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

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

structural characterization and condition for measurement statistics preservation of a unital quantum operation we investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation acting on finite dimensional quantum states by reducing this problem to the problem of simultaneous diagonalization of the kraus operators associated with we can completely characterize the kinds of quantum states that are fixed by our work has several applications it gives a simple proof of the structural characterization of a unital quantum operation that acts on finite dimensional quantum states a result not explicitly mentioned in earlier studies it also provides a necessary and sufficient condition for determining what kind of measurement statistics is preserved by a unital quantum operation finally our result clarifies and extends the work of st rmer by giving a proof of a reduction theorem on the unassisted and entanglement assisted classical capacities coherent information and minimal output renyi entropy of a unital channel acting on a finite dimensional quantum state

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

additivity and distinguishability of random unitary channels a random unitary channel is one that is given by a convex combination of unitary channels it is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p norm can be equivalently restated in terms of random unitary channels this is done by constructing a random unitary approximation to a general quantum channel this approximation can be constructed efficiently and so it is also applied to the computational problem of distinguishing quantum circuits it is shown that the problem of distinguishing random unitary circuits is as hard as the problem of distinguishing general mixed state circuits which is complete for the class of problems having quantum interactive proof systems

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

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

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

inequalities for quantum entropy a review with conditions for equality this article presents self contained proofs of the strong subadditivity inequality for von neumann s quantum entropy s and some related inequalities for the quantum relative entropy most notably its convexity and its monotonicity under stochastic maps moreover the approach presented here which is based on klein s inequality and lieb s theorem that the function a tr ek log a is concave allows one to obtain conditions for equality in the case of strong subadditivity which states that s 123 s 2 s 12 s 23 where the subscripts denote subsystems of a composite system equality holds if and only if log 123 log 12 log 2 log 23 using the fact that the holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy we show that equality can be attained for that bound only when the states in the ensemble commute the article concludes with an appendix giving a short description of epstein s elegant proof of lieb s the

positive semidefinite diagonal minus tail forms are sums of squares by a diagonal minus tail form of even degree we understand a real homogeneous polynomial f x1 xn f x d x t x where the diagonal part d x is a sum of terms of the form b i x i 2d with all bi 0 and the tail t x a sum of terms a i 1i 2 cdots i n x 1 i 1 cdots x n i n with a i 1i 2 cdots i n 0 and at least two i 1 we show that an arbitrary change of the signs of the tail terms of a positive semidefinite diagonal minus tail form will result in a sum of squares of polynomials we also give an easily tested sufficient condition for a polynomial to be a sum of squares of polynomials sos and show that the class of polynomials passing this test is wider than the class passing lasserre s recent conditions another sufficient condition for a polynomial to be sos like lasserre s piecewise linear in its coefficients is also given

capacity bounds via operator space methods we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity we prove that for generalized dephasing channels the coherent information and reverse coherent information coincides it also implies an alternative approach for the strong super additivity and strong converse of generalized dephasing channels using the operator space technique our argument is based on an improved renyi relative entropy estimate via analyzing the channel s stinespring space we also apply this estimate to new examples of quantum channels arising from quantum group co representation and kitave s quantum computation model in particular we find concrete examples of non degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity

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

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

the influence of the environment on levy random search efficiency fractality and memory effects an open problem in the field of random searches relates to optimizing the search efficiency in fractal environments here we address this issue through a systematic study of levy searches in landscapes encompassing several degrees of target aggregation and fractality for scarce resources non destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets in contrast no such universal behavior occurs in the destructive case with forbidden revisits in which the optimal strategy strongly depends on the degree of target aggregation we also investigate how the presence of memory and learning skills of the searcher affect the search efficiency by considering a limiting model in which the searcher learns through recent experience to recognize food rich areas we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches instead there is an optimal extent of memory dependent on specific details of the search space and stochastic dynamics which maximizes the search efficiency this finding suggests a more general result namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects

positive semi definiteness and sum of squares property of fourth order four dimensional hankel tensors a symmetric positive semi definite psd tensor which is not sum of squares sos is called a psd non sos pns tensor is there a fourth order four dimensional pns hankel tensor the answer for this question has both theoretical and practical significance under the assumptions that the generating vector v of a hankel tensor a is symmetric and the fifth element v 4 of v is fixed at 1 we show that there are two surfaces m 0 and n 0 with the elements v 2 v 6 v 1 v 3 v 5 of v as variables such that m 0 n 0 a is sos if and only if v 0 m 0 and a is psd if and only if v 0 n 0 where v 0 is the first element of v if m 0 n 0 for a point p v 2 v 6 v 1 v 3 v 5 there are no fourth order four dimensional pns hankel tensors with symmetric generating vectors for such v 2 v 6 v 1 v 3 v 5 then we call such p a pns free point we prove that a 45 degree planar closed convex cone a segment a ray and an additional point are pns free numerical tests check various grid points and report that they are all pns free

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

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

the asymptotic entanglement cost of preparing a quantum state we give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state is equal to limn ef n n where ef is the entanglement of formation

sos approximations of nonnegative polynomials via simple high degree perturbations we show that every real polynomial f nonnegative on 1 1 n can be approximated in the l1 norm of coefficients by a sequence of polynomials f epsilon r that are sums of squares s o s this complements the existence of s o s approximations in the denseness result of berg christensen and ressel as we provide a very simple and explicit approximation sequence then we show that if the moment problem holds for a basic closed semi algebraic set k s subset mathbb r n with nonempty interior then every polynomial nonnegative on ks can be approximated in a similar fashion by elements from the corresponding preordering finally we show that the degree of the perturbation in the approximating sequence depends on epsilon as well as the degree and the size of coefficients of the nonnegative polynomial f but not on the specific values of its coefficients

unital quantum channels convex structure and revivals of birkhoff s theorem the set of doubly stochastic quantum channels and its subset of mixtures of unitaries are investigated we provide a detailed analysis of their structure together with computable criteria for the separation of the two sets when applied to o d covariant channels this leads to a complete characterization and reveals a remarkable feature instances of channels which are not in the convex hull of unitaries can become elements of this set by either taking two copies of them or supplementing with a completely depolarizing channel these scenarios imply that a channel whose noise initially resists any environment assisted attempt of correction can become perfectly correctable

lower bounds for polynomials using geometric programming we make use of a result of hurwitz j reine angew math 108 1891 pp 266 268 and reznick math ann 283 1989 pp 431 464 and a consequence of this result due to fidalgo and kovacec math z 269 2011 pp 629 645 to establish in theorem 2 3 a new sufficient condition for a polynomial f in mathbb r x 1 dots x n of even degree to be a sum of squares theorem 2 3 yields as special cases the results of ghasemi and marshall in arch math basel 95 2010 pp 343 353 and consequently also those of fidalgo and kovacec and lasserre arch math basel 89 2007 pp 390 398 we apply theorem 2 3 to obtain a new lower bound f textrm gp for f and we explain how f textrm gp can be computed using geometric programming the lower bound f textrm gp is generally not as good as the lower bound f textrm sos introduced by lasserre siam j optim 11 2001 pp 796 817 and parrilo and sturmfels algorithmic and quantitative real algebraic geometry ams provid

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