quantum hypothesis testing with group symmetry the asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system we consider various asymptotic error exponents in connection with the problems of the chernoff bound the hoeffding bound and stein s lemma and derive bounds on these quantities in terms of their corresponding statistical distance measures a special emphasis is put on the comparison of the performances of group invariant and unrestricted measurements

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 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

quantum estimation for non differentiable models state estimation is a classical problem in quantum information in optimization of an estimation scheme to find a lower bound to the error of the estimator is a very important step so far all the proposed tractable lower bounds use a derivative of the density matrix however sometimes we are interested in quantities with singularity e g concurrence etc in this paper lower bounds to a mean square error of an estimator are derived for a quantum estimation problem without smoothness assumptions our main idea is to replace the derivative by difference as is done in classical estimation theory we applied the inequalities to several examples and derived an optimal estimator for some of them

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

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

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

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

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

an example of problem arising in a finite difference context direct and inverse problem for the discrete analog of the equation ψxx uψ σψy the direct and inverse spectral problem for the discrete analog of the equation xx u y is solved in the framework of theory the time evolution of the spectral data for the simplest nonlinear differential difference equations associated to this linear problem is derived

discrimination between states of a quantum system by observations let m be a von neumann algebra with normal states and and let i ai m be a net of positive unital mappings of finite dimensional algebras the paper studies the existence of a net i m ai such that i i and i i a necessary and sufficient condition is given for the existence of i which may fail to hold when ai are commutative and does not commute with reformulation in terms of relative entropy is given and the study of the entropic version is extended to the case when m and are infinite tensor products all these mathematical problems are motivated by quantum statistical considerations

making good codes for classical quantum channel coding via quantum hypothesis testing in this correspondence we give an alternative proof of the direct part of the classical quantum channel coding theorem the holevo schumacher westmoreland hsw theorem using ideas of quantum hypothesis testing in order to show the existence of good codes we invoke a limit theorem relevant to the quantum stein s lemma in quantum hypothesis testing as the law of large numbers used in the classical case we also apply a greedy construction of good codes using a packing procedure of noncommutative operators consequently we derive an upper bound on the coding error probability which is used to give an alternative proof of the hsw theorem this approach elucidates how the holevo information applies to the classical quantum channel coding problems

on zero error communication via quantum channels in the presence of noiseless feedback we initiate the study of zero error communication via quantum channels when the receiver and the sender have at their disposal a noiseless feedback channel of unlimited quantum capacity generalizing shannon s zero error communication theory with instantaneous feedback we first show that this capacity is only a function of the linear span of choi kraus operators of the channel which generalizes the bipartite equivocation graph of a classical channel and which we dub non commutative bipartite graph then we go on to show that the feedback assisted capacity is non zero allowing for a constant amount of activating noiseless communication if and only if the non commutative bipartite graph is non trivial and give a number of equivalent characterizations this result involves a far reaching extension of the conclusive exclusion of quantum states we then present an upper bound on the feedback assisted zero error capacity motivated by a conjecture originally made by shannon and proved later by ahlswede we demonstrate that this bound to have many good properties including being additive and given by a minimax formula we also prove a coding theorem showing that this quantity is the entanglement assisted capacity against an adversarially chosen channel from the set of all channels with the same choi kraus span which can also be interpreted as the feedback assisted unambiguous capacity the proof relies on a generalization of the postselection lemma de finetti reduction that allows to reflect additional constraints and which we believe to be of independent interest this capacity is a relaxation of the feedback assisted zero error capacity however we have to leave open the question of whether they coincide in general we illustrate our ideas with a number of examples including classical quantum channels and weyl diagonal channels and close with an extensive discussion of open questions

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

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

universal coding for classical quantum channel we construct a universal code for a stationary and memoryless classical quantum channel as a quantum version of the universal coding by csiszar and korner our code is constructed utilizing a combination of irreducible representations a decoder introduced through the quantum information spectrum and the packing lemma

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

quantum hypothesis testing for gaussian states quantum analogues of χ 2 t and f tests we consider quantum counterparts of testing problems for which the optimal tests are the 2 t and f tests these quantum counterparts are formulated as quantum hypothesis testing problems concerning gaussian state families and they contain nuisance parameters which have group symmetry the quantum hunt stein theorem removes some of these nuisance parameters but other difficulties remain in order to remove them we combine the quantum hunt stein theorem and other reduction methods to establish a general reduction theorem that reduces a complicated quantum hypothesis testing problem to a fundamental quantum hypothesis testing problem using these methods we derive quantum counterparts of the 2 t and f tests as optimal tests in the respective settings

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

typical support and sanov large deviations of correlated states discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support the exponential growth rate of which is given by the maximal ergodic entropy in the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy according to sanov s theorem we give an extension to the correlated case referring to the newly introduced class of hp states

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

multivariate trace inequalities we prove several trace inequalities that extend the golden thompson and the araki lieb thirring inequality to arbitrarily many matrices in particular we strengthen lieb s triple matrix inequality as an example application of our four matrix extension of the golden thompson inequality we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub additivity of the von neumann entropy in terms of recoverability we find the first explicit remainder terms that are tight in the commutative case our proofs rely on complex interpolation theory as well as asymptotic spectral pinching providing a transparent approach to treat generic multivariate trace inequalities

an information spectrum approach to classical and quantum hypothesis testing for simple hypotheses the information spectrum analysis made by han for classical hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and quantum hypothesis testing the results are also applied to fixed length source coding when loosening the normalizing condition for probability distributions and for quantum states we establish general formulas for several quantities relating to the asymptotic optimality of tests codes in terms of classical and quantum information spectra

two approaches to obtain the strong converse exponent of quantum hypothesis testing for general sequences of quantum states we present two general approaches to obtain the strong converse exponent of simple quantum hypothesis testing for correlated quantum states one approach requires that the states satisfy a certain factorization property typical examples of such states are the temperature states of translation invariant finite range interactions on a spin chain the other approach requires the differentiability of a regularized renyi alpha divergence in the parameter alpha typical examples of such states include temperature states of non interacting fermionic lattice systems and classical irreducible markov chains in all cases we get that the strong converse exponent is equal to the hoeffding anti divergence which in turn is obtained from the regularized renyi divergences of the two states

pretty strong converse for the quantum capacity of degradable channels we exhibit a possible road toward a strong converse for the quantum capacity of degradable channels in particular we show that all degradable channels obey what we call a pretty strong converse when the code rate increases above the quantum capacity the fidelity makes a discontinuous jump from 1 to at most 1 2 asymptotically a similar result can be shown for the private classical capacity furthermore we can show that if the strong converse holds for symmetric channels which have quantum capacity zero then degradable channels obey the strong converse the above mentioned asymptotic jump of the fidelity at the quantum capacity then decreases from 1 to 0

hypothesis testing for gaussian states on bosonic lattices the asymptotic state discrimination problem with simple hypotheses is considered for a cubic lattice of bosons a complete solution is provided for the problems of the chernoff and the hoeffding bounds and stein s lemma in the case when both hypotheses are gaussian states with gauge and translation invariant quasifree parts

hypothesis testing with the general source the asymptotically optimal hypothesis testing problem with general sources as the null and alternative hypotheses is studied under exponential type error constraints on the first kind of error probability our fundamental philosophy is to convert all of the hypothesis testing problems to the pertinent computation problems in the large deviation probability theory this methodologically new approach enables us to establish compact general formulas of the optimal exponents of the second kind of error and correct testing probabilities for the general sources including all nonstationary and or nonergodic sources with arbitrary abstract alphabet countable or uncountable these general formulas are presented from the information spectrum point of view

models of active learning in group structured state spaces we investigate the problem of learning the transition dynamics of deterministic discrete state environments we assume that an agent exploring such an environment is able to perform actions from a finite set of actions in the environment and to sense the state changes the question investigated is whether the agent can learn the dynamics without visiting all states such a goal is unrealistic in general hence we assume that the environment has structural properties an agent might exploit in particular we assume that the set of all action sequences forms an algebraic group we introduce a learning model in different variants and study under which circumstances the corresponding group structured environments can be learned efficiently by experimenting with group generators actions it turns out that for some classes of such environments the choice of actions given to the agent determines if efficient learning is possible negative results are presented even without efficiency constraints for rather general classes of groups showing that even with group structure learning an environment from partial information is far from trivial however positive results for special subclasses of abelian groups turn out to be a good starting point for the design of efficient learning algorithms based on structured representations

adversarial hypothesis testing and a quantum stein s lemma for restricted measurements recall the classical hypothesis testing setting with two sets of probability distributions p and q one receives either n i i d samples from a distribution p in p or from a distribution q in q and wants to decide from which set the points were sampled it is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum likelihood ratio test which does not depend on p or q but only on the sets p and q we consider an adaptive generalization of this model where the choice of p in p and q in q can change in each sample in some way that depends arbitrarily on the previous samples in other words in the k th round an adversary having observed all the previous samples in rounds 1 ldots k 1 chooses p k in p and q k in q with the goal of confusing the hypothesis test we prove that even in this case the optimal exponential error rate can be achieved by a simple maximum likelihood test that depends only on p and q we then show that the adversarial model has applications in hypothesis testing for quantum states using restricted measurements for example it can be used to study the problem of distinguishing entangled states from the set of all separable states using only measurements that can be implemented with local operations and classical communication locc the basic idea is that in our setup the deleterious effects of entanglement can be simulated by an adaptive classical adversary we prove a quantum stein s lemma in this setting in many circumstances the optimal hypothesis testing rate is equal to an appropriate notion of quantum relative entropy between two states in particular our arguments yield an alternate proof of li and winter s recent strengthening of strong subadditivity for von neumann entropy

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

flexible constrained de finetti reductions and applications de finetti theorems show how sufficiently exchangeable states are well approximated by convex combinations of independent identically distributed states recently it was shown that in many quantum information applications a more relaxed de finetti reduction i e only a matrix inequality between the symmetric state and one of the de finetti forms is enough and that it leads to more concise and elegant arguments here we show several uses and general flexible applicability of a constrained de finetti reduction in quantum information theory which was recently discovered by duan severini and winter in particular we show that the technique can accommodate other symmetries commuting with the permutation action and permutation invariant linear constraints we then demonstrate that in some cases it is also fruitful with convex constraints in particular separability in a bipartite setting this is a constraint particularly interesting in the context of the complexity class qma 2 of interactive quantu

hypothesis testing on invariant subspaces of the symmetric group part i quantum sanov s theorem and arbitrarily varying sources we report a proof of the quantum sanov theorem by the elementary application of basic facts about representations of the symmetric group together with a complete characterization of the optimal error exponent in a situation where the null hypothesis is given by an arbitrarily varying quantum source instead our approach differs from previous ones in two ways first it supports a reasoning inspired by the method of types second the measurement scheme we propose to distinguish the two alternatives not only does that job asymptotically perfectly but also yields additional information about the null hypothesis an example of that is given the measurement is composed of projections onto permutation invariant subspaces thus providing a direct link between one of the most basic tasks in quantum information on the one hand and fundamental objects in representation theory on the other we additionally connect to the representation theory by proving a relation between kostka numbers and quantum states and to state estimation via a generalization of a well known spectral estimation theorem to non i i d sequences

a generalization of quantum stein s lemma given many independent and identically distributed i i d copies of a quantum system described either by the state or called null and alternative hypotheses respectively what is the optimal measurement to learn the identity of the true state in asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying instead of while requiring that the probability that is identified in the place of is bounded by a small fixed number quantum stein s lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of and

error exponents in hypothesis testing for correlated states on a spin chain we study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum chernoff and hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain our results apply to states satisfying a certain factorization property typical examples are the global gibbs states of translation invariant finite range interactions as well as certain finitely correlated states