common randomness in information theory and cryptography i secret sharing as the first part of a study of problems involving common randomness at distance locations information theoretic models of secret sharing generating a common random key at two terminals without letting an eavesdropper obtain information about this key are considered the concept of key capacity is defined single letter formulas of key capacity are obtained for several models and bounds to key capacity are derived for other models

squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof new measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states one is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state s extension and takes the infimum over such extensions additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related the second one is based on taking classical extensions this scheme is generalized which enables to construct measures of entanglement based on the mixed convex roof of a quantity which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states as one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key the findings are finally related to analogous results in classical key agreement

secret key and private key constructions for simple multiterminal source models we propose an approach for constructing secret and private keys based on the long known slepian wolf code due to wyner for correlated sources connected by a virtual additive noise channel our work is motivated by results of csiszar and narayan which highlight innate connections between secrecy generation by multiple terminals that observe correlated source signals and slepian wolf near lossless data compression explicit procedures for such constructions and their substantiation are provided the performance of low density parity check channel codes in devising a new class of secret keys is examined

quantum key distribution based on private states unconditional security over untrusted channels with zero quantum capacity in this paper we prove unconditional security for a quantum key distribution qkd protocol based on distilling pbits twisted ebits from an arbitrary untrusted state that is claimed to contain distillable key our main result is that we can verify security using only public communication via parameter estimation of the given untrusted state the technique applies even to bound entangled states thus extending qkd to the regime where the available quantum channel has zero quantum capacity we also show how to convert our purification based qkd schemes to prepare measure schemes

converses for secret key agreement and secure computing we consider information theoretic secret key sk agreement and secure function computation by multiple parties observing correlated data with access to an interactive public communication channel our main result is an upper bound on the sk length which is derived using a reduction of binary hypothesis testing to multiparty sk agreement building on this basic result we derive new converses for multiparty sk agreement furthermore we derive converse results for the oblivious transfer problem and the bit commitment problem by relating them to sk agreement finally we derive a necessary condition for the feasibility of secure computation by trusted parties that seek to compute a function of their collective data using an interactive public communication that by itself does not give away the value of the function in many cases we strengthen and improve upon previously known converse bounds our results are single shot and use only the given joint distribution of the correlated observations for the case when the correlated observations consist of independent and identically distributed in time sequences we derive strong versions of previously known converses

uniform random number generation from markov chains non asymptotic and asymptotic analyses in this paper we derive non asymptotic achievability and converse bounds on the random number generation with without side information our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length we also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds which implies that our bounds are asymptotically tight in those regimes we also show the second order rates of those problems and derive single letter forms of the variances characterizing the second order rates furthermore we address the relative entropy rate and the modified mutual information rate for these problems

the dynamics of a fisher kpp nonlocal diffusion model with free boundaries abstract we introduce and study a class of free boundary models with nonlocal diffusion which are natural extensions of the free boundary models in 16 and elsewhere where local diffusion is used to describe the population dispersal with the free boundary representing the spreading front of the species we show that this nonlocal problem has a unique solution defined for all time and then examine its long time dynamical behavior when the growth function is of fisher kpp type we prove that a spreading vanishing dichotomy holds though for the spreading vanishing criteria significant differences arise from the well known local diffusion model in 16

series parallel languages and the bounded width property abstract in order to model concurrency we extend automata theory from the usual languages of words to sets of labelled series parallel posets or sp languages we use the algebraic properties of the set of series parallel posets to investigate the notion of recognizability based on finite index congruences and a notion of regularity based on a new class of branching automata the main results of the paper concern the sp languages having bounded width we show that such languages are recognizable if and only if they are regular in addition the regular bounded width sp languages are exactly those which can be described by series rational expressions that is expressions using the letters the operations of sequential and parallel product and sequential iterations we also give a strictly algebraic characterization of the bounded width property in terms of depth nilpotent series parallel algebras

a semidefinite program for distillable entanglement we show that the maximum fidelity obtained by a positive partial transpose p p t distillation protocol is given by the solution to a certain semidefinite program this gives a number of new lower and upper bounds on p p t distillable entanglement and thus new upper bounds on 2 locally distillable entanglement in the presence of symmetry the semidefinite program simplifies considerably becoming a linear program in the case of isotropic and werner states using these techniques we determine the p p t distillable entanglement of asymmetric werner states and maximally correlated states we conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1 local distillation

useful states and entanglement distillation we derive general upper bounds on the distillable entanglement of a mixed state under one way and two way local operations and classical communication locc in both cases the upper bound is based on a convex decomposition of the state into useful and useless quantum states by useful we mean a state whose distillable entanglement is non negative and equal to its coherent information and thus given by a single letter tractable formula on the other hand useless states are undistillable i e their distillable entanglement is zero we prove that in both settings the distillable entanglement is convex on such decompositions hence an upper bound on the distillable entanglement is obtained from the contributions of the useful states alone being equal to the convex combination of their coherent informations optimizing over all such decompositions of the input state yields our upper bound the useful and useless states are given by degradable and antidegradable states in the one way locc setting and by maximally correlated and positive partial transpose ppt states in the two way locc setting respectively we also illustrate how our method can be extended to quantum channels interpreting our upper bound as a convex roof extension we show that it reduces to a particularly simple non convex optimization problem for the classes of isotropic states and werner states in the one way locc setting this non convex optimization yields an upper bound on the quantum capacity of the qubit depolarizing channel that is strictly tighter than previously known bounds for large values of the depolarizing parameter in the two way locc setting the non convex optimization achieves the ppt relative entropy of entanglement for both isotropic and werner states

distillation of entanglement by projection on permutationally invariant subspaces we consider distillation of entanglement from two qubit states which are mixtures of two pure entangled states and one pure product state which is orthogonal to them we distill entanglement from such states by projecting n copies of the state on a permutationally invariant subspace and then applying one way hashing protocol we find analytical expressions for the rate of the protocol and show that for wide range of parameters this protocol achieves higher rates than previous ones we also generalize this method to higher dimensional systems to get analytical expression for two qubit case we faced a mathematical problem of diagonalizing a family of matrices enjoying some symmetries w r t the symmetric group we have solved this problem in two ways i directly by the use of schur weyl decomposition and young symmetrizers and ii showing that the problem is equivalent to a problem of diagonalizing adjacency matrices in a particular instance of a so called algebraic association scheme

large deviation analysis for quantum security via smoothing of renyi entropy of order 2 it is known that the security evaluation can be done by smoothing of renyi entropy of order 2 in the classical and quantum settings when we apply universal 2 hash functions using the smoothing of renyi entropy of order 2 we derive security bounds for l 1 distinguishability and modified mutual information criterion under the classical and quantum setting and have derived these exponential decreasing rates these results are extended to the case when we apply e almost dual universal 2 hash functions furthermore we apply this analysis to the secret key generation with error correction

efficient quantum polar codes requiring no preshared entanglement we construct an explicit quantum coding scheme which achieves a communication rate not less than the coherent information when used to transmit the quantum information over a noisy quantum channel for pauli and erasure channels we also present efficient encoding and decoding algorithms for this communication scheme based on polar codes essentially linear in the blocklength but which do not require the sender and receiver to share any entanglement before the protocol begins due to the existence of degeneracies in the involved error correcting codes it is indeed possible that the rate of the scheme exceeds the coherent information we provide a simple criterion which indicates such performance finally we discuss how the scheme can be used for secret key distillation as well as private channel coding

trading classical communication quantum communication and entanglement in quantum shannon theory in this paper we give tradeoffs between classical communication quantum communication and entanglement for processing information in the shannon theoretic setting we first prove a unit resource capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation the optimal strategy mixes the three fundamental protocols of teleportation superdense coding and entanglement distribution we then provide an achievable rate region and a matching multiletter converse for the direct static capacity theorem this theorem applies to the scenario where a large number of copies of a noisy bipartite state are available in addition to consumption or generation of the above three noiseless resources our coding strategy involves a protocol that we name the classically assisted state redistribution protocol and the three fundamental protocols we finally provide an achievable rate region and a matching multiletter converse for the direct dynamic capacity theorem this theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources our coding strategy combines the classically enhanced father protocol with the three fundamental unit protocols

exponential decay of correlations implies area law we prove that a finite correlation length i e exponential decay of correlations implies an area law for the entanglement entropy of quantum states defined on a line the entropy bound is exponential in the correlation length of the state thus reproducing as a particular case hastings s proof of an area law for groundstates of 1d gapped hamiltonians

asynchronous mappings and asynchronous cellular automata the aim of this paper is the study of asynchronous automata a special kind of automata which encode the independency relation between actions and which enable their concurrent execution these automata introduced by zielonka rairo inform theor appl 21 99 135 1987 constitute a natural extension of finite automata to the case of asynchronous parallelism their behaviour is described by trace languages subsets of partially commutative monoids the main result concerning this class of automata states that they accept exactly all recognizable trace languages in this paper we give new improved constructions of asynchronous automata in the final part of the paper we present a distributed system of messages with bounded time stamps based on asynchronous automata

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

secret key agreement by public discussion from common information the problem of generating a shared secret key s by two parties knowing dependent random variables x and y respectively but not sharing a secret key initially is considered an enemy who knows the random variable z jointly distributed with x and y according to some probability distribution p sub xyz can also receive all messages exchanged by the two parties over a public channel the goal of a protocol is that the enemy obtains at most a negligible amount of information about s upper bounds on h s as a function of p sub xyz are presented lower bounds on the rate h s n as n to infinity are derived for the case in which x x sub 1 x sub n y y sub 1 y sub n and z z sub 1 z sub n result from n independent executions of a random experiment generating x sub i y sub i and z sub i for i 1 n it is shown that such a secret key agreement is possible for a scenario in which all three parties receive the output of a binary symmetric source over independent binary symmetric channels even when the enemy s channel is superior to the other two channels

general paradigm for distilling classical key from quantum states in this paper we develop a formalism for distilling a classical key from a quantum state in a systematic way expanding on our previous work on a secure key from bound entanglement horodecki et al 2005 more detailed proofs discussion and examples are provided of the main results namely we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory with the only change being that instead of distilling einstein podolsky rosen epr pairs the parties distill private states the form of these general private states are given and we show that there are a number of useful ways of expressing them some of the private states can be approximated by certain states which are bound entangled thus distillable entanglement is not a requirement for a private key we find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel pqc we also find a general class of states which have negative partial transpose are npt but which appear to be bound entangled the relative entropy distance is shown to be an upper bound on the rate of a key this allows us to compute the exact value of a distillable key for a certain class of private states

a resource framework for quantum shannon theory quantum shannon theory is loosely defined as a collection of coding theorems such as classical and quantum source compression noisy channel coding theorems entanglement distillation etc which characterize asymptotic properties of quantum and classical channels and states in this paper we advocate a unified approach to an important class of problems in quantum shannon theory consisting of those that are bipartite unidirectional and memoryless

applications of ldpc codes to the wiretap channel with the advent of quantum key distribution qkd systems perfect i e information theoretic security can now be achieved for distribution of a cryptographic key qkd systems and similar protocols use classical error correcting codes for both error correction for the honest parties to correct errors and privacy amplification to make an eavesdropper fully ignorant from a coding perspective a good model that corresponds to such a setting is the wire tap channel introduced by wyner in 1975 in this correspondence we study fundamental limits and coding methods for wire tap channels we provide an alternative view of the proof for secrecy capacity of wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel we also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel and propose specific practical codes in some cases our designs achieve the secrecy capacity and in others the codes provide security at rates below secrecy capacity for the special case of a noiseless main channel and binary erasure channel we consider encoder and decoder design for codes achieving secrecy on the wiretap channel we show that it is possible to construct linear time decodable secrecy codes based on low density parity check ldpc codes that achieve secrecy

a lossy source coding interpretation of wyner s common information wyner s common information was originally defined for a pair of dependent discrete random variables its significance is largely reflected in and also confined to several existing interpretations in various source coding problems this paper attempts to expand its practical significance by providing a new operational interpretation in the context of the gray wyner network it is established that wyner s common information has a new lossy source coding interpretation specifically it is established that under suitable conditions wyner s common information equals to the smallest common message rate when the total rate is arbitrarily close to the rate distortion function with joint decoding for the gray wyner network a surprising observation is that such equality holds independent of the values of distortion constraints as long as the distortions are within some distortion region the new lossy source coding interpretation provides the first meaningful justification for defining wyner s common information for continuous random variables and the result can also be extended to that of multiple variables examples are given for characterizing the rate distortion region for the gray wyner lossy source coding problem and for identifying conditions under which wyner s common information equals that of the smallest common rate as a by product the explicit expression for the common information between a pair of gaussian random variables is obtained

a father protocol for quantum broadcast channels a new protocol for quantum broadcast channels based on the fully quantum slepian wolf protocol is presented the protocol yields an achievable rate region for entanglement assisted transmission of quantum information through a quantum broadcast channel that can be considered the quantum analogue of marton s region for classical broadcast channels the protocol can be adapted to yield achievable rate regions for unassisted quantum communication and for entanglement assisted classical communication in the case of unassisted transmission the region we obtain has no independent constraint on the sum rate only on the individual transmission rates regularized versions of all three rate regions are provably optimal

geometrical analysis of physically allowed quantum cloning transformations for quantum cryptography the security of quantum key distribution qkd relies on the no cloning theorem which allows no to copy perfectly a quantum system an eavesdropping activity on the quantum channel perturbs the state of the quantum states which results in noise at the receiver the physical layer detection of the eavesdropping activity of the quantum channel requires tomography which is intractable in experiment an adequate and equivalent answer for the problem can be proposed through the logical layer we propose an efficient algorithmical tool to study the eavesdropping activity on the quantum channel and characterize the properties of a quantum cloning based attack for dv and cvqkd protocols the physically allowed quantum cloning transformations on a quantum system can be described in terms of information geometry we propose a computational geometrical method to analyze the cloning activity on the quantum channel and to characterize the noise properties the security analysis studies the dv discrete variable and cv continuous variable qkd schemes through the four state bb84 and six state dvqkd protocols and the two way cvqkd protocol the proposed geometrical method provides a useful tool to analyze the most powerful attacks against quantum cryptography and the effects of the physically allowed quantum cloning transformations

one shot classical data compression with quantum side information and the distillation of common randomness or secret keys the task of compressing classical information in the one shot scenario is studied in the setting where the decompressor additionally has access to some given quantum side information in this hybrid classical quantum version of the famous slepian wolf problem the smooth max entropy is found to govern the number of bits into which classical information can be compressed so that it can be reliably recovered from the compressed version and quantum side information combining this result with known results on privacy amplification then yields tight bounds on the amount of common randomness and secret key that can be recovered in one shot from hybrid classical quantum systems using one way classical communication

computational complexity of continuous variable quantum key distribution the continuous variable quantum key distribution has been considered to have the potential to provide high secret key rate however in present experimental demonstrations the secret key can be distilled only under very small loss rates here by calculating explicitly the computational complexity with the channel transmission we show that under high loss rate it is hard to distill the secret key in present continuous variable scheme and one of its advantages the potential of providing high secret key rate may therefore be limited

note on a two species competition diffusion model with two free boundaries abstract in guo and wu 2015 and wu 2015 the authors studied a two species competition diffusion model with two free boundaries these two free boundaries describing the spreading fronts of two competing species respectively may intersect each other as time evolves the existence uniqueness and long time behavior of global solution have been established in this note we discuss the conditions for spreading and vanishing and more accurate limits of u v as t when spreading occurs some new results and simpler proofs will be provided

distillation of secret key and entanglement from quantum states we study and solve the problem of distilling a secret key from quantum states representing correlation between two parties alice and bob and an eavesdropper eve via oneway public discussion we

perfect omniscience perfect secrecy and steiner tree packing we consider perfect secret key generation for a pairwise independent network model in which every pair of terminals share a random binary string with the strings shared by distinct terminal pairs being mutually independent the terminals are then allowed to communicate interactively over a public noiseless channel of unlimited capacity all the terminals as well as an eavesdropper observe this communication the objective is to generate a perfect secret key shared by a given set of terminals at the largest rate possible and concealed from the eavesdropper first we show how the notion of perfect omniscience plays a central role in characterizing perfect secret key capacity second a multigraph representation of the underlying secrecy model leads us to an efficient algorithm for perfect secret key generation based on maximal steiner tree packing this algorithm attains capacity when all the terminals seek to share a key and in general attains at least half the capacity third when a single helper terminal assists the remaining user terminals in generating a perfect secret key we give necessary and sufficient conditions for the optimality of the algorithm also a weak helper is shown to be sufficient for optimality

low dimensional quite noisy bound entanglement with a cryptographic key we provide a class of bound entangled states that have a positive distillable secure key rate the smallest state of this kind is 4 4 our class is a generalization of the class presented in horodecki et al 2008 ieee trans inf theory 54 2621 5 it is much wider containing in particular states from the boundary of ppt entangled states all of the states in the previous class were of this kind and also states inside the set of ppt entangled states even approaching the separable states this generalization comes at a price for the wider class a positive key rate requires in general apart from the one way devetak winter protocol used in the previous case also the recurrence preprocessing and thus is effectively a two way protocol we also analyze the amount of noise that can be admixtured to the states of our class without losing the key distillability property which may be crucial for experimental realization the wider class contains key distillable states with higher entropy up to 3 524 as opposed to 2 564 for the previous class

non asymptotic entanglement distillation entanglement distillation an essential quantum information processing task refers to the conversion from multiple copies of noisy entangled states to a smaller number of highly entangled states in this paper we study the non asymptotic fundamental limits for entanglement distillation we investigate the optimal tradeoff between the distillation rate the number of prepared states and the error tolerance first we derive the one shot distillable entanglement under completely positive partial transpose preserving operations as a semidefinite program and demonstrate an exact characterization via the quantum hypothesis testing relative entropy second we establish efficiently computable second order estimations of the distillation rate for general quantum states in particular we provide explicit as well as approximate evaluations for various quantum states of practical interest including pure states mixture of bell states maximally correlated states and isotropic states

a few steps more towards npt bound entanglement in this paper existence of bound entangled states with nonpositive partial transpose npt is considered as one knows existence of such states would in particular imply nonadditivity of distillable entanglement moreover it would rule out a simple mathematical description of the set of distillable states the particular state known to be 1 copy nondistillable and supposed to be bound entangled is considered the problem of its two copy distillability which boils down to show that maximal overlap of some projector q with schmidt rank two states does not exceed 1 2 called the half property is studied first it is shown that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies then the problem in attacked twofold way a the half property is proved for some wide classes of schmidt rank two states b the overlap for all schmidt rank two states is bounded from above by c 3 4 moreover the problem has been translated into the following matrix analysis problem bound the sum of the squares of the two largest singular values of matrix a i i b with a b traceless 4 4 matrices and tr afa tr bfb 1 4

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

operational resource theory of coherence we establish an operational theory of coherence or of superposition in quantum systems by focusing on the optimal rate of performance of certain tasks namely we introduce the two basic concepts coherence distillation and coherence cost in the processing quantum states under so called incoherent operations baumgratz cramer and plenio phys rev lett 113 140401 2014 we then show that in the asymptotic limit of many copies of a state both are given by simple single letter formulas the distillable coherence is given by the relative entropy of coherence in other words we give the relative entropy of coherence its operational interpretation and the coherence cost by the coherence of formation which is an optimization over convex decompositions of the state an immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state but no coherence could be distilled from it further we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states

the commitment capacity of the gaussian channel is infinite we prove that the commitment capacity of the power constrained gaussian channel i e the optimal rate at which this channel can be used for implementing commitment schemes is infinite

asynchronous cellular automata for promsets abstract this paper extends to pomsets without auto concurrency the fundamental notion of asynchronous cellular automata aca which was originally introduced for traces by zielonka we generalize to pomsets the notion of asynchronous mapping introduced by cori metivier and zielonka and we show how to construct a deterministic aca from an asynchronous mapping then we investigate the relation between the expressiveness of monadic second order logic nondeterministic acas and deterministic acas we can generalize buchi s theorem for finite words to a class of pomsets without auto concurrency which satisfy a natural axiom this axiom ensures that an asynchronous cellular automaton works on the pomset as a concurrent read and exclusive owner write machine more precisely in this class nondeterministic acas deterministic acas and monadic second order logic have the same expressive power then we consider a class where deterministic acas are strictly weaker than nondeterministic ones but in this class nondeterministic acas still capture monadic second order logic finally it is shown that even this equivalence does not hold in the class of all pomsets since there the class of recognizable pomset languages is not closed under complementation

tight exponential analysis of universally composable privacy amplification and its applications motivated by the desirability of universal composability we analyze in terms of l1 distinguishability the task of secret key generation from a joint random variable under this secrecy criterion using the renyi entropy of order 1 s for s 0 1 we derive a new upper bound of eve s distinguishability under the application of the universal2 hash functions it is also shown that this bound gives the tight exponential rate of decrease in the case of independent and identical distributions the result is applied to the wiretap channel model and to secret key generation distillation by public discussion

entanglement transmission and generation under channel uncertainty universal quantum channel coding we determine the optimal rates of universal quantum codes for entanglement transmission and generation under channel uncertainty in the simplest scenario the sender and receiver are provided merely with the information that the channel they use belongs to a given set of channels so that they are forced to use quantum codes that are reliable for the whole set of channels this is precisely the quantum analog of the compound channel coding problem we determine the entanglement transmission and entanglement generating capacities of compound quantum channels and show that they are equal moreover we investigate two variants of that basic scenario namely the cases of informed decoder or informed encoder and derive corresponding capacity results

on the definition and the properties of the principal eigenvalue of some nonlocal operators in this article we study some spectral properties of the linear operator mathcal l omega a defined on the space c bar omega by mathcal l omega varphi a varphi int omega k x y varphi y dy a x varphi x where omega subset mathbb r n is a domain possibly unbounded a is a continuous bounded function and k is a continuous non negative kernel satisfying an integrability condition we focus our analysis on the properties of the generalised principal eigenvalue lambda p mathcal l omega a defined by lambda p mathcal l omega a sup lambda in mathbb r exists varphi in c bar omega varphi 0 textit such that mathcal l omega varphi a varphi lambda varphi le 0 text in omega we establish some new properties of this generalised principal eigenvalue lambda p namely we prove the equivalence of different definitions of the principal eigenvalue we also study the behaviour of lambda p mathcal l omega a with respect to some scaling of k for kernels k of the type k x y j x y with j a compactly supported probability density we also establish some asymptotic properties of lambda p left mathcal l sigma m omega frac 1 sigma m a right where mathcal l sigma m omega is defined by displaystyle mathcal l sigma m omega varphi frac 1 sigma 2 n int omega j left frac x y sigma right varphi y dy in particular we prove that lim sigma to 0 lambda p left mathcal l sigma 2 omega frac 1 sigma 2 a right lambda 1 left frac d 2 j 2n delta a right where d 2 j int mathbb r n j z z 2 dz and lambda 1 denotes the dirichlet principal eigenvalue of the elliptic operator in addition we obtain some convergence results for the corresponding eigenfunction varphi p sigma

the conditional common information in classical and quantum secret key distillation in this paper we consider two extensions of the gacs korner common information to three variables the conditional common information cci and the coarse grained conditional common information ccci both quantities are shown to be useful technical tools in the study of classical and quantum resource transformations in particular the ccci is shown to have an operational interpretation as the optimal rate of secret key extraction from an eavesdropped classical source p xyz when alice x and bob y are unable to communicate but share common randomness with the eavesdropper eve z moving to the quantum setting we consider two different ways of generating a tripartite quantum state from classical correlations p xyz 1 coherent encodings sum xyz sqrt p xyz xyz rangle and 2 incoherent encodings sum xyz p xyz xyz rangle langle xyz we study how well can alice and bob extract secret key from these quantum sources using quantum operations compared with the extraction of key from the underlying classical sources p xyz using classical operations while the power of quantum mechanics increases alice and bob s ability to generate shared randomness it also equips eve with a greater arsenal of eavesdropping attacks therefore it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one we first demonstrate that the classical key rate of p xyz is equivalent to the quantum key rate for an incoherent quantum encoding of the distribution for coherent encodings we next show that the classical and quantum rates are generally incomparable and in fact their difference can be arbitrarily large in either direction finally we introduce a zoo of entangled tripartite states all characterized by the conditional common information of their encoded probability distributions remarkably for these states almost all entanglement measures such as alice and bob s entanglement cost squashed entanglement and relative entropy of entanglement can be sharply bounded or even exactly expressed in terms of the conditional common information in the latter case we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated

a partial trace semantics for petri nets abstract we introduce the notion of partial trace a partial trace is an equivalence class of some labelled partial orders over a dependence alphabet sd partial traces arise in a natural way by the synchronization of semi traces they form a monoid which is shown to be free partially commutative we prove an embedding theorem which shows that any partial trace has a canonical representation as a tuple of words we then apply this concept to petri nets we define the behavior of a p t system in terms of partial traces we consider local morphisms between petri nets and we show how this relates to the partial trace behavior in particular we obtain the desired result that the partial trace behavior of a synchronized system is the synchronization of its local behavior

strong secrecy from channel resolvability we analyze physical layer security based on the premise that the coding mechanism for secrecy over noisy channels is tied to the notion of channel resolvability instead of considering capacity based constructions which associate to each message a subcode that operates just below the capacity of the eavesdropper s channel we consider channel resolvability based constructions which associate to each message a subcode that operates just above the resolvability of the eavesdropper s channel building upon the work of csiszar and hayashi we provide further evidence that channel resolvability is a powerful and versatile coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels specifically we show that at least for symmetric wiretap channels random capacity based constructions fail to achieve the strong secrecy capacity while channel resolvability based constructions achieve it we then leverage channel resolvability to establish the secrecy capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics finally we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information csi mixed channels and compound channels with receiver csi as well as the secret key capacity of source models for secret key agreement by tying secrecy to channel resolvability we obtain achievable rates for strong secrecy metrics with simple proofs

quantum feedback channels in shannon information theory the capacity of a memoryless communication channel cannot be increased by the use of feedback in quantum information theory the no cloning theorem means that noiseless copying and feedback of quantum information cannot be achieved in this correspondence quantum feedback is defined as the unlimited use of a noiseless quantum channel from receiver to sender given such quantum feedback it is shown to provide no increase in the entanglement assisted capacities of a memoryless quantum channel in direct analogy to the classical case it is also shown that in various cases of nonassisted capacities feedback may increase the capacity of memoryless quantum channels

secure multiplex coding attaining channel capacity in wiretap channels it is known that a message can be transmitted safely against any wiretapper via a noisy channel without a secret key if the coding rate is less than the so called secrecy capacity cs which is usually smaller than the channel capacity c in order to remove the loss c cs we propose a multiplex coding scheme with plural independent messages in this paper it is shown that the proposed multiplex coding scheme can attain the channel capacity as the total rate of the plural messages and the perfect secrecy for each massage several bounds of achievable multiplex coding rate region are derived for general wiretap channels in the sense of information spectral methods by extending hayashi s proof in which the coding of the channel resolvability is applied to wiretap channels furthermore the exact region for deterministic coding is determined for stationary memoryless full rank wiretap channels

free boundary problems with nonlocal and local diffusions i global solution abstract we study a class of free boundary problems of ecological models with nonlocal and local diffusions which are natural extensions of free boundary problems of reaction diffusion systems in there local diffusions are used to describe the population dispersal with the free boundary representing the spreading front of the species we prove that such kind of nonlocal and local diffusion problems has a unique global solution and then show that a spreading vanishing dichotomy holds moreover criteria of spreading and vanishing and long time behavior of solution when spreading happens are established for the classical lotka volterra competition and prey predator models compared with free boundary problems of reaction diffusion systems with local diffusions 10 24 25 with nonlocal diffusions 9 as well as with nonlocal and local diffusions 14 one equation is cauchy problem and the other one is free boundary problem the present paper involves some new difficulties which should be overcome by use of new techniques this is part i of a two part series where we prove the existence uniqueness regularity and estimates of global solution the spreading vanishing dichotomy criteria of spreading and vanishing and long time behavior of solution when spreading happens will be studied in the separate part ii 17

squashed entanglement an additive entanglement measure in this paper we present a new entanglement monotone for bipartite quantum states its definition is inspired by the so called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions we derive certain properties of the new measure which we call squashed entanglement it is a lower bound on entanglement of formation and an upper bound on distillable entanglement furthermore it is convex additive on tensor products and superadditive in general continuity in the state is the only property of our entanglement measure which we cannot provide a proof for we present some evidence however that our quantity has this property the strongest indication being a conjectured fannes type inequality for the conditional von neumann entropy this inequality is proved in the classical case

coding theorem and strong converse for quantum channels we present a new proof of holevo s 1973 1977 coding theorem for transmitting classical information through quantum channels and its strong converse the technique is largely inspired by wolfwitz s 1964 combinatorial approach using types of sequences as a byproduct of our approach which is independent of previous ones both in the coding theorem and the converse we can give a new proof of holevo s information bound

the equational theory of pomsets abstract pomsets have been introduced as a model of concurrency since a pomset is a string in which the total order has been relaxed to be a partial order in this paper we view them as a generalization of strings and investigate their algebraic properties in particular we investigate the axiomatic properties of pomsets sets of pomsets and ideals of pomsets under such operations as concatenation parallel composition union and their associated closure operations we find that the equational theory of sets pomsets under concatenation parallel composition and union is finitely axiomatizable whereas the theory of languages under the analogous operations is not a similar result is obtained for ideals of pomsets which incorporate the notion of subsumption which is also known as augmentation finally we show that the addition of any closure operation parallel or serial leads to nonfinite axiomatizability of the resulting equational theory

on quantum fidelities and channel capacities for the discrete memoryless quantum channel we show the equivalence of two different notions of quantum channel capacity that which uses the entanglement fidelity as its criterion for success in transmission and that which uses the minimum fidelity of pure states in a subspace of the input hilbert space as its criterion as a corollary any source with entropy rate less than the capacity may be transmitted with high entanglement fidelity we also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel this enables us to simplify a known upper bound for the channel capacity it also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity

distilling entanglement from arbitrary resources we obtain the general formula for the optimal rate at which singlets can be distilled from any given noisy and arbitrarily correlated entanglement resource by means of local operations and classical communication locc our formula obtained by employing the quantum information spectrum method reduces to that derived by devetak and winter proc r soc london ser a 461 207 2005 in the special case of an independent and identically distributed resource the proofs rely on a one shot version of the so called hashing bound which in turn provides bounds on the one shot distillable entanglement under general locc

low dimensional bound entanglement with one way distillable cryptographic key in this paper we provide a class of bound entangled states that have positive distillable secure key rate the smallest state of this kind is 4 otimes 4 which shows that peculiar security contained in bound entangled states does not need high dimensional systems we show that for these states a positive key rate can be obtained by one way devetak winter dw protocol subsequently the volume of bound entangled key distillable states for m otimes n hilbert space with m n 4 is shown to be nonzero we provide a scheme of verification of cryptographic quality of experimentally prepared state in terms of local observables proposed set of seven collective settings is proven to be optimal in number of settings

exact random coding secrecy exponents for the wiretap channel we analyze the exact exponential decay rate of the expected amount of information leaked to the wiretapper in wyner s wiretap channel setting using wiretap channel codes constructed from both i i d and constant composition random codes our analysis for those sampled from i i d random coding ensemble shows that the previously known achievable secrecy exponent using this ensemble is indeed the exact exponent for an average code in the ensemble furthermore our analysis on wiretap channel codes constructed from the ensemble of constant composition random codes leads to an exponent which in addition to being the exact exponent for an average code is larger than the achievable secrecy exponent that has been established so far in the literature for this ensemble which in turn was known to be smaller than that achievable by wiretap channel codes sampled from i i d random coding ensemble we show examples where the exact secrecy exponent for the wiretap channel codes constructed from random constant composition codes is larger than that of those constructed from i i d random codes and examples where the exact secrecy exponent for the wiretap channel codes constructed from i i d random codes is larger than that of those constructed from constant composition random codes we hence conclude that unlike the error correction problem there is no general ordering between the two random coding ensembles in terms of their secrecy exponent

cryptographic distinguishability measures for quantum mechanical states this paper mostly expository in nature surveys four measures of distinguishability for quantum mechanical states this is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions namely the probability of an identification error the kolmogorov distance the bhattacharyya coefficient and the shannon 1948 distinguishability as defined through mutual information these measures have a long history of use in statistical pattern recognition and classical cryptography we obtain several inequalities that relate the quantum distinguishability measures to each other one of which may be crucial for proving the security of quantum cryptographic key distribution in another vein these measures and their connecting inequalities are used to define a single notion of cryptographic exponential indistinguishability for two families of quantum states this is a tool that may prove useful in the analysis of various quantum cryptographic protocols

the smooth entropy formalism for von neumann algebras we discuss information theoretic concepts on infinite dimensional quantum systems in particular we lift the smooth entropy formalism as introduced by renner and collaborators for finite dimensional systems to von neumann algebras for the smooth conditional min and max entropy we recover similar characterizing properties and information theoretic operational interpretations as in the finite dimensional case we generalize the entropic uncertainty relation with quantum side information of tomamichel and renner and discuss applications to quantum cryptography in particular we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von neumann algebra

coding theorems for a 2 2 threshold scheme with detectability of impersonation attacks coding theorems on a 2 2 threshold scheme with an opponent are discussed in an asymptotic setup where the opponent tries to impersonate one of the two participants a situation is considered where n secrets s n from a memoryless source is blockwisely encoded to two shares and the two shares are decoded to s n with permitting negligible decoding error we introduce correlation level of the two shares and characterize the minimum attainable rates of the shares and a uniform random number for realizing a 2 2 threshold scheme that is secure against the impersonation attack by the opponent it is shown that if the correlation level between the two shares equals to ell geq 0 the minimum attainable rates coincide with h s ell where h s denotes the entropy of the source and the maximum attainable exponent of the success probability of the impersonation attack equals to ell it is also shown that a simple scheme using an ordinary 2 2 threshold scheme attains all the bounds as well

extrinsic and intrinsic data in quantum measurements asymptotic convex decomposition of positive operator valued measures we study the problem of separating the data produced by a given quantum measurement on states from a memoryless source which is unknown except for its average state described by a positive operator valued measure povm into a meaningful intrinsic and a not meaningful extrinsic part we are able to give an asymptotically tight separation of this form with the intrinsic data quantified by the holevo mutual information of a certain state ensemble associated to the povm and the source in a model that can be viewed as the asymptotic version of the convex decomposition of povms into extremal ones this result is applied to a similar separation therorem for quantum instruments and quantum operations in their kraus form finally we comment on links to related subjects we stress the difference between data and information in particular by pointing out that information typically is strictly less than data derive the holevo bound from our main result and look at its classical case we show that this includes the solution to the problem of extrinsic intrinsic data separation with a known source then compare with the well known notion of sufficient statistics the result on decomposition of quantum operations is used to exhibit a new aspect of the concept of entropy exchange of an open dynamics an appendix collects several estimates for mixed state fidelity and trace norm distance that seem to be new in particular a construction of canonical purification of mixed states that turns out to be valuable to analyze their fidelity

strong converse for identification via quantum channels we present a simple proof of the strong converse for identification via discrete memoryless quantum channels based on a novel covering lemma the new method is a generalization to quantum communication channels of ahlswede s 1979 1992 approach to classical channels it involves a development of explicit large deviation estimates to the case of random variables taking values in self adjoint operators on a hilbert space this theory is presented separately in an appendix and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems

quantum broadcast channels we consider quantum channels with one sender and two receivers used in several different ways for the simultaneous transmission of independent messages we begin by extending the technique of superposition coding to quantum channels with a classical input to give a general achievable region we also give outer bounds to the capacity regions for various special cases from the classical literature and prove that superposition coding is optimal for a class of channels we then consider extensions of superposition coding for channels with a quantum input where some of the messages transmitted are quantum instead of classical in the sense that the parties establish bipartite or tripartite ghz entanglement we conclude by using state merging to give achievable rates for establishing bipartite entanglement between different pair of parties with the assistance of free classical communication

systems with nonlocal vs local diffusions and free boundaries abstract we study a class of free boundary problems of ecological models with nonlocal and local diffusions which are natural extensions of free boundary problems of reaction diffusion systems in there local diffusions are used to describe the population dispersal with the free boundary representing the spreading front of the species we first prove the existence uniqueness and regularity of global solution for the classical competition prey predator and mutualist models we show that a spreading vanishing dichotomy holds and establish the criteria of spreading and vanishing and long time behavior of the solution

universal quantum state merging we determine the optimal entanglement rate of quantum state merging when assuming that the state is unknown except for its membership in a certain set of states we find that merging is possible at the lowest rate allowed by the individual states additionally we establish a lower bound for the classical cost of state merging under state uncertainty to this end we give an elementary proof for the cost in case of a perfectly known state which makes no use of the resource framework as applications of our main result we determine the capacity for one way entanglement distillation if the source is not perfectly known moreover we give another achievability proof for the entanglement generation capacity over compound quantum channels