one and a half quantum de finetti theorems when n k systems of an n partite permutation invariant state are traced out the resulting state can be approximated by a convex combination of tensor product states this is the quantum de finetti theorem in this paper we show that an upper bound on the trace distance of this approximation is given by 2 frac kd 2 n where d is the dimension of the individual system thereby improving previously known bounds our result follows from a more general approximation theorem for representations of the unitary group consider a pure state that lies in the irreducible representation u mu nu subset u mu otimes u nu of the unitary group u d for highest weights and let be the state obtained by tracing out u then is close to a convex combination of the coherent states u mu g v mu rangle where g in u d and v mu rangle is the highest weight vector in u

geometry of sets of quantum maps a generic positive map acting on a high dimensional system is not completely positive we investigate the set a of positive trace preserving maps acting on density matrices of size n and a sequence of its nested subsets the sets of maps which are b decomposable c completely positive and d extended by identity impose positive partial transpose and e are superpositive working with the hilbert schmidt euclidean measure we derive tight explicit two sided bounds for the volumes of all five sets a sample consequence is the fact that as n increases a generic positive map becomes not decomposable and a fortiori not completely positive due to the jamiolkowski isomorphism the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets such as states with positive partial transpose or separable states or supersets our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity high dimensional probability and geometry of banach spaces some of which are not standard

on the structure of the body of states with positive partial transpose we show that the convex set of separable mixed states of the 2 2s ystem is a body of a constant height this fact is used to prove that the probability of finding a random state to be separable equals twice the probability of finding a random boundary state to be separable provided that the random states are generated uniformly with respect to the hilbert schmidt euclidean measure an analogous property holds for the set of positive partial transpose states for an arbitrary bipartite system

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

a most compendious and facile quantum de finetti theorem in its most basic form the finite quantum de finetti theorem states that the reduced k partite density operator of an n partite symmetric state can be approximated by a convex combination of k fold product states variations of this result include renner s exponential approximation by almost product states a theorem which deals with certain triples of representations of the unitary group and the result of d cruz et al e print quant ph 0606139 phys rev lett 98 160406 2007 for infinite dimensional systems we show how these theorems follow from a single general de finetti theorem for representations of symmetry groups each instance corresponding to a particular choice of symmetry group and representation of that group this gives some insight into the nature of the set of approximating states and leads to some new results including an exponential theorem for infinite dimensional systems

quantum de finetti theorems under local measurements with applications quantum de finetti theorems are a useful tool in the study of correlations in quantum multipartite states in this paper we prove two new quantum de finetti theorems both showing that under tests formed by local measurements in each of the subsystems one can get an exponential improvement in the error dependence on the dimension of the subsystems we also obtain similar results for non signaling probability distributions we give several applications of the results to quantum complexity theory polynomial optimization and quantum information theory the proofs of the new quantum de finetti theorems are based on information theory in particular on the chain rule of mutual information the results constitute improvements and generalizations of a recent de finetti theorem due to brandao christandl and yard

approximating the set of separable states using the positive partial transpose test the positive partial transpose test is one of the main criteria for detecting entanglement and the set of states with positive partial transpose is considered as an approximation of the set of separable states however we do not know to what extent this criterion as well as the approximation is efficient in this paper we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states more precisely we prove that as the dimension of the space tends to infinity the maximum trace distance of a positive partial transpose state from separable states tends to 1 using similar techniques we show that the same result holds for other well known separability criteria such as reduction criterion majorization criterion and symmetric extension criterion we also bring in evidence that the sets of positive partial transpose states and separable states have totally different shapes

asymptotic error rates in quantum hypothesis testing we consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero this leads to the identification of the quantum generalisation of the classical chernoff distance which is the corresponding quantity in classical symmetric hypothesis testing

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

a quantum version of sanov s theorem we present a quantum version of sanov s theorem focussing on a hypothesis testing aspect of the theorem there exists a sequence of typical subspaces for a given set of stationary quantum product states asymptotically separating them from another fixed stationary product state analogously to the classical case the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set while in the classical case the separating subsets can be chosen universally in the sense that they depend only on the chosen set of i i d processes in the quantum case the choice of the separating subspaces depends additionally on the reference state

spectral families of projections in hardy spaces strongly continuous one parameter groups of isometries in the reflexive hardy spaces of the disc d and the half plane are considered in the light of the author s previous joint result with h benzinger and t a gillespie which generalizes stone s theorem for unitary groups to arbitrary banach spaces it is shown that every such group tt of hardy space isometries has a spectral decomposition with respect to a suitable projection valued function on the real line r as in the classical statement of stone s theorem in hilbert space the relevant type of projection valued function is known as a spectral family this circle of ideas is intimately bound up with harmonic analysis particularly in hpr in particular if the group tt acts in hpd and is associated as in forelli s theorem with a group of parabolic mobius transformations of d then it can be analyzed by way of the translation group on hpr the stone type spectral family of the latter is shown to be obtained by restriction of the m riesz projections to hpr by this means a concrete description of the stone type spectral family for a parabolic isometric group on hpd is obtained a pleasant by product of the parabolic case is absorption of the classical paley wiener theorem for hpr 1 p 2 into the framework of the generalized stone s theorem

limitations of semidefinite programs for separable states and entangled games semidefinite programs sdps are a framework for exact or approximate optimization with widespread application in quantum information theory we introduce a new method for using reductions to construct integrality gaps for sdps meaning instances where the sdp value is far from the true optimum these are based on new limitations on the sum of squares sos hierarchy in approximating two particularly important sets in quantum information theory where previously no omega 1 round integrality gaps were known integrality gaps for the 2 rightarrow 4 norm had previously been sought due to its connection to small set expansion sse and unique games ug in both cases no go theorems were previously known based on computational assumptions such as the exponential time hypothesis eth which asserts that 3 sat requires exponential time to solve our unconditional results achieve the same parameters as all of these previous results for separable states or as some of the previous results for quantum correlations in some cases we can make use of the framework of lee raghavendra steurer lrs to establish integrality gaps for any sdp extended formulation not only the sos hierarchy our hardness result on separable states also yields a dimension lower bound of approximate disentanglers answering a question of watrous and aaronson et al these results can be viewed as limitations on the monogamy principle the ppt test the ability of tsirelson type bounds to restrict quantum correlations as well as the sdp hierarchies of doherty parrilo spedalieri navascues pironio acin and berta fawzi scholz indeed a wide range of past work in quantum information can be described as using an sdp on one of the above two problems and our results put broad limits on these lines of argument

spectral decompositions of one parameter groups of isometries on hardy spaces spectral decompositions of strongly continuous one parameter groups of surjective isometries on hardy spaces of the disk d and the torus t2 are examined a concrete descriptiofi of the pointwise action of these decompositions is presented mainly in the parabolic case leading to a complete description of the action of the partial sum operators of m riesz when carried from lp r to hp d 1 p 2 the pointwise action of the spectral decompositions of these isometric groups on hp t2 1 p oo is also examined and concrete descriptions are derived mainly in the parabolic case

nonadditivity of renyi entropy and dvoretzky s theorem the goal of this note is to show that the analysis of the minimum output p renyi entropy of a typical quantum channel essentially amounts to applying milman s version of dvoretzky s theorem about almost euclidean sections of high dimensional convex bodies this conceptually simplifies the nonconstructive argument by hayden winter disproving the additivity conjecture for the minimal output p renyi entropy for p 1

correlated entanglement distillation and the structure of the set of undistillable states we consider entanglement distillation under the assumption that the input states are allowed to be correlated among each other we hence replace the usually considered independent and identically distributed hypothesis by the weaker assumption of merely having identical reductions we find that whether a state is then distillable or not is only a property of these reductions and not of the correlations that are present in the input state this is shown by establishing an appealing relation between the set of copy correlated undistillable states and the standard set of undistillable states the former turns out to be the convex hull of the latter as an example of the usefulness of our approach to the study of entanglement distillation we prove a new activation result which generalizes earlier findings it is shown that for every entangled state and every k there exists a copy correlated k undistillable state such that is single copy distillable finally the relation of our results to the conjecture about the existence of bound entangled states with a nonpositive partial transpose is discussed

on the bures volume of separable quantum states we obtain two sided estimates for the bures volume of an arbitrary subset of the set of n n density matrices in terms of the hilbert schmidt volume of that subset for general subsets our results are essentially optimal for large n as applications we derive in particular nontrivial lower and upper bounds for the bures volume of sets of separable states and for sets of states with positive partial transpose

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

vector valued wavelets and vector filter banks in this paper we introduce vector valued multiresolution analysis and vector valued wavelets for vector valued signal spaces we construct vector valued wavelets by using paraunitary vector filter bank theory in particular we construct vector valued meyer wavelets that are band limited we classify and construct vector valued wavelets with sampling property as an application of vector valued wavelets multiwavelets can be constructed from vector valued wavelets we show that certain linear combinations of known scalar valued wavelets may yield multiwavelets we then present discrete vector wavelet transforms for discrete time vector valued or blocked signals which can be thought of as a family of unitary vector transforms

smooth entropies and the quantum information spectrum many of the traditional results in information theory such as the channel coding theorem or the source coding theorem are restricted to scenarios where the underlying resources are independent and identically distributed i i d over a large number of uses to overcome this limitation two different techniques the information spectrum method and the smooth entropy framework have been developed independently they are based on new entropy measures called spectral entropy rates and smooth entropies respectively that generalize shannon entropy in the classical case and von neumann entropy in the more general quantum case here we show that the two techniques are closely related more precisely the spectral entropy rate can be seen as the asymptotic limit of the smooth entropy our results apply to the quantum setting and thus include the classical setting as a special case

how often is a random quantum state k entangled the set of trace preserving positive maps acting on density matrices of size d forms a convex body we investigate its nested subsets consisting of kpositive maps where k 2 d working with the measure induced by the hilbert schmidt distance we derive asymptotically tight bounds for the volumes of these sets our results strongly suggest that the inner set of k 1 positive maps forms a small fraction of the outer set of k positive maps these results are related to analogous bounds for the relative volume of the sets of k entangled states describing a bipartite d d system

a de finetti representation for finite symmetric quantum states consider a symmetric quantum state on an n fold product space that is the state is invariant under permutations of the n subsystems we show that conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems the state in the remaining subsystems is close to having product form this immediately generalizes the so called de finetti representation to the case of finite symmetric quantum states

distinguishing separable and entangled states we show how to design families of operational criteria that distinguish entangled from separable quantum states the simplest of these tests corresponds to the well known peres horodecki positive partial transpose ppt criterion and the more complicated tests are strictly stronger the new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs we successfully applied the results to many low dimensional states from the literature where the ppt test fails as a by product of the criteria we provide an explicit construction of the corresponding entanglement witnesses

the chernoff lower bound for symmetric quantum hypothesis testing we consider symmetric hypothesis testing in quantum statistics where the hypotheses are density operators on a finite dimensional complex hilbert space representing states of a finite quantum system we prove a lower bound on the asymptotic rate exponents of bayesian error probabilities the bound represents a quantum extension of the chernoff bound which gives the best asymptotically achievable error exponent in classical discrimination between two probability measures on a finite set in our framework the classical result is reproduced if the two hypothetic density operators commute recently it has been shown elsewhere phys rev lett 98 2007 160504 that the lower bound is achievable also in the generic quantum noncommutative case this implies that our result is one part of the definitive quantum chernoff bound

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

partial parameterization of orthogonal wavelet matrix filters in this paper we propose a procedure which allows the construction of a large family of fir dxd matrix wavelet filters by exploiting the one to one correspondence between qmf systems and orthogonal operators which commute with the shifts by two a characterization of the class of filters of full rank type that can be obtained with such procedure is given in particular we restrict our attention to a special construction based on the representation of so 2d in terms of the elements of its lie algebra explicit expressions for the filters in the case d 2 are given as a result of a local analysis of the parameterization obtained from perturbing the haar system

separable states are more disordered globally than locally a remarkable feature of quantum entanglement is that an entangled state of two parties alice a and bob b may be more disordered locally than globally that is s a s a b where s is the von neumann entropy it is known that satisfaction of this inequality implies that a state is nonseparable in this paper we prove the stronger result that for separable states the vector of eigenvalues of the density matrix of system ab is majorized by the vector of eigenvalues of the density matrix of system a alone this gives a strong sense in which a separable state is more disordered globally than locally and a new necessary condition for separability of bipartite states in arbitrary dimensions

spectral decompositions of hyperbolic isometric groups in hardy spaces of the disk abstract we introduce a mechanism that can be used to achieve concrete descriptions of the pointwise action of the spectral decompositions for the hyperbolic type groups of isometries on hp d 1

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

unextendible product bases and bound entanglement an unextendible product basis upb for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state we give examples of upbs and show that the uniform mixed state over the subspace complementary to any upb is a bound entangled state we exhibit a tripartite 2 3 2 3 2 upb whose complementary mixed state has tripartite entanglement but no bipartite entanglement i e all three corresponding 2 3 4 bipartite mixed states are unentangled we show that members of a upb are not perfectly distinguishable by local positive operator valued measurements and classical communication s0031 9007 99 09360 6

short wavelets and matrix dilation equations scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter in a two band orthogonal filter bank for multifilters those coefficients are matrices this gives a new block structure for the filter bank and leads to multiple scaling functions and wavelets geronimo hardin and massopust see j approx theory vol 78 p 373 401 1994 constructed two scaling functions that have extra properties not previously achieved the functions spl phi sub 1 and spl phi sub 2 are symmetric linear phase and they have short support two intervals or less while their translates form an orthogonal family for any single function spl phi apart from haar s piecewise constants those extra properties are known to be impossible the novelty is to introduce 2 spl times 2 matrix coefficients while retaining orthogonality of the multiwavelets this note derives the properties of spl phi sub 1 and spl phi sub 2 from the matrix dilation equation that they satisfy then our main step is to construct associated wavelets two wavelets for two scaling functions the properties were derived by geronimo hardin and massopust from the iterated interpolation that led to spl phi 1 and spl phi sub 2 one pair of wavelets was found earlier by direct solution of the orthogonality conditions using mathematica our construction is in parallel with recent progress by hardin and geronimo to develop the underlying algebra from the matrix coefficients in the dilation equation in another language to build the 4 spl times 4 paraunitary polyphase matrix in the filter bank the short support opens new possibilities for applications of filters and wavelets near boundaries

one shot rates for entanglement manipulation under non entangling maps we obtain expressions for the optimal rates of one shot entanglement manipulation under operations which generate a negligible amount of entanglement as the optimal rates for entanglement distillation and dilution in this paradigm we obtain the max and min relative entropies of entanglement the two logarithmic robustnesses of entanglement and smoothed versions thereof this gives a new operational meaning to these entanglement measures moreover by considering the limit of many identical copies of the shared entangled state we partially recover the recently found reversibility of entanglement manipulation under the class of operations which asymptotically do not generate entanglement

an improved semidefinite programming hierarchy for testing entanglement we present a stronger version of the doherty parrilo spedalieri dps hierarchy of approximations for the set of separable states unlike dps our hierarchy converges exactly at a finite number of rounds for any fixed input dimension this yields an algorithm for separability testing that is singly exponential in dimension and polylogarithmic in accuracy our analysis makes use of tools from algebraic geometry but our algorithm is elementary and differs from dps only by one simple additional collection of constraints