one shot coherence dilution manipulation and quantification of quantum resources are fundamental problems in quantum physics in the asymptotic limit coherence distillation and dilution have been proposed by manipulating infinite identical copies of states in the nonasymptotic setting finite data size effects emerge and the practically relevant problem of coherence manipulation using finite resources has been left open this letter establishes the one shot theory of coherence dilution which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations dephasing covariant incoherent operations incoherent operations or strictly incoherent operations we introduce several coherence monotones with concrete operational interpretations that estimate the one shot coherence cost the minimum amount of maximally coherent states needed for faithful coherence dilution furthermore we derive the asymptotic coherence dilution results with maximally incoherent operations incoherent operations and strictly incoherent operations as special cases our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources such as quantum key distribution and random number generation

splitting scheme for the stability of strong shock profile abstract in this paper we use the laplace transform and dirichlet neumann map to give a systematical scheme to study the small wave perturbation of general shock profile with general amplitude here we use certain non classical shock waves for a rotationally invariant system of viscous conservation laws to demonstrate this scheme we derive an explicit solution and show that it converges pointwise to another over compressive profile exponentially when the perturbations of the initial data to a given over compressive shock profile are sufficiently small

quantum achievability proof via collision relative entropy in this paper we provide a simple framework for deriving one shot achievable bounds for some problems in quantum information theory our framework is based on the joint convexity of the exponential of the collision relative entropy and is a partial quantum generalization of the technique of yassaee et al from classical information theory based on this framework we derive one shot achievable bounds for the problems of communication over classical quantum channels quantum hypothesis testing and classical data compression with quantum side information we argue that our one shot achievable bounds are strong enough to give the asymptotic achievable rates of these problems even up to the second order

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

general formulas for capacity of classical quantum channels the capacity of a classical quantum channel or in other words the classical capacity of a quantum channel is considered in the most general setting where no structural assumptions such as the stationary memoryless property are made on a channel a capacity formula as well as a characterization of the strong converse property is given just in parallel with the corresponding classical results of verdu han 1994 which are based on the so called information spectrum method the general results are applied to the stationary memoryless case with or without cost constraint on inputs whereby a deep relation between the channel coding theory and the hypothesis testing for two quantum states is elucidated

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

identifying regression outliers and mixtures graphically regressions in practice can include outliers and other unknown subpopulation structures for example mixtures of regressions occur if there is an omitted categorical predictor like gender or location and different regressions occur within each category the theory of regression graphics based on central subspaces can be used to construct graphical solutions to long standing problems of this type it is argued that in practice the central subspace automatically expands to incorporate outliers and regression mixtures thus methods of estimating the central subspace can be used to identify these phenomena without specifying a model examples illustrating the power of the theory are presented

tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state we consider asymptotic hypothesis testing or state discrimination with asymmetric treatment of errors between an arbitrary fixed bipartite pure state psi rangle and the white noise state the completely mixed state under one way local operations and classical communications locc two way locc and separable positive operator valued measures povms as a result we derive the hoeffding bounds under two way locc povms and separable povms further we derive stein s lemma type of optimal error exponents under one way locc two way locc and separable povms up to the third order which clarifies the difference between one way and two way locc povm our results clarify the relationship between the entanglement of renyi entropy and the hypothesis testing under locc since the entanglement of renyi entropy appears in the formula of both the hoeffding bounds and stein s lemma type of error exponents this paper gives a very rare example in which the optimal performance under the infinite round two way locc is also equal to that under separable operations and can be attained with two round communication but not with the one way locc

one shot coherence distillation we characterize the distillation of quantum coherence in the one shot setting that is the conversion of general quantum states into maximally coherent states under different classes of quantum operations we show that the maximally incoherent operations mio and the dephasing covariant incoherent operations dio have the same power in the task of one shot coherence distillation we establish that the one shot distillable coherence under mio and dio is efficiently computable with a semidefinite program which we show to correspond to a quantum hypothesis testing problem further we introduce a family of coherence monotones generalizing the robustness of coherence as well as the modified trace distance of coherence and show that they admit an operational interpretation in characterizing the fidelity of distillation under different classes of operations by providing an explicit formula for these quantities for pure states we show that the one shot distillable coherence under mio dio strictly incoherent operations and incoherent operations is equal for all pure states

α z renyi relative entropies we consider a two parameter family of renyi relative entropies d z that are quantum generalisations of the classical renyi divergence d p q this family includes many known relative entropies or divergences such as the quantum relative entropy the recently defined quantum renyi divergences as well as the quantum renyi relative entropies all its members satisfy the quantum generalizations of renyi s axioms for a divergence we consider the range of the parameters z for which the data processing inequality holds we also investigate a variety of limiting cases for the two parameters obtaining explicit formulas for each one of them

fast very robust methods for the detection of multiple outliers abstract a few repeats of a simple forward search from a random starting point are shown to provide sufficiently robust parameter estimates to reveal masked multiple outliers the stability of the patterns obtained is exhibited by the stalactite plot the robust estimators used are least median of squares for regression and the minimum volume ellipsoid for multivariate outliers the forward search also has potential as an algorithm for calculation of these parameter estimates for large problems parallel computing provides appreciable reduction in computational time

strong converse exponents for a quantum channel discrimination problem and quantum feedback assisted communication this paper studies the difficulty of discriminating between an arbitrary quantum channel and a replacer channel that discards its input and replaces it with a fixed state the results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination we show that in this particular setting the most general adaptive discrimination strategies provide no asymptotic advantage over non adaptive tensor power strategies this conclusion follows by proving a quantum stein s lemma for this channel discrimination setting showing that a constant bound on the type i error leads to the type ii error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels the strong converse part of the lemma states that any attempt to make the type ii error decay to zero at a rate faster than the channel relative entropy implies that the type i error necessarily converges to one we then refine this latter result by identifying the optimal strong converse exponent for this task as a consequence of these results we can establish a strong converse theorem for the quantum feedback assisted capacity of a channel sharpening a result due to bowen furthermore our channel discrimination result demonstrates the asymptotic optimality of a non adaptive tensor power strategy in the setting of quantum illumination as was used in prior work on the topic the sandwiched renyi relative entropy is a key tool in our analysis finally by combining our results with recent results of hayashi and tomamichel we find a novel operational interpretation of the mutual information of a quantum channel mathcal n as the optimal type ii error exponent when discriminating between a large number of independent instances of mathcal n and an arbitrary worst case replacer channel chosen from the set of all replacer channels

decoding quantum information via the petz recovery map we obtain a lower bound on the maximum number of qubits qn e n which can be transmitted over n uses of a quantum channel n for a given non zero error threshold e to obtain our result we first derive a bound on the one shot entanglement transmission capacity of the channel and then compute its asymptotic expansion up to the second order in our method to prove this achievability bound the decoding map used by the receiver on the output of the channel is chosen to be the petz recovery map also known as the transpose channel our result in particular shows that this choice of the decoder can be used to establish the coherent information as an achievable rate for quantum information transmission applying our achievability bound to the 50 50 erasure channel which has zero quantum capacity we find that there is a sharp error threshold above which qn e n scales as n

strong converse and stein s lemma in quantum hypothesis testing the hypothesis testing problem for two quantum states is treated we show a new inequality between the errors of the first kind and the second kind which complements the result of hiai and petz 1991 to establish the quantum version of stein s lemma the inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy which yields the strong converse in quantum hypothesis testing finally we discuss the relation between the bound and the power exponent derived by han and kobayashi 1989 in classical hypothesis testing

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

quantum conditional mutual information and approximate markov chains a state on a tripartite quantum system a otimes b otimes c forms a markov chain if it can be reconstructed from its marginal on a otimes b by a quantum operation from b to b otimes c we show that the quantum conditional mutual information i a c b of an arbitrary state is an upper bound on its distance to the closest reconstructed state it thus quantifies how well the markov chain property is approximated

classical communication over a quantum interference channel calculating the capacity of interference channels is a notorious open problem in classical information theory such channels have two senders and two receivers and each sender would like to communicate with a partner receiver the capacity of such channels is known exactly in the settings of very strong and strong interference while the han kobayashi coding strategy gives the best known achievable rate region in the general case here we introduce and study the quantum interference channel a natural generalization of the interference channel to the setting of quantum information theory we restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results though generalizations of our results to channels with quantum inputs are straightforward we are able to determine the exact classical capacity of this channel in the settings of very strong and strong interference by exploiting winter s successive decoding strategy and a novel two sender quantum simultaneous decoder respectively we provide a proof that a han kobayashi strategy is achievable with holevo information rates up to a conjecture regarding the existence of a three sender quantum simultaneous decoder this conjecture holds for a special class of quantum multiple access channels with average output states that commute and we discuss some other variations of the conjecture that hold finally we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates

a decoupling approach to classical data transmission over quantum channels most coding theorems in quantum shannon theory can be proven using the decoupling technique to send data through a channel one guarantees that the environment gets no information about it uhlmann s theorem then ensures that the receiver must be able to decode while a wide range of problems can be solved this way one of the most basic coding problems remains impervious to a direct application of this method sending classical information through a quantum channel we will show that this problem can in fact be solved using decoupling ideas specifically by proving a dequantizing theorem which ensures that the environment is only classically correlated with the sent data our techniques naturally yield a generalization of the holevo schumacher westmoreland theorem to the one shot scenario where a quantum channel can be applied only once

lower bounds for quantum parameter estimation the laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of unknown parameters of a quantum system in this paper we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test in group covariant scenarios we find that there is an ultimate bound for any estimation scheme which depends only on the representation theoretic data of the probe system and we evaluate its asymptotics in the limit of many systems establishing a general heisenberg limit for region estimation we apply our results to several examples in particular to phase estimation where our bounds allow us to recover the well known heisenberg and shot noise scaling

moderate deviation analysis for classical quantum channels and quantum hypothesis testing in this paper we study the tradeoffs between the error probabilities of classical quantum channels and the blocklength n when the transmission rates approach the channel capacity at a rate lower than 1 sqrt n a research topic known as moderate deviation analysis we show that the optimal error probability vanishes under this rate convergence our main technical contributions are a tight quantum sphere packing bound obtained via chaganty and sethuraman s concentration inequality in strong large deviation theory and asymptotic expansions of error exponent functions moderate deviation analysis for quantum hypothesis testing is also established the converse directly follows from our channel coding result while the achievability relies on a martingale inequality

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

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

stability of systems of viscous conservation laws we consider systems of conservation laws and give conditions for nonlinear stability of viscous shock profiles the analysis applies to classical shocks of arbitrary strength 1998 john wiley sons inc

a smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels we use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel we provide lower and upper bounds on the optimal type ii error of quantum hypothesis testing in terms of the smooth max relative entropy of the two states representing the two hypotheses then using a relative entropy version of the quantum asymptotic equipartition property qaep we can recover the strong converse rate of the i i d hypothesis testing problem in the asymptotics on the other hand combining stein s lemma with our bounds we obtain a stronger e independent version of the relative entropy qaep similarly we provide bounds on the one shot e error classical capacity of a quantum channel in terms of a smooth max relative entropy variant of its holevo capacity using these bounds and the e independent version of the relative entropy qaep we can recover both the holevo schumacher westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding as well as its strong converse counterpart

coding for classical quantum channels with rate limited side information at the encoder information spectrum approach we study the hybrid classical quantum version of the channel coding problem for the famous gel fand pinsker channel in the classical setting for this channel the conditional distribution of the channel output given the channel input is a function of a random parameter called the channel state we study this problem when a rate limited version of the channel state is available at the encoder for the classical quantum gel fand pinsker channel we establish the capacity region for this problem in the information spectrum setting the capacity region is quantified in terms of spectral sup classical mutual information rate and spectral inf quantum mutual information rate

one shot lossy quantum data compression we provide a framework for one shot quantum rate distortion coding in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level we obtain a one shot characterization of the minimum qubit compression size for an entanglement assisted quantum rate distortion code in terms of the smooth max information a quantity previously employed in the one shot quantum reverse shannon theorem next we show how this characterization converges to the known expression for the entanglement assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source finally we give a tight finite blocklength characterization for the entanglement assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbolwise distortion constraint

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

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

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

semidefinite programming strong converse bounds for classical capacity we investigate the classical communication over quantum channels when assisted by no signaling and positive partial transpose preserving ppt codes for which both the optimal success probability of a given transmission rate and the one shot epsilon error capacity are formalized as semidefinite programs sdps based on this we obtain improved sdp finite blocklength converse bounds of general quantum channels for entanglement assisted codes and unassisted codes furthermore we derive two sdp strong converse bounds for the classical capacity of general quantum channels for any code with a rate exceeding either of the two bounds of the channel the success probability vanishes exponentially fast as the number of channel uses increases in particular applying our efficiently computable bounds we derive an improved upper bound on the classical capacity of the amplitude damping channel we also establish the strong converse property for the classical and private capacities of a new class of quantum channels we finally study the zero error setting and provide efficiently computable upper bounds on the one shot zero error capacity of a general quantum channel

secret key agreement general capacity and second order asymptotics we revisit the problem of secret key agreement using interactive public communication for two parties and propose a new secret key agreement protocol the protocol attains the secret key capacity for general observations and attains the second order asymptotic term in the maximum length of a secret key for independent and identically distributed observations in contrast to the previously suggested secret key agreement protocols the proposed protocol uses interactive communication in fact the standard one way communication protocol used prior to this paper fails to attain the asymptotic results above our converse proofs rely on a recently established upper bound for secret key lengths both our lower and upper bounds are derived in a single shot setup and the asymptotic results are obtained as corollaries

one shot classical quantum capacity and hypothesis testing the one shot classical capacity of a quantum channel quantifies the amount of classical information that can be transmitted through a single use of the channel such that the error probability is below a certain threshold in this work we show that this capacity is well approximated by a relative entropy type measure defined via hypothesis testing combined with a quantum version of stein s lemma our results give a conceptually simple proof of the well known holevo schumacher westmoreland theorem for the capacity of memoryless channels more generally we obtain tight capacity formulas for arbitrary not necessarily memoryless channels

a tight upper bound for the third order asymptotics for most discrete memoryless channels this paper shows that the logarithm of the e error capacity average error probability for n uses of a discrete memoryless channel dmc is upper bounded by the normal approximation plus a third order term that does not exceed 1 2 logn o 1 if the e dispersion of the channel is positive this matches a lower bound by y polyanskiy 2010 for dmcs with positive reverse dispersion if the e dispersion vanishes the logarithm of the e error capacity is upper bounded by n times the capacity plus a constant term except for a small class of dmcs and e 1 2

second order asymptotics for quantum hypothesis testing in the asymptotic theory of quantum hypothesis testing the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way this is well known as the direct part and strong converse of quantum stein s lemma here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind and hence we obtain the second order asymptotics for quantum hypothesis testing this actually implies quantum stein s lemma as a special case meanwhile our analysis also yields tight bounds for the case of finite sample size these results have potential applications in quantum information theory our method is elementary based on basic linear algebra and probability theory it deals with the achievability part and the optimality part in a unified fashion

a limit of the quantum renyi divergence recently an interesting quantity called the quantum renyi divergence or sandwiched renyi relative entropy was defined for pairs of positive semi definite operators and it depends on a parameter and acts as a parent quantity for other relative entropies which have important operational significance in quantum information theory the quantum relative entropy and the min and max relative entropies there is however another relative entropy called the 0 relative renyi entropy which plays a key role in the analysis of various quantum information processing tasks in the one shot setting we prove that the 0 relative renyi entropy is obtainable from the quantum renyi divergence only if and have equal supports this along with existing results in the literature suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived the quantum renyi divergence with parameter 1 2 and the relative renyi entropy with 0 1

stability of viscous shocks on finite intervals consider the cauchy problem for a system of viscous conservation laws with a solution consisting of a thin viscous shock layer connecting smooth regions we expect the time dependent behavior of such a solution to involve two processes one process consists of the large scale evolution of the solution this process is well modeled by the corresponding inviscid equations the other process is the adjustment in shape and position of the shock layer to the large scale solution the time scale of the second process is much faster than the first 1 v compared to 1 the second process can be divided into two parts adjustment of the shape and of the position during this adjustment the end states are essentially constant in order to answer the question of stability we have developed a technique where the two processes can be separated to isolate the fast process we consider the region in the vicinity of the shock layer the equations are augmented with special boundary conditions that reflect the slow change of the end states we show that for the isolated fast process the perturbations decay exponentially in time

finite blocklength and moderate deviation analysis of hypothesis testing of correlated quantum states and application to classical quantum channels with memory martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning here we apply techniques borrowed from this field to quantum hypothesis testing which is the problem of discriminating quantum states belonging to two different sequences rho n n and sigma n n we obtain achievability bounds on the finite blocklength type ii stein and hoeffding errors which for i i d states are in general tighter than the corresponding bounds obtained by audenaert mosonyi and verstraete we also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a non homogeneous factorization property examples of such sequences include gibbs states of spin chains with translation invariant finite range interaction as well as finitely correlated quantum states we apply our results to find bounds on the capacity of a certain class of classical quantum channels with memory which satisfy a so called channel factorization property both in the finite blocklength and moderate deviation regimes

second order asymptotics for source coding dense coding and pure state entanglement conversions we introduce two variants of the information spectrum relative entropy defined by tomamichel and hayashi which have the particular advantage of satisfying the data processing inequality i e monotonicity under quantum operations this property allows us to obtain one shot bounds for various information processing tasks in terms of these quantities moreover these relative entropies have a second order asymptotic expansion which in turn yields tight second order asymptotics for optimal rates of these tasks in the independent and identically distributed setting the tasks studied in this paper are fixed length quantum source coding noisy dense coding entanglement concentration pure state entanglement dilution and transmission of information through a classical quantum channel in the latter case we retrieve the second order asymptotics obtained by tomamichel and tan our results also yield the known second order asymptotics of fixed length classical source coding derived by hayashi the second order asymptotics of entanglement concentration and dilution provide a refinement of the inefficiency of these protocols a quantity which in the case of entanglement dilution was studied by harrow and lo we prove how the discrepancy between the optimal rates of these two processes in the second order implies the irreversibility of entanglement concentration established by kumagai and hayashi in addition the spectral divergence rates of the information spectrum approach isa can be retrieved from our relative entropies in the asymptotic limit this enables us to directly obtain the more general results of the isa from our one shot bounds

finite blocklength converse bounds for quantum channels we derive upper bounds on the rate of transmission of classical information over quantum channels by block codes with a given blocklength and error probability for both entanglement assisted and unassisted codes in terms of a unifying framework of quantum hypothesis testing with restricted measurements our bounds do not depend on any special property of the channel such as memorylessness and generalize both a classical converse of polyanskiy poor and verdu as well as a quantum converse of renner and wang and have a number of desirable properties in particular our bound on entanglement assisted codes is a semidefinite program and for memoryless channels its large blocklength limit is the well known formula for entanglement assisted capacity due to bennett shor smolin and thapliyal

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

thermodynamic capacity of quantum processes thermodynamics imposes restrictions on what state transformations are possible in the macroscopic limit of asymptotically many independent copies of a state as for instance in the case of an ideal gas the possible transformations become reversible and are fully characterized by the free energy in this letter we present a thermodynamic resource theory for quantum processes that also becomes reversible in the macroscopic limit a property that is especially rare for a resource theory of quantum channels we identify a unique single letter and additive quantity the thermodynamic capacity that characterizes the thermodynamic value of a quantum channel in the sense that the work required to simulate many repetitions of a quantum process employing many repetitions of another quantum process becomes equal to the difference of the respective thermodynamic capacities on a technical level we provide asymptotically optimal constructions of universal implementations of quantum processes a challenging aspect of this construction is the apparent necessity to coherently combine thermal engines that would run in different thermodynamic regimes depending on the input state our results have applications in quantum shannon theory by providing a generalized notion of quantum typical subspaces and by giving an operational interpretation to the entropy difference of a channel

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

one shot marton inner bound for classical quantum broadcast channel we consider the problem of communication over a classical quantum broadcast channel with one sender and two receivers generalizing the classical inner bounds shown by marton and the recent quantum asymptotic version shown by savov and wilde we obtain one shot inner bounds in the quantum setting our bounds are stated in terms of hypothesis testing and one shot max divergences these results give a full justification of the claims of savov and wilde in the classical quantum asymptotic iid setting the techniques also yield similar bounds in the information spectrum setting we obtain these results using a different analysis of the random codebook argument our method yields a classical one shot marton bound with a common message and a classical one shot mutual covering lemma based on rejection sampling

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

duality of channels and codes for any given channel w with classical inputs and possibly quantum outputs a dual classical input channel w perp can be defined by embedding the original into a channel mathcal n with quantum inputs and outputs here we give new uncertainty relations for a general class of entropies that lead to very close relationships between the original channel and its dual moreover we show that channel duality can be combined with duality of linear codes whereupon the uncertainty relations imply that the performance of a given code over a given channel is entirely characterized by the performance of the dual code on the dual channel this has several applications in the context of polar codes it implies that the rates of polarization to ideal and useless channels must be identical duality also relates the tasks of channel coding and privacy amplification implying that the finite blocklength performance of extractors and codes is precisely linked and those optimal rate extractors can be transformed into capacity achieving codes and vice versa finally duality also extends to the exit function of any channel and code here it implies that for any channel family if the exit function for a fixed code has a sharp transition then it must be such that the rate of the code equals the capacity at the transition this gives a different route to proving a code family achieves capacity by establishing sharp exit function transitions

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

superadditivity of quantum channel coding rate with finite blocklength joint measurements the maximum rate at which classical information can be reliably transmitted per use of a quantum channel strictly increases in general with n the number of channel outputs that are detected jointly by the quantum joint detection receiver jdr this phenomenon is known as superadditivity of the maximum achievable information rate over a quantum channel we study this phenomenon for a pure state classical quantum channel and provide a lower bound on c n n the maximum information rate when the jdr is restricted to making joint measurements over no more than n quantum channel outputs while allowing arbitrary classical error correction we also show the appearance of a superadditivity phenomenon of mathematical resemblance to the aforesaid problem in the channel capacity of a classical discrete memoryless channel when a concatenated coding scheme is employed and the inner decoder is forced to make hard decisions on n length inner codewords using this correspondence we develop a unifying framework for the above two notions of superadditivity and show that for our lower bound to c n n to be equal to a given fraction of the asymptotic capacity c of the respective channel n must be proportional to v c 2 where v is the respective channel dispersion quantity

semidefinite programming converse bounds for quantum communication we derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one shot and asymptotic regime first we derive one shot semidefinite programming sdp converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel which improve the previous bound from tomamichel berta renes nat commun 7 2016 as applications we study quantum communication over depolarizing channels and amplitude damping channels with finite resources second we find an sdp strong converse bound for the quantum capacity of an arbitrary quantum channel which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases furthermore we prove that the sdp strong converse bound improves the partial transposition bound introduced by holevo and werner third we prove that this sdp strong converse bound is equal to the so called max rains information which is an analog to the rains information introduced in tomamichel wilde winter ieee trans inf theory 63 715 2017 our sdp strong converse bound is weaker than the rains information but it is efficiently computable for general quantum channels

on converse bounds for classical communication over quantum channels we explore several new converse bounds for classical communication over quantum channels in both the one shot and asymptotic regimes first we show that the matthews wehner meta converse bound for entanglement assisted classical communication can be achieved by activated no signaling assisted codes suitably generalizing a result for classical channels second we derive a new efficiently computable meta converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel as applications we provide a finite resource analysis of classical communication over quantum erasure channels including the second order and moderate deviation asymptotics third we explore the asymptotic analogue of our new meta converse the upsilon information of the channel we show that its regularization is an upper bound on the classical capacity which is generally tighter than the entanglement assisted capacity and other known efficiently computable strong converse bounds for covariant channels we show that the upsilon information is a strong converse bound

moderate deviation analysis for classical communication over quantum channels we analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases to characterise the fundamental tradeoff between decoding error code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by alt g and wagner as well as polyanskiy and verdu we derive such a tradeoff for classical quantum as well as image additive channels in terms of the channel capacity and the channel dispersion giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data to derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime due to the central importance of the latter task we expect that our techniques will find further applications in the analysis of other quantum information processing tasks

on the convergence to steady state of solutions of nonlinear hyperbolic parabolic systems the cauchy problem for time dependent quasi linear partial differential equations is considered and the nonlinear stability of steady state solutions is investigated