duality between smooth min and max entropies in classical and quantum information theory operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures called smooth min entropy and smooth max entropy respectively while both entropies are equal to the von neumann entropy in certain special cases e g asymptotically for many independent repetitions of the given data their values can differ arbitrarily in the general case in this paper a recently discovered duality relation between nonsmooth min and max entropies is extended to the smooth case more precisely it is shown that the smooth min entropy of a system a conditioned on a system b equals the negative of the smooth max entropy of a conditioned on a purifying system c this result immediately implies that certain operational quantities such as the amount of compression and the amount of randomness that can be extracted from given data are related we explain how such relations have applications in cryptographic security proofs

the quantum capacity of channels with arbitrarily correlated noise we study optimal rates for quantum communication over a single use of a channel which itself can correspond to a finite number of uses of a channel with arbitrarily correlated noise the corresponding capacity is often referred to as the one shot quantum capacity in this paper we prove bounds on the one shot quantum capacity of an arbitrary channel this allows us to compute the quantum capacity of a channel with arbitrarily correlated noise in the limit of asymptotically many uses of the channel in the memoryless case we explicitly show that our results reduce to known expressions for the quantum capacity

recursive markov chains stochastic grammars and monotone systems of nonlinear equations we define recursive markov chains rmcs a class of finitely presented denumerable markov chains and we study algorithms for their analysis informally an rmc consists of a collection of finite state markov chains with the ability to invoke each other in a potentially recursive manner rmcs offer a natural abstract model for probabilistic programs with procedures they generalize in a precise sense a number of well studied stochastic models including stochastic context free grammars scfg and multi type branching processes mt bp we focus on algorithms for reachability and termination analysis for rmcs what is the probability that an rmc started from a given state reaches another target state or that it terminatesq these probabilities are in general irrational and they arise as least fixed point solutions to certain monotone systems of nonlinear equations associated with rmcs we address both the qualitative problem of determining whether the probabilities are 0 1 or in between and the quantitative problems of comparing the probabilities with a given bound or approximating them to desired precision we show that all these problems can be solved in pspace using a decision procedure for the existential theory of reals we provide a more practical algorithm based on a decomposed version of multi variate newton s method and prove that it always converges monotonically to the desired probabilities we show this method applies more generally to any monotone polynomial system we obtain polynomial time algorithms for various special subclasses of rmcs among these for scfgs and mt bps equivalently for 1 exit rmcs the qualitative problem can be solved in p time for linearly recursive rmcs the probabilities are rational and can be computed exactly in p time we show that our pspace upper bounds cannot be substantially improved without a breakthrough on long standing open problems the square root sum problem and an arithmetic circuit decision problem that captures p time on the unit cost rational arithmetic ram model we show that these problems reduce to the qualitative problem and to the approximation problem to within any nontrivial error for termination probabilities of general rmcs and to the quantitative decision problem for termination extinction of scfgs mt bps

on the power of quantum memory we address the question whether quantum memory is more powerful than classical memory in particular we consider a setting where information about a random n bit string x is stored in s classical or quantum bits for s n i e the stored information is bound to be only partial later a randomly chosen predicate f about x has to be guessed using only the stored information the maximum probability of correctly guessing f x is then compared for the cases where the storage device is classical or quantum mechanical respectively we show that despite the fact that the measurement of quantum bits can depend arbitrarily on the predicate f the quantum advantage is negligible already for small values of the difference n s our setting generalizes the setting of ambainis et al who considered the problem of guessing an arbitrary bit i e one of the n bits of x an implication for cryptography is that privacy amplification by universal hashing remains essentially equally secure when the adversary s memory is allowed to be quantum rather than only classical since privacy amplification is a main ingredient of many quantum key distribution qkd protocols our result can be used to prove the security of qkd in a generic way

security analysis of varepsilon almost dual universal 2 hash functions smoothing of min entropy versus smoothing of renyi entropy of order 2 recently varepsilon almost dual universal2 hash functions have been proposed as a new and wider class of hash functions using this class of hash functions several efficient hash functions were proposed this paper evaluates the security performance when we apply this kind of hash functions we evaluate the security in several kinds of setting based on the l 1 distinguishability criterion and the modified mutual information criterion the obtained evaluation is based on smoothing of renyi entropy of order 2 and or min entropy we clarify the difference between these two methods

a hierarchy of information quantities for finite block length analysis of quantum tasks we consider two fundamental tasks in quantum information theory data compression with quantum side information as well as randomness extraction against quantum side information we characterize these tasks for general sources using so called one shot entropies these characterizations in contrast to earlier results enable us to derive tight second order asymptotics for these tasks in the i i d limit more generally our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain the one shot entropies most accurately describe an operational quantity yet they tend to be difficult to calculate for large systems we show that they asymptotically agree up to logarithmic terms with entropies related to the quantum and classical information spectrum which are easier to calculate in the i i d limit our technique also naturally yields bounds on operational quantities for finite block lengths

leftover hashing against quantum side information the leftover hash lemma states that the output of a two universal hash function applied to an input with sufficiently high entropy is almost uniformly random in its standard formulation the lemma refers to a notion of randomness that is usually implicitly defined with respect to classical side information here a strictly more general version of the leftover hash lemma that is valid even if side information is represented by the state of a quantum system is shown our result applies to almost two universal families of hash functions the generalized leftover hash lemma has applications in cryptography e g for key agreement in the presence of an adversary who is not restricted to classical information processing

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

expressing infinite matrices as products of involutions abstract we take into consideration ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 we show that for any field k every element of this group can be expressed as a product of at most five involutions moreover we prove that if k is a field of characteristic different from 2 then four involutions suffice

yang mills measure and the master field on the sphere we study the yang mills measure on the sphere with unitary structure group in the limit where the structure group has high dimension we show that the traces of loop holonomies converge in probability to a deterministic limit which is known as the master field on the sphere the values of the master field on simple loops are expressed in terms of the solution of a variational problem we show that given its values on simple loops the master field is characterized on all loops of finite length by a system of differential equations known as the makeenko migdal equations we obtain a number of further properties of the master field on specializing to families of simple loops our results identify the high dimensional limit in non commutative distribution of the brownian bridge in the group of unitary matrices starting and ending at the identity

relating different quantum generalizations of the conditional renyi entropy recently a new quantum generalization of the renyi divergence and the corresponding conditional renyi entropies was proposed here we report on a surprising relation between conditional renyi entropies based on this new generalization and conditional renyi entropies based on the quantum relative renyi entropy that was used in previous literature our result generalizes the well known duality relation h a b h a c 0 of the conditional von neumann entropy for tripartite pure states to renyi entropies of two different kinds as a direct application we prove a collection of inequalities that relate different conditional renyi entropies and derive a new entropic uncertainty relation

a random matrix model from two dimensional yang mills theory we use fat graphs to give a unified treatment of various asymptotic freeness results in particular a random matrix model from two dimensional yang mills theory on the plane is presented

computing with very weak random sources we give an efficient algorithm to extract randomness from a very weak random source using a small additional number t of truly random bits our work extends that of nisan and zuckerman j comput system sci 52 1996 pp 43 52 in that t remains small even if the entropy rate is well below constant a key application of this is in running randomized algorithms using such a very weak source of randomness for any fixed gamma 0 we show how to simulate rp algorithms in time n o log n using the output of a ds with min entropy r gamma such a weak random source is asked once for r bits it outputs an r bit string according to any probability distribution that places probability at most 2 r gamma on each string if gamma 1 2 our simulation also works for bpp for gamma 1 1 k 1 our simulation takes time n o logk n log k is the logarithm iterated k times we also give a polynomial time bpp simulation using chor goldreich sources of min entropy r omega 1 which is optimal we present applications to time space tradeoffs expander constructions and to the hardness of approximation of independent interest is our randomness efficient leftover hash lemma a key tool for extracting randomness from weak random sources

generalized relative entropies and the capacity of classical quantum channels we provide lower and upper bounds on the information transmission capacity of one single use of a classical quantum channel the lower bound is expressed in terms of the hoeffding capacity which we define similarly to the holevo capacity but replacing the relative entropy with the hoeffding distance similarly our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max relative entropy in the definition of the divergence radius of a channel

quantum to classical randomness extractors the goal of randomness extraction is to distill almost perfect randomness from a weak source of randomness when the source yields a classical string x many extractor constructions are known yet when considering a physical randomness source x is itself ultimately the result of a measurement on an underlying quantum system when characterizing the power of a source to supply randomness it is hence natural to ask how much classical randomness we can extract from a quantum system to tackle this question we here take on the study of quantum to classical randomness extractors qc extractors we provide constructions of qc extractors based on measurements in a full set of mutually unbiased bases mubs and certain single qubit measurements the latter are particularly appealing since they are not only easy to implement but also appear throughout quantum cryptography we proceed to prove an upper bound on the maximum amount of randomness that we could hope to extract from any quantum state some of our qc extractors almost match this bound we show two applications of our results first we show that any qc extractor gives rise to entropic uncertainty relations with respect to quantum side information such relations were previously only known for two measurements in particular we obtain strong relations in terms of the von neumann shannon entropy as well as the min entropy for measurements in almost unitary two designs a full set of mubs and single qubit measurements in three mubs each second we resolve the central open question in the noisy storage model by linking security to the quantum capacity of the adversary s storage device more precisely we show that any two party cryptographic primitives can be implemented securely as long as the adversary s storage device has sufficiently low quantum capacity our protocol does not need any quantum storage to implement and is technologically feasible using present day technology

extracting kolmogorov complexity with applications to dimension zero one laws we apply results on extracting randomness from independent sources to extract kolmogorov complexity for any a e 0 given a string x with k x a x we show how to use a constant number of advice bits to efficiently compute another string y y w x with k y 1 e y this result holds for both unbounded and space bounded kolmogorov complexity we use the extraction procedure for space bounded complexity to establish zero one laws for the strong dimensions of complexity classes within espace the unbounded extraction procedure yields a zero one law for the constructive strong dimensions of turing degrees

unbalanced expanders and randomness extractors from parvaresh vardy codes we give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree which is polylogarithmic in the number of vertices both the degree and the number of right hand vertices are polynomially close to optimal whereas the previous constructions of ta shma et al 2007 required at least one of these to be quasipolynomial in the optimal our expanders have a short and self contained description and analysis based on the ideas underlying the recent list decodable error correcting codes of parvaresh and vardy 2005 our expanders can be interpreted as near optimal randomness condensers that reduce the task of extracting randomness from sources of arbitrary min entropy rate to extracting randomness from sources of min entropy rate arbitrarily close to 1 which is a much easier task using this connection we obtain a new self contained construction of randomness extractors that is optimal up to constant factors while being much simpler than the previous construction of lu et al 2003 and improving upon it when the error parameter is small e g 1 poly n

quantum entropic security and approximate quantum encryption an encryption scheme is said to be entropically secure if an adversary whose min entropy on the message is upper bounded cannot guess any function of the message similarly an encryption scheme is entropically indistinguishable if the encrypted version of a message whose min entropy is high enough is statistically indistinguishable from a fixed distribution we present full generalizations of these two concepts to the encryption of quantum states in which the quantum conditional min entropy as introduced by renner is used to bound the adversary s prior information on the message a proof of the equivalence between quantum entropic security and quantum entropic indistinguishability is presented we also provide proofs of security for two different ciphers in this model and a proof for a lower bound on the key length required by any such cipher these ciphers generalize existing schemes for approximate quantum encryption to the entropic security model

strong converse theorems using renyi entropies we use a renyi entropy method to prove strong converse theorems for certain information theoretic tasks which involve local operations and quantum or classical communication between two parties these include state redistribution coherent state merging quantum state splitting measurement compression with quantum side information randomness extraction against quantum side information and data compression with quantum side information the method we employ in proving these results extends ideas developed by sharma preprint arxiv 1404 5940 quant ph 2014 which he used to give a new proof of the strong converse theorem for state merging for state redistribution we prove the strong converse property for the boundary of the entire achievable rate region in the e q plane where e and q denote the entanglement cost and quantum communication cost respectively in the case of measurement compression with quantum side information we prove a strong converse theorem for the classical communication cost which is a new result extending the previously known weak converse for the remaining tasks we provide new proofs for strong converse theorems previously established using smooth entropies for each task we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a renyi generalization of the optimal rate hence we identify candidates for the strong converse exponents for each task discussed in this paper to prove our results we establish various new entropic inequalities which might be of independent interest these involve conditional entropies and mutual information derived from the sandwiched renyi divergence in particular we obtain novel bounds relating these quantities as well as the renyi conditional mutual information to the fidelity of two quantum states

better short seed quantum proof extractors we construct a strong extractor against quantum storage that works for every min entropy k has logarithmic seed length and outputs w k bits provided that the quantum adversary has at most bk qubits of memory for any b 12 the construction works by first condensing the source with minimal entropy loss and then applying an extractor that works well against quantum adversaries when the source is close to uniform we also obtain an improved construction of a strong quantum proof extractor in the high min entropy regime specifically we construct an extractor that uses a logarithmic seed length and extracts w n bits from any source over 0 1 n provided that the min entropy of the source conditioned on the quantum adversary s state is at least 1 b n for any b 12

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

simple extractors for all min entropies and a new pseudorandom generator a randomness extractor is an algorithm that given a sample from a distribution with sufficiently high min entropy and a short random seed produces an output that is statistically indistinguishable from uniform min entropy is a measure of the amount of randomness in a distribution we present a simple self contained extractor construction that produces good extractors for all min entropies our construction is algebraic and builds on a new polynomial based approach introduced by ta shma et al 2001b using our improvements we obtain for example an extractor with output length m k log n o 1 and seed length 1 p log n for an arbitrary 0 1 where n is the input length and k is the min entropy of the input distribution a pseudorandom generator is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform our technique also gives a new way to construct pseudorandom generators from functions that require large circuits our pseudorandom generator construction is not based on the nisan wigderson generator nisan and wigderson 1994 and turns worst case hardness directly into pseudorandomness the parameters of our generator match those in impagliazzo and wigderson 1997 and sudan et al 2001 and in particular are strong enough to obtain a new proof that p bpp if e requires exponential size circuits our construction also gives the following improvements over previous work we construct an optimal hitting set generator that stretches o log n random bits into s 1 pseudorandom bits when given a function on log n bits that requires circuits of size s this yields a quantitatively optimal hardness versus randomness tradeoff for both rp and bpp and solves an open problem raised in impagliazzo et al 1999 we give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits this technique also give a quantitatively optimal hardness versus randomness tradeoff for am and the first hardness amplification result for nondeterministic circuits

sampling of min entropy relative to quantum knowledge let x1 xn be a sequence of n classical random variables and consider a sample xs1 xsr of r n positions selected at random then except with exponentially in r small probability the min entropy hmin xs1 xsr of the sample is not smaller than roughly a fraction r n of the overall entropy hmin x1 xn which is optimal here we show that this statement originally proved in s vadhan lncs 2729 springer 2003 for the purely classical case is still true if the min entropy hmin is measured relative to a quantum system because min entropy quantifies the amount of randomness that can be extracted from a given random variable our result can be used to prove the soundness of locally computable extractors in a context where side information might be quantum mechanical in particular it implies that key agreement in the bounded storage model using a standard sample and hash protocol is fully secure against quantum adversaries thus solving a long standing open problem

central limit theorem for the heat kernel measure on the unitary group we prove that for a finite collection of real valued functions f1 fn on the group of complex numbers of modulus 1 which are derivable with lipschitz continuous derivative the distribution of trf1 trfn under the properly scaled heat kernel measure at a given time on the unitary group u n has gaussian fluctuations as n tends to infinity with a covariance for which we give a formula and which is of order n 1 in the limit where the time tends to infinity we prove that this covariance converges to that obtained by p diaconis and s n evans in a previous work on uniformly distributed unitary matrices finally we discuss some combinatorial aspects of our results

robust fuzzy extractors and authenticated key agreement from close secrets consider two parties holding samples from correlated distributions w and w prime respectively where these samples are within distance t of each other in some metric space the parties wish to agree on a close to uniformly distributed secret key r by sending a single message over an insecure channel controlled by an all powerful adversary who may read and modify anything sent over the channel we consider both the keyless case where the parties share no additional secret information and the keyed case where the parties share a long term secret ssr sk ssr ext that they can use to generate a sequence of session keys r j using multiple pairs w j w prime j the former has applications to e g biometric authentication while the latter arises in e g the bounded storage model with errors we show solutions that improve upon previous work in several respects the best prior solution for the keyless case with no errors i e t 0 requires the min entropy of w to exceed 2n 3 where n is the bit length of w our solution applies whenever the min entropy of w exceeds the minimal threshold n 2 and yields a longer key

the operational meaning of min and max entropy in this paper we show that the conditional min entropy h min a b of a bipartite state rhoab is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the b part of rhoab are allowed in the special case where a is classical this overlap corresponds to the probability of guessing a given b in a similar vein we connect the conditional max entropy h max a b to the maximum fidelity of rhoab with a product state that is completely mixed on a in the case where a is classical this corresponds to the security of a when used as a secret key in the presence of an adversary holding b because min and max entropies are known to characterize information processing tasks such as randomness extraction and state merging our results establish a direct connection between these tasks and basic operational problems for example they imply that the logarithm of the probability of guessing a given b is a lower bound on the number of uniform secret bits that can be extracted from a relative to an adversary holding b

quantum proof randomness extractors via operator space theory quantum proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion furthermore they are also a useful tool in quantum shannon theory it is known that some extractor constructions are quantum proof whereas others are provably not gavinsky et al stoc 07 we argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries we first phrase the definition of extractors as a bounded norm condition between normed spaces and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces from this we show that very high min entropy extractors as well as extractors with small output are always approximately quantum proof we also study a generalization of extractors called randomness condensers we phrase the definition of condensers as a bounded norm condition and the definition of quantum proof condensers as a completely bounded norm condition seeing condensers as bipartite graphs we then find that the bounded norm condition corresponds to an instance of a well studied combinatorial problem called bipartite densest subgraph furthermore using the characterization in terms of operator spaces we can associate to any condenser a bell inequality two player game such that classical and quantum strategies are in one to one correspondence with classical and quantum attacks on the condenser hence we get for every quantum proof condenser which includes in particular quantum proof extractors a bell inequality that cannot be violated by quantum mechanics

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

the elements of the orthogonal group ωn v as products of commutators of symmetries abstract we will assume throughout thatfis a field of characteristic char f 2 and thatvis a non degenerate quadratic space overfof finite dimension dim v n the orthogonal group ofvis denotedon v and n v is its commutator subgroup john hsia in reference to the classical fact that every element of n v is a product of commutators of symmetries asked the following very basic question forfa local field does there exist a boundk depending only onn such that every element of n v is a product ofksuch commutators or fewer a corollary of the results of this article answers this question completely for a non dyadicf every element in n v is a product of n 2 such commutators except for a handful of elements which can be listed all are certain types of involutions whenn 6 where n 2 1 factors are required see theorem 4 of course hsia s question can be asked for anyf and most of the analysis is carried out in the more general context

the quantum reverse shannon theorem based on one shot information theory the quantum reverse shannon theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel s entanglement assisted classical capacity in this paper we provide a new proof of this theorem which has previously been proved by bennett devetak harrow shor and winter our proof has a clear structure being based on two recent information theoretic results one shot quantum state merging and the post selection technique for quantum channels

randomness extraction via a quantum generalization of the conditional collision entropy randomness extraction against side information is the art of distilling from a given source a key which is almost uniform conditioned on the side information this paper provides randomness extraction against quantum side information whose extractable key length is given by a quantum generalization of the collision entropy which is smoothed and conditioned differently from how this is done in existing schemes based on the fact that the collision entropy is not subadditive its optimization with respect to additional side information is introduced and is shown to be asymptotically optimal the lower bound derived there for general states is expressed as the difference between two unconditional entropies and its evaluation reduces to an eigenvalue problem of two states which are the entire state and the marginal state of side information

expressing infinite matrices over rings as products of involutions abstract let k be an arbitrary field and r be an arbitrary associative ring with identity 1 slowik in 12 proved that each matrix of ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 can be expressed as a product of at most five involutions in this article we extend the investigate to an arbitrary associative ring r with identity 1 our conclusion is that every element of ut r can be expressed as a product of at most four involutions we also prove that for the complex field every element of t c the group of upper triangular infinite matrices whose entries lying on the main diagonal satisfy a a 1 can be expressed as a product of at most three coninvolutions matrices satisfying a a i

smooth max information as one shot generalization for mutual information we study formal properties of smooth max information a generalization of von neumann mutual information derived from the max relative entropy recent work suggests that it is a useful quantity in one shot channel coding quantum rate distortion theory and the physics of quantum many body systems max information can be defined in multiple ways we demonstrate that different smoothed definitions are essentially equivalent up to logarithmic terms in the smoothing parameters these equivalence relations allow us to derive new chain rules for the max information in terms of min and max entropies thus extending the smooth entropy formalism to mutual information

the spectral edge of unitary brownian motion the brownian motion u n t t ge 0 on the unitary group converges as a process to the free unitary brownian motion u t t ge 0 as n rightarrow infty in this paper we prove that it converges strongly as a process not only in distribution but also in operator norm in particular for a fixed time t 0 we prove that the unitary brownian motion has a spectral edge there are no outlier eigenvalues in the limit we also prove an extension theorem any strongly convergent collection of random matrix ensembles independent from a unitary brownian motion also converge strongly jointly with the brownian motion we give an application of this strong convergence to the jacobi process

a fully quantum asymptotic equipartition property the classical asymptotic equipartition property is the statement that in the limit of a large number of identical repetitions of a random experiment the output sequence is virtually certain to come from the typical set each member of which is almost equally likely in this paper a fully quantum generalization of this property is shown where both the output of the experiment and side information are quantum an explicit bound on the convergence is given which is independent of the dimensionality of the side information this naturally leads to a family of reacutenyi like quantum conditional entropies for which the von neumann entropy emerges as a special case

distillation of greenberger horne zeilinger states by combinatorial methods we prove a lower bound on the rate of greenberger horne zeilinger states distillable from pure multipartite states by local operations and classical communication locc our proof is based on a modification of a combinatorial argument used in the fast matrix multiplication algorithm of coppersmith and winograd previous use of methods from algebraic complexity in quantum information theory concerned transformations with stochastic locc slocc resulting in an asymptotically vanishing success probability in contrast our new protocol works with an asymptotically vanishing error

the bounded storage model in the presence of a quantum adversary an extractor is a function that is used to extract randomness given an imperfect random source x and a uniform seed y the output e x y is close to uniform we study properties of such functions in the presence of prior quantum information about x with a particular focus on cryptographic applications we prove that certain extractors are suitable for key expansion in the bounded storage model where the adversary has a limited amount of quantum memory for extractors with one bit output we show that the extracted bit is essentially equally secure as in the case where the adversary has classical resources we prove the security of certain constructions that output multiple bits in the bounded storage model

fuzzy extractors how to generate strong keys from biometrics and other noisy data we provide formal definitions and efficient secure techniques for turning noisy information into keys usable for any cryptographic application and in particular reliably and securely authenticating biometric data our techniques apply not just to biometric information but to any keying material that unlike traditional cryptographic keys is 1 not reproducible precisely and 2 not distributed uniformly we propose two primitives a fuzzy extractor reliably extracts nearly uniform randomness r from its input the extraction is error tolerant in the sense that r will be the same even if the input changes as long as it remains reasonably close to the original thus r can be used as a key in a cryptographic application a secure sketch produces public information about its input w that does not reveal w and yet allows exact recovery of w given another value that is close to w thus it can be used to reliably reproduce error prone biometric inputs without incurring the security risk inherent in storing them we define the primitives to be both formally secure and versatile generalizing much prior work in addition we provide nearly optimal constructions of both primitives for various measures of closeness of input data such as hamming distance edit distance and set difference

schur weyl duality and the heat kernel measure on the unitary group abstract we investigate a relation between the brownian motion on the unitary group and the most natural random walk on the symmetric group based on schur weyl duality we use this relation to establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure this expectation turns out to be the generating series of certain paths in the cayley graph of the symmetric group using our expansion we recover asymptotic results of xu biane and voiculescu we give an interpretation of our main expansion in terms of random ramified coverings of a disk

a continuous semigroup of notions of independence between the classical and the free one in this paper we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non commutative random variables these notions are related to the liberation process introduced by d voiculescu to each notion of independence correspond new convolutions of probability measures for which we establish formulae and of which we compute simple examples we prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

the large n limit of the segal bargmann transform on un we study the two parameter segal bargmann transform b n on the unitary group un for large n acting on matrix valued functions that are equivariant under the adjoint action of the group the transform has a meaningful limit gs t as n which can be identified as an operator on the space of complex laurent polynomials we introduce the space of trace polynomials and use it to give effective computational methods to determine the action of the heat operator and thus the segal bargmann transform we prove several concentration of measure and limit theorems giving a direct connection from the finite dimensional transform b n to its limit gs t we characterize the operator gs t through its inverse action on the standard polynomial basis finally we show that in the case s t the limit transform gt t is the free hall transform g t introduced by biane

decomposition of matrices into commutators of involutions abstract we consider the group sl n f of all n n matrices with determinant 1 over a field f we prove that if f is the complex number field or the real number field every matrix a in sl n f is a product of at most two commutators of involutions moreover two is the smallest such number

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

entanglement cost of quantum channels the entanglement cost of a quantum channel is the minimal rate at which entanglement between sender and receiver is needed in order to simulate many copies of a quantum channel in the presence of free classical communication in this paper we show how to express this quantity as a regularized optimization of the entanglement formation over states that can be generated between sender and receiver our formula is the channel analog of a well known formula for the entanglement cost of quantum states in terms of the entanglement of formation and shares a similar relation to the recently shattered hope for additivity the entanglement cost of a quantum channel can be seen as the analog of the quantum reverse shannon theorem in the case where free classical communication is allowed the techniques used in the proof of our result are then also inspired by a recent proof of the quantum reverse shannon theorem and feature the one shot formalism for quantum information theory the postselection technique for quantum channels as well as sion s minimax theorem we discuss two applications of our result first we are able to link the security in the noisy storage model to a problem of sending quantum rather than classical information through the adversary s storage device this not only improves the range of parameters where security can be shown but also allows us to prove security for storage devices for which no results were known before second our result has consequences for the study of the strong converse quantum capacity here we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity

one shot decoupling if a quantum system a which is initially correlated to another system e undergoes an evolution separated from e then the correlation to e generally decreases here we study the conditions under which the correlation disappears almost completely resulting in a decoupling of a from e we give a criterion for decoupling in terms of two smooth entropies one quantifying the amount of initial correlation between a and e and the other characterizing the mapping that describes the evolution of a the criterion applies to arbitrary such mappings in the general one shot setting furthermore the criterion is tight for mappings that satisfy certain natural conditions one shot decoupling has a number of applications both in physics and information theory e g as a building block for quantum information processing protocols as an example we give a one shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption production

extraction of optimally unbiased bits from a biased source we explore the problem of transforming n independent and identically biased 1 1 valued random variables x sub 1 x sub n into a single 1 1 random variable f x sub 1 x sub n so that this result is as unbiased as possible in general no function f produces a completely unbiased result we perform the first study of the relationship between the bias b of these x sub i and the rate at which f x sub 1 x sub n can converge to an unbiased 1 1 random variable as n spl rarr spl infin a 1 1 random variable has bias b if e x sub i b fixing a bias b we explore the rate at which the output bias e f x sub 1 x sub n can tend to zero for a function f 1 1 spl rarr 1 1 this is accomplished by classifying the behavior of the natural normalized quantity spl xi b spl delta inf sub f lim sub n spl rarr spl infin n spl radic e f x sub 1 x sub n this infimum taken over all such f we show that for rational b spl xi b 1 s where 1 b 2 r s r and s relatively prime developing the theory of uniform distribution of sequences to suit our problem we then explore the case where b is irrational we prove a new metrical theorem concerning multidimensional diophantine approximation type from which we show that for lebesgue almost all biases b spl xi b 0 finally we show that algebraic biases exhibit curious boundary behavior falling into two classes class 1 those algebraics b for which spl xi b 0 and furthermore c sub 1 spl les spl xi b spl les c sub 2 where c sub 1 and c sub 2 are positive constants depending only on b s algebraic characteristics class 2 those algebraics b for which there exist n 0 and f 1 1 sup n spl rarr 1 1 so that e f x sub 1 x sub n 0 notice that this classification excludes the possibility that n spl radic e f x sub 1 x sub n limits to zero for algebraics for rational and algebraic biases we also study the computational problem by restricting f to be a polynomial time computable function finally we discuss natural extensions where output distributions other than the uniform distribution on 1 1 are sought

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

min and max relative entropies and a new entanglement monotone two new relative entropy quantities called the min and max relative entropies are introduced and their properties are investigated the well known min and max entropies introduced by renner are obtained from these we define a new entanglement monotone which we refer to as the max relative entropy of entanglement and which is an upper bound to the relative entropy of entanglement we also generalize the min and max relative entropies to obtain smooth min and max relative entropies these act as parent quantities for the smooth renyi entropies eth zurich ph d dissertation 2005 and allow us to define the analogues of the mutual information in the smooth renyi entropy framework further the spectral divergence rates of the information spectrum approach are shown to be obtained from the smooth min and max relative entropies in the asymptotic limit

free convolution operators and free hall transform abstract we define an extension of the polynomial calculus on a w probability space by introducing an algebra c x i i i which contains polynomials this extension allows us to define transition operators for additive and multiplicative free convolution it also permits us to characterize the free segal bargmann transform and the free hall transform introduced by biane in a manner which is closer to classical definitions finally we use this extension of polynomial calculus to prove two asymptotic results on random matrices the convergence for each fixed time as n tends to of the distribution of the brownian motion on the linear group gl n c to the distribution of a free multiplicative circular brownian motion and the convergence of the classical hall transform on u n to the free hall transform

dual universality of hash functions and its applications to quantum cryptography in this paper we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography we begin by establishing the one to one correspondence between a linear function family f and a code family c and thereby defining e almost dual universal2 hash functions as a generalization of the conventional universal2 hash functions then we show that this generalized and thus broader class of hash functions is in fact sufficient for the security of quantum cryptography this result can be explained in two different formalisms first by noting its relation to the biased family introduced by dodis and smith we demonstrate that renner s two universal hashing lemma is generalized to our class of hash functions next we prove that the proof technique by shor and preskill can be applied to quantum key distribution qkd systems that use our generalized class of hash functions for privacy amplification while shor preskill formalism requires an implementer of a qkd system to explicitly construct a linear code of the calderbank shor steane css type this result removes the existing difficulty of the construction of a linear code of css code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function we also show that a similar result applies to the quantum wire tap channel finally we compare our results in the two formalisms and show that in typical qkd scenarios the shor preskill type argument gives better security bounds in terms of the trace distance and holevo information than the method based on the biased family

decomposition of infinite matrices into products of commutators of involutions abstract let r be an associative ring with identity 1 in which 2 is invertible we prove that each matrix of ut r the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to 1 can be expressed as a product of at most two commutators of involutions let f be an arbitrary field with char f 2 using this result each element of the commutator subgroup of the vershik kerov group over f can be decomposed into a product of at most two commutators of involutions

each 2n by 2n complex symplectic matrix is a product of n 1 commutators of j symmetries abstract a 2 n 2 n complex matrix a is symplectic if a 0 i i 0 a 0 i i 0 if a is symplectic and rank a i 1 then it is called a j symmetry for each n we prove that every 2 n 2 n symplectic matrix m is a product of n 1 commutators of j symmetries and this number cannot be smaller for some m