decentralized filtered dynamic inversion for uncertain minimum phase systems decentralized filtered dynamic inversion is a control method for uncertain linear time invariant systems that are minimum phase have multi input multi output decentralized subsystems and are potentially subject to unknown disturbances this controller requires limited model information specifically knowledge of the relative degree and an estimate of the first nonzero markov parameter for each local subsystem decentralized filtered dynamic inversion is effective for command following and rejection of unknown disturbances we derive the decentralized filtered dynamic inversion controller and analyzes the closed loop stability and performance i e command following error we show that for sufficiently large choice of a single control parameter the closed loop system is asymptotically stable and the average power of the performance is arbitrarily small

quantum reverse hypercontractivity its tensorization and application to strong converses in this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log sobolev inequalities next we prove a generalization of the stroock varopoulos inequality in the non commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2 log sobolev and 1 log sobolev inequalities respectively we then prove some tensorization type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers finally as an application of these results we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting we prove strong converse bounds for the problems of quantum hypothesis testing and classical quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

quantum correlations and distinguishability of quantum states a survey of various concepts in quantum information is given with a main emphasis on the distinguishability of quantum states and quantum correlations covered topics include generalized and least square measurements state discrimination quantum relative entropies the bures distance on the set of quantum states the quantum fisher information the quantum chernoff bound bipartite entanglement the quantum discord and geometrical measures of quantum correlations the article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them and for mathematicians looking for an up to date introductory course on these subjects which are mainly developed in the physics literature

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

local hypothesis testing between a pure bipartite state and the white noise state in this paper we treat a local discrimination problem in the framework of asymmetric hypothesis testing we choose a known bipartite pure state psi rangle as an alternative hypothesis and the completely mixed state as a null hypothesis as a result we analytically derive an optimal type 2 error and an optimal positive operator valued measure povm for one way local operations and classical communication locc povm and separable povm for two way locc povm we study a family of simple three step locc protocols and show that the best protocol in this family has strictly better performance than any one way locc protocol in low dimensional systems when there may exist differences between two way locc povm and one way locc povm

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

on the quantum renyi relative entropies and related capacity formulas following csiszar s approach in classical information theory it is shown that the quantum relative entropies with parameter 0 1 can be represented as generalized cutoff rates and hence a direct operational interpretation of the quantum relative entropies are provided it is also shown that various generalizations of the holevo capacity defined in terms of the relative entropies coincide for the parameter range 0 2 and an upper bound on the one shot e capacity of a classical quantum channel in terms of these capacities is given

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

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

optimal sequence of quantum measurements in the sense of stein s lemma in quantum hypothesis testing we derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing we discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0 as an asymptotically optimal measurement we propose a projection measurement characterized by the irreducible representation theory of the special linear group sl especially in the spin 1 2 system it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction as a by product we obtain another proof of quantum stein s lemma in addition an asymptotically optimal measurement is constructed in the quantum gaussian case and it is physically meaningful

on certain properties of the relative entropy of states of operator algebras the aim of the present paper is to establish certain properties of the relative entropy functional in the general context of c algebras most of these properties have been observed under more restrictive conditions in earlier papers the celebrated strong subadditivity of the entropy gives such an example the general approach is applied to prove the existence of the mean relative entropy for a noncommutative stationary markov chain with conditional expectations

continuity and relative hamiltonians let n n 1 be a norm convergent sequence of normal states on a von neumann algebraa with n let kn n 1 be a strongly convergent sequence of self adjoint elements ofa withkn k it is shown that the sequence omega n k n n geqq 1 of perturbed states converges in norm to a related result holds forc algebras a counter example is provided to show that it is not sufficient to assume weak convergence of n n 1 even whenkn k for alln however conditions are given which together with weak convergence are sufficient relative entropy methods are used and a relative entropy inequality is proved

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

on the relative entropy fora any subset of the bounded operators on a hilbert space containing the unit and and restrictions of states on toa ent a the entropy of relative to given the information ina is defined and given an axiomatic characterisation it is compared with ent the relative entropy introduced by umegaki and generalised by various authors which is defined only fora an algebra it is proved that ent and ent s agree on pairs of normal states on an injective von neumann algebra it is also proved that ent always has all the most important properties known for ent s monotonicity concavity w upper semicontinuity etc

classical capacities of compound and averaged quantum channels we determine the capacity of compound classical quantum channels as a consequence we obtain the capacity formula for the averaged classical quantum channels the capacity result for compound channels demonstrates as in the classical setting the existence of reliable universal classical quantum codes in scenarios where the only a priori information about the channel used for the transmission of information is that it belongs to a given set of memoryless classical quantum channels our approach is based on a universal classical approximation of the quantum relative entropy which in turn relies on a universal hypothesis testing result

discriminating quantum states the multiple chernoff distance we consider the problem of testing multiple quantum hypotheses rho 1 otimes n ldots rho r otimes n where an arbitrary prior distribution is given and each of the r hypotheses is n copies of a quantum state it is known that the minimal average error probability p e decays exponentially to zero that is p e exp xi n o n however this error exponent xi is generally unknown except for the case that r 2 in this paper we solve the long standing open problem of identifying the above error exponent by proving nussbaum and szkola s conjecture that xi min i neq j c rho i rho j the right hand side of this equality is called the multiple quantum chernoff distance and c rho i rho j max 0 leq s leq1 log operatorname tr rho i s rho j 1 s has been previously identified as the optimal error exponent for testing two hypotheses rho i otimes n versus rho j otimes n the main ingredient of our proof is a new upper bound for the average error probability for testing an ensemble of finite dimensional but otherwise general quantum states this upper bound up to a states dependent factor matches the multiple state generalization of nussbaum and szkola s lower bound specialized to the case r 2 we give an alternative proof to the achievability of the binary hypothesis chernoff distance which was originally proved by audenaert et al

asymptotics of quantum relative entropy from a representation theoretical viewpoint in this paper it is proved that the quantum relative entropy d can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive operator valued measures povms the sequence of povms depends on but is independent of the choice of

asymptotics of varadhan type and the gibbs variational principle for a large class of quantum models of mean field type the thermodynamic limit of the free energy density is proved to be given by the gibbs variational principle the latter is shown to be equivalent to a non commutative version of varadhan s asymptotic formula

a study of locc detection of a maximally entangled state using hypothesis testing we study how well we can answer the question is the given quantum state equal to a certain maximally entangled state using locc in the context of hypothesis testing under several locality and invariance conditions optimal tests will be derived for several special cases by using basic theory of group representations some optimal tests are realized by performing quantum teleportation and checking whether the state is teleported we will also give a finite process for realizing some optimal tests the performance of the tests will be numerically compared

strengthened monotonicity of relative entropy via pinched petz recovery map the quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy it is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states in this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version the recovery map is a convex combination of rotated petz recovery maps and perfectly reverses the quantum channel on the second state as a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved by fawzi and renner our proof only relies on the elementary properties of pinching maps and the operator logarithm

entropy and quantum kolmogorov complexity a quantum brudno s theorem in classical information theory entropy rate and algorithmic complexity per symbol are related by a theorem of brudno in this paper we prove a quantum version of this theorem connecting the von neumann entropy rate and two notions of quantum kolmogorov complexity both based on the shortest qubit descriptions of qubit strings that run by a universal quantum turing machine reproduce them as outputs

coding theorems for compound problems via quantum renyi divergences recently a new notion of quantum renyi divergences has been introduced by muller lennert dupuis szehr fehr and tomamichel and wilde winter and yang which found a number of applications in strong converse theorems here we show that these new renyi divergences are also useful tools to obtain coding theorems in the direct domain of various problems we demonstrate this by giving new and considerably simplified proofs for the achievability parts of stein s lemma with composite null hypothesis universal state compression and the classical capacity of compound classical quantum channels based on single shot error bounds already available in the literature and simple properties of the quantum renyi divergences the novelty of our proofs is that the composite compound coding theorems can be almost directly obtained from the single shot error bounds essentially with the same effort as for the case of simple null hypothesis single source single channel

exponential error rates in multiple state discrimination on a quantum spin chain we consider decision problems on finite sets of hypotheses represented by pairwise different shift invariant states on a quantum spin chain the decision in favor of one of the hypotheses is based on outcomes of generalized measurements performed on local states on blocks of finite size we assume the existence of the mean quantum chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum chernoff distance we establish that this minimum specifies an asymptotic bound on the exponential rate of decay of the averaged probability of rejecting the true state in increasing block size if the mean quantum chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem this assumption is in particular fulfilled by shift invariant product states independent and identically distributed states further we provide a constructive proof for the existence of a sequence of quan

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

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

quantum sphere packing bounds with polynomial prefactors we study lower bounds on the optimal error probability in classical coding over classical quantum channels at rates below the capacity commonly termed quantum sphere packing bounds winter and dalai have derived such bounds for classical quantum channels however the exponents in their bounds only coincide when the channel is classical in this paper we show that these two exponents admit a variational representation and are related by the golden thompson inequality reaffirming that dalai s expression is stronger in general classical quantum channels second we establish a finite blocklength sphere packing bound for classical quantum channels which significantly improves dalai s prefactor from the order of subexponential to polynomial furthermore the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o log n n indicating our sphere packing bound is almost exact in the high rate regime finally for a special class of symmetric classical quantum channels we can completely characterize its optimal error probability without the constant composition code assumption the main technical contributions are two converse hoeffding bounds for quantum hypothesis testing and the saddle point properties of error exponent functions

classical capacity of quantum channels with general markovian correlated noise the classical capacity of a quantum channel with arbitrary markovian correlated noise is evaluated for the general case of a channel with long term memory which corresponds to a markov chain which does not converge to equilibrium the capacity is expressed in terms of the communicating classes of the markov chain for an irreducible and aperiodic markov chain the channel is forgetful and one retrieves the known expression kretschmann and werner in phys rev a 72 062323 2005 for the capacity

two quantum analogues of fisher information from a large deviation viewpoint of quantum estimation we discuss two quantum analogues of the fisher information the symmetric logarithmic derivative fisher information and kubo mori bogoljubov fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of consistent superefficient estimators as another comparison it is shown that the difference between them is characterized by the change of the order of limits

preservation of a quantum renyi relative entropy implies existence of a recovery map it is known that a necessary and sufficient condition for equality in the data processing inequality dpi for the quantum relative entropy is the existence of a recovery map we show that equality in dpi for a sandwiched renyi relative entropy with is also equivalent to this property for the proof we use an interpolating family of l p norms with respect to a state

a complementary result of kantorovich type order preserving inequalities by micic pecaric seo abstract we shall discuss operator inequalities which are obtained by elementary lemma lemma 3 1 associated with holder mccarthy and kantorovich inequalities firstly we shall give the following complementary result to micic et al linear algebra appl 360 2003 15 let a and b be two strictly positive operators on a hilbert space h such that m1i a m1i 0 and m2i b m2i 0 where m1 m1 0 m2 m2 0 and a b a if p 1 and q 1 then the following inequality holds q 1 q 1 q q m 2 p m 2 p q m 2 m 2 m 2 m 2 p m 2 m 2 p q 1 a q b p for m 2 p 1 q m 2 p m 2 p m 2 m 2 m 2 p 1 q b if p m 1 m 1 p m 1 m 1 p q 1 m 1 m 1 q 1 m 1 p m 1 p q m 1 m 1 p m 1 m 1 p q b q a p for m 1 p 1 q m 1 p m 1 p m 1 m 1 m 1 p 1 q we remark that a is shown in linear algebra appl 360 2003 15 as an extension of two variable version of our previous one variable one j inequal appl 2 1998 137 secondly we shall show the following extension of two parameters type of an extension of fujii et al sci math 1 1998 307 on the determinant of an operator let t be strictly positive operators on a hilbert space h such that mi t mi 0 then the following inequality holds s h p q x t q t p x x x t p for p 0 and q 0 wheresh p q is defined by s h p q m p q h q h p 1 e log h q h p 1 if q h p 1 log h qh p andthe determinant x t for strictly positive operator t at a unit vector x in hilbert space h is defined by x t exp log t x x as an application of this result we shall give an alternative proof of two variable version of characterization of the chaotic order in linear algebra appl ibid theorem 4 4

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

a note on operator inequalities of tsallis relative operator entropy tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized shannon inequalities were shown in the previous paper after the review of some fundamental properties of tsallis relative operator entropy some operator inequalities related to tsallis relative operator entropy are shown in the present paper our inequalities give the upper and lower bounds of tsallis relative operator entropy the operator equality on tsallis relative operator entropy is also shown by considering the tensor product this relation generalizes the pseudoadditivity for tsallis entropy as a corollary of our operator equality derived from the tensor product manipulation we show several operator inequalities including the superadditivity and the subadditivity for tsallis relative operator entropy our results are generalizations of the superadditivity and the subadditivity for tsallis entropy