely positive trace preserving or stochastic map we provide strong support for the conjecture that this minimal entropy is additive namely that the minimum entropy can be achieved when product states are transmitted explicitly we prove that for a tensor product of two unital stochastic maps on qubit states using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states we give a separate argument based on the geometry of the image of the set of density matrices under stochastic maps which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps we also show that the maximal norm of the output states is multiplicative for most product maps on n qubit states including all those for which at least one map is unital for the class of unital channels on c sup 2 we show that additivity of minimal entropy implies that the holevo see ieee trans inform theory vol 44 p 269 73 1998 and russian math surv p 1295 1331 1999 capacity of the channel is additive over two inputs achievable with orthogonal states and equal to the shannon capacity this implies that superadditivity of the capacity is possible only for nonunital channels

effective transport in random shear flows we obtain an effective transport description for the superdiffusive motion of random walkers in stratified flow by projection of the process on the direction of stratification the effective dimensionally reduced motion is shown to describe a correlated random walk characterized by the lagrangian velocity correlation we analyze the projected motion through exact analytical solutions for the distribution density for an arbitrary correlated gaussian noise and derive an evolution equation for the one point and conditional two point displacement densities the latter gives an explicit effective equation for superdiffusive transport in stratified random flow and demonstrates that the displacement density has a gaussian scaling form for all times

random quantum channels ii entanglement of random subspaces renyi entropy estimates and additivity problems in this paper we obtain new bounds for the minimum output entropies of random quantum channels these bounds rely on random matrix techniques arising from free probability theory we then revisit the counterexamples developed by hayden and winter to get violations of the additivity equalities for minimum output renyi entropies we show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the p renyi entropy for some sequences of random quantum channels we compute almost surely the limit of their schatten s 1 to s p norms

power laws in surface physics the deep the shallow and the useful abstract the growth and dynamics of solid surfaces displays a multitude of power law relationships which are often associated with geometric self similarity in many cases the mechanisms behind these power laws are comparatively trivial and require little more than dimensional analysis for their derivation the information of interest to surface physicists then resides in the prefactors this point will be illustrated by recent experimental and theoretical work on the growth induced roughening of thin films and step fluctuations on vicinal surfaces the conventional distinction between trivial and nontrivial power laws will be critically examined in general and specifically in the context of persistence of step fluctuations

simplifying additivity problems using direct sum constructions we study the additivity problems for the classical capacity of quantum channels the minimal output entropy and its convex closure we show for each of them that additivity for arbitrary pairs of channels holds if and only if it holds for arbitrary equal pairs which in turn can be taken to be unital in a similar sense weak additivity is shown to imply strong additivity for any convex entanglement monotone the implications are obtained by considering direct sums of channels or states for which we show how to obtain several information theoretic quantities from their values on the summands this provides a simple and general tool for lifting additivity results

persistence of a rouse polymer chain under transverse shear flow we consider a single rouse polymer chain in two dimensions in the presence of a transverse shear flow along the x direction and calculate the persistence probability p 0 t that the x coordinate of a bead in the bulk of the chain does not return to its initial position up to time t we show that the persistence decays at late times as a power law p 0 t ensuremath sim t ensuremath ensuremath theta with a nontrivial exponent ensuremath theta the analytical estimate of ensuremath theta 0 359 dots obtained using an independent interval approximation is in excellent agreement with the numerical value ensuremath theta ensuremath approx 0 360 ifmmode pm else textpm fi 0 001

spectral properties of truncated toeplitz operators by equivalence after extension abstract we study truncated toeplitz operators in model spaces k p for 1 p with essentially bounded symbols in a class including the algebra c r h as well as sums of analytic and anti analytic functions satisfying a separation condition using their equivalence after extension to toeplitz operators with 2 2 matrix symbols we establish fredholmness and invertibility criteria for truncated toeplitz operators with separated symbols and in particular we identify a class of operators for which semi fredholmness is equivalent to invertibility for symbols in c r h we extend to all p 1 the spectral mapping theorem for the essential spectrum stronger results are obtained in the case of operators with rational symbols or if the underlying model space is finite dimensional

on the extreme points of quantum channels abstract let l m n denote the convex set of completely positive trace preserving operators from c m m to c n n i e quantum channels we give a necessary condition for l l m n to be an extreme point we show that generically this condition is also sufficient we characterize completely the extreme points of l 2 2 and l 3 2 i e quantum channels from qubits to qubits and from qutrits to qubits

algorithmic superactivation of asymptotic quantum capacity of zero capacity quantum channels the superactivation of zero capacity quantum channels makes it possible to use two zero capacity quantum channels with a positive joint capacity for their output currently we have no theoretical background to describe all possible combinations of superactive zero capacity channels hence there may be many other possible combinations in practice to discover such superactive zero capacity channel pairs we must analyze an extremely large set of possible quantum states channel models and channel probabilities there is still no extremely efficient algorithmic tool for this purpose this paper shows an efficient algorithmical method of finding such combinations our method can be a very valuable tool for improving the results of fault tolerant quantum computation and possible communication techniques over very noisy quantum channels

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

persistence of randomly coupled fluctuating interfaces we study the persistence properties in a simple model of two coupled interfaces characterized by heights h 1 and h 2 respectively each growing over a d dimensional substrate the first interface evolves independently of the second and can correspond to any generic growing interface e g of the edwards wilkinson or of the kardar parisi zhang variety the evolution of h 2 however is coupled to h 1 via a quenched random velocity field in the limit d ensuremath rightarrow 0 our model reduces to the matheron de marsily model in two dimensions for d 1 our model describes a rouse polymer chain in two dimensions advected by a transverse velocity field we show analytically that after a long waiting time t 0 ensuremath rightarrow ensuremath infty the stochastic process h 2 at a fixed point in space but as a function of time becomes a fractional brownian motion with a hurst exponent h 2 1 ensuremath ensuremath beta 1 2 where ensuremath beta 1 is the growth exponent characterizing the first interface the associated persistence exponent is shown to be ensuremath theta s 2 1 ensuremath h 2 ensuremath beta 1 2 these analytical results are verified by numerical simulations

channel capacities via p summing norms abstract in this paper we show how the metric theory of tensor products developed by grothendieck perfectly fits in the study of channel capacities a central topic in shannon s information theory furthermore in the last years shannon s theory has been fully generalized to the quantum setting and revealed qualitatively new phenomena in comparison in this paper we consider the classical capacity of quantum channels with restricted assisted entanglement these capacities include the classical capacity and the unlimited entanglement assisted classical capacity of a quantum channel our approach to restricted capacities is based on tools from functional analysis and in particular the notion of p summing maps going back to grothendieck s work pisier s noncommutative vector valued l p spaces allow us to establish the new connection between functional analysis and information theory in the quantum setting

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

multipoint concentration statistics for transport in stratified random velocity fields we determine the full multipoint concentration statistics for superdiffusive transport in a steady stratified random velocity field using a lagrangian approach we derive explicit analytical expressions for the multipoint moments of concentration and specifically for the concentration variance which is a measure for concentration uncertainty the multipoint concentration moments are fully characterized by the lagrangian mean velocity and by the one and two particle velocity correlations while the relative variance at the center of mass of the mean concentration is constant it increases exponentially with time and distance from the center of mass this implies that small concentration values are particularly uncertain which can pose a serious practical concern as these are typically the earliest and latest to arrive at a point

asymptotically well behaved input states do not violate additivity for conjugate pairs of random quantum channels it is now well known that with high probability the additivity of minimum output entropy does not hold for pairs of the random quantum channel and its complex conjugate we investigate asymptotic behavior of output states of r tensor powers of such pairs as the dimension of inputs grows we compute the limit output states for any sequence of well behaved inputs which consist of a large class of input states having a nice set of parameters then we show that among these input states tensor products of bell states give asymptotically the least output entropy giving positive mathematical evidence towards additivity of above pairs of channels

multiplicativity properties of entrywise positive maps abstract multiplicativity of certain maximal p q norms of a tensor product of linear maps on matrix algebras is proved in situations in which the condition of complete positivity cp is either augmented by or replaced by the requirement that the entries of a matrix representative of the map are non negative ep in particular for integer t multiplicativity holds for the maximal 2 2t norm of a product of two maps whenever one of the pair is ep for the maximal 1 t norm for pairs of cp maps when one of them is also ep and for the maximal 1 2t norm for the product of an ep and a 2 positive map similar results are shown in the infinite dimensional setting of convolution operators on l 2 r with the pointwise positivity of an integral kernel replacing entrywise positivity of a matrix these results apply in particular to gaussian bosonic channels

schur complements and state space realizations motivated by state space realizations of transfer functions from system theory a number of operations on schur complements are introduced and studied these operations are equivalence extension multiplication inversion and factorization together they form an algebraic framework which is of independent interest and also useful in solving problems in analysis

equivalence after extension for compact operators on banach spaces in recent years the coincidence of the operator relations equivalence after extension and schur coupling was settled for the hilbert space case by showing that equivalence after extension implies equivalence after one sided extension in the present paper we investigate consequences of equivalence after extension for compact banach space operators we show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension in analogy with the necessary and sufficient conditions for compact hilbert space operators to be equivalent after extension in terms of their singular values we prove under certain additional conditions the necessity of a similar relationship between the s numbers of two compact banach space operators that are equivalent after extension for arbitrary s functions we investigate equivalence after extension for operators on lp spaces we show that two operators that act on different lp spaces cannot be equivalent after one sided extension such operators can still be equivalent after extension for instance all invertible operators are equivalent after extension however if one of the two operators is compact then they cannot be equivalent after extension this contrasts the hilbert space case where equivalence after one sided extension and equivalence after extension are in fact identical relations finally for general banach spaces x and y we investigate consequences of an operator on x being equivalent after extension to a compact operator on y we show that in this case a closed finite codimensional subspace of y must embed into x and that certain general banach space properties must transfer from x to y we also show that no operator on x can be equivalent after extension to an operator on y if x and y are essentially incomparable banach spaces

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

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

towards a state minimizing the output entropy of a tensor product of random quantum channels we consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels depending on natural assumptions that we make on the states the eigenvalues of their outputs have new properties which we describe our motivation is provided by the additivity questions in quantum information theory and we build on the idea that a bell state sent through a product of conjugated random channels has at least one large eigenvalue we generalize this setting in two directions first we investigate general entangled pure inputs and show that bell states give the least entropy among those inputs in the asymptotic limit we then study mixed input states and obtain new multi scale random matrix models that allow to quantify the difference of the outputs eigenvalues between a quantum channel and its complementary version in the case of a non pure input

persistence of a particle in the matheron de marsily velocity field we show that the longitudinal position x t of a particle in a d 1 dimensional layered random velocity field the matheron de marsily model can be identified as a fractional brownian motion fbm characterized by a variable hurst exponent h d 1 ensuremath d 4 for dl2 and h d 1 2 for dg2 the fbm becomes marginal at d 2 moreover using the known first passage properties of fbm we prove analytically that the disorder averaged persistence the probability of no zero crossing of the process x t up to time t has a power law decay for large t with an exponent ensuremath theta d 4 for dl2 and ensuremath theta 1 2 for dg 2 with logarithmic correction at d 2 results that were earlier derived by redner based on heuristic arguments and supported by numerical simulations s redner phys rev e 56 4967 1997

complementary schur complements abstract for square matrices the relationship is discussed between the notion of schur complement and the equivalent concepts of matricial coupling and equivalence after extension special attention is paid to the hermitian and skew hermitian cases

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