remainder terms for some quantum entropy inequalities we consider three von neumann entropy inequalities subadditivity pinsker s inequality for relative entropy and the monotonicity of relative entropy for these we state conditions for equality and we prove some new error bounds away from equality including an improved version of pinsker s inequality

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

structure of states which satisfy strong subadditivity of quantum entropy with equality we give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von neumann entropy by combining a result of petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by koashi and imoto we show that such states will have the form of a so called short quantum markov chain which in turn implies that two of the systems are independent conditioned on the third in a physically meaningful sense this characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the holevo bound on accessible information

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

adaptive probability based latin hypercube designs adaptive sampling is an effective method developed mainly for regular regions however experimental regions in irregular shapes are commonly observed in practice motivated by a data center thermal management study a new class of adaptive designs is proposed to accommodate a specific type of irregular region because the adaptive procedure introduces biases into conventional estimators several design unbiased estimators are given for estimating the population mean efficient and easy to compute unbiased estimators are also introduced the proposed method is applied to obtain an adaptive sensor placement plan to monitor and study the thermal distribution in a data center all of the supplemental materials used in this work are available online

the space complexity of recognizing well parenthesized expressions in the streaming model the index function revisited we show an p n t lower bound for the space required by any unidirectional constanterror randomized t pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well parenthesized this proves a conjecture due to magniez mathieu and nayak 2009 and rigorously establishes the peculiar power of bi directional streams over unidirectional ones observed in the algorithms they present the lower bound is obtained by analysing the information that is necessarily revealed by the players about their respective inputs in a two party communication protocol for a variant of the index function

detectability in conventional and adaptive sampling in this paper a simple but very general method is given for estimating a population total with any sampling design when objects in sampled units are observed with imperfect detectability a problem characteristic of many surveys of natural and human populations in the most general case the method consists of dividing the value of the variable of interest associated with each detected object by the detection probability for that object and then proceeding to use the estimation method that would ordinarily be used under the design if there were no detectability problems examples illustrating the method include simple random sampling conventional unequal probability sampling and adaptive cluster sampling

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

an adaptive procedure for sampling animal populations summary we consider an adaptive stratified sampling procedure for animal populations in which the sample size in a given stratum or primary unit depends on the observations obtained in previous strata or primary units this method provides a more efficient way of sampling sparse but highly clustered populations two sampling approaches are considered the first is a design unbiased strategy as the estimate of population density is unbiased through design induced factors such as the random selection of sites in each stratum the second is called a model unbiased strategy as the density estimate is unbiased under the model assumed for the spatial distribution of the population the theory is demonstrated by reference to sampling from a shrimp population

conditional decoupling of quantum information insights from quantum information theory show that correlation measures based on quantum entropy are fundamental tools that reveal the entanglement structure of multipartite states in that spirit groisman popescu and winter phys rev a 72 032317 2005 showed that the quantum mutual information i a b quantifies the minimal rate of noise needed to erase the correlations in a bipartite state of quantum systems ab here we investigate correlations in tripartite systems abe in particular we are interested in the minimal rate of noise needed to apply to the systems ae in order to erase the correlations between a and b given the information in system e in such a way that there is only negligible disturbance on the marginal be we present two such models of conditional decoupling called deconstruction and conditional erasure cost of tripartite states abe our main result is that both are equal to the conditional quantum mutual information i a b e establishing it as an operational measure for tripartite quantum correlations

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

adaptive cluster sampling designs with primary and secondary units summary in adaptive cluster sampling designs an initial probability sample is selected and whenever the observed value of the variable of interest satisfies a given condition units in the neighborhood of that observation are added to the sample in this paper the initial design is selected in terms of primary units while subsequent sampling is in terms of secondary units such initial designs include systematic sampling strip sampling and other forms of classical cluster sampling but because of the subsequent addition to the sample of secondary units in the neighborhood of any secondary unit that satisfies the condition of interest the final clusters of units obtained through the procedure may be quite different in shape from the initial primary units the methods described in this paper apply to such sampling situations as whale surveys in which the research vessel temporarily leaves the selected transect to close in on sighted whales surveys of rare bird species in which initial observations are made at systematically selected sites and additional observations are made in the vicinity of any site at which sufficiently high abundance is observed and aerial walrus surveys in which the aircraft searches to either side of the preselected transect line whenever a congregation of animals is encountered because conventional estimators of the population mean and total are biased with such a procedure estimators that are unbiased under the adaptive designs are presented in this paper variance formulae and unbiased estimators of variance are also given the designs are illustrated using a point pattern representing locations of individuals or objects in a spatially aggregated population for such a population the adaptive designs can be substantially more efficient than their conventional counterparts

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

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

continuity of quantum conditional information we prove continuity of quantum conditional information s 12 2 with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party this can e g be used to prove the continuity of squashed entanglement

adaptive cluster sampling with networks selected without replacement summary in the adaptive cluster design introduced by thompson 1990 a finite population of units under investigation is partitioned into networks on the basis of a specified condition for adding neighbourhoods to a sampled unit an initial sample of units is taken and a network may be sampled more than once in this paper we introduce a modification of the design in which networks are sampled only once two unbiased estimators are considered and the rao blackwell theorem is used to improve them in terms of efficiency the various estimators are compared using two examples

adjusted two stage adaptive cluster sampling an adjusted two stage sampling procedure is discussed for adaptive cluster sampling where some networks are large and others are small a two stage sample is drawn from the large networks and a single stage sample is drawn from the rest the simple random sampling srs procedure is used at the second stage an estimator for the population mean along with its properties is discussed

quotient spaces and critical points of invariant functions for c actions consider a linear action of the group c on x c n 1 we study the fundamental algebraic properties of the sheaves of invariant and basic differential forms for such an action and use these to define an algebraic notion of multiplicity for critical points of functions which are invariant under the c action we also prove a theorem relating the cohomology of the milnor fibre of the critical point on the quotient space with this algebraic multiplicity we also include an appendix showing how to use cech complexes to compute local cohomology

on quantum renyi entropies a new generalization and some properties the renyi entropies constitute a family of information measures that generalizes the well known shannon entropy inheriting many of its properties they appear in the form of unconditional and conditional entropies relative entropies or mutual information and have found many applications in information theory and beyond various generalizations of renyi entropies to the quantum setting have been proposed most prominently petz s quasi entropies and renner s conditional min max and collision entropy however these quantum extensions are incompatible and thus unsatisfactory we propose a new quantum generalization of the family of renyi entropies that contains the von neumann entropy min entropy collision entropy and the max entropy as special cases thus encompassing most quantum entropies in use today we show several natural properties for this definition including data processing inequalities a duality relation and an entropic uncertainty relation

an equivariant version of grauert s oka principle for example a theorem of oka asserts that every holomorphic line bundle over a domain of holomorphy in 12 is holomorphicaily trivial if and only if it is topologically trivial in g3 grauer t proves that the category of bundles over a stein space x with a complex lie group as a structure group is the same up to isomorphism if considered in the topological or complex analytic category we would like to underl ine that grauert s method of proof also has far reaching consequences see e g fr1 fr2 in this article we prove the equivariant version of grauert s oka principle for a compact lie group of holomorphic t ransformations on x the precise statements can be found in w w and w for example as a consequence of the main results we obtain

biometrika centenary sample surveys summary despite karl pearson s stated aim that the evolutionist has to become in the widest sense of the words a registrar general for all forms of life the papers published in biometrika on sample surveys have been mainly theoretical and do not start to appear until the 1940s there has been little work on sampling in the life sciences with the exception of capture recapture methods the major areas have been in variable probability designs the foundations of inference for finite populations analytic surveys and the role of selection mechanisms in inference in recent years the work on survey analysis has been the outcome of an interaction between sample selection mechanisms and mainstream theory with interesting consequences for both areas

homotopy principles for equivariant isomorphisms let g be a reductive complex lie group acting holomorphically on stein manifolds x and y let px x qx and py y qy be the quotient mappings when is there an equivariant biholomorphism of x and y a necessary condition is that the categorical quotients qx and qy are biholomorphic and that the biholomorphism sends the luna strata of qx isomorphically onto the corresponding luna strata of qy fix we demonstrate two homotopy principles in this situation the first result says that if there is a g diffeomorphism x y inducing which is g biholomorphic on the reduced fibres of the quotient mappings then is homotopic through g diffeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y the second result roughly says that if we have a g homeomorphism x y which induces a continuous family of gequivariant biholomorphisms of the fibres px 1 q and py 1 q for q qx and if x satisfies an auxiliary property which holds for most x then is homotopic through g homeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y our results improve upon those of our earlier paper j reine angew math 706 2015 193 214 and use new ideas and techniques

locality at the boundary implies gap in the bulk for 2d peps proving that the parent hamiltonian of a projected entangled pair state peps is gapped remains an important open problem we take a step forward in solving this problem by showing two results first we identify an approximate factorization condition on the boundary state of rectangular subregions that is sufficient to prove that the parent hamiltonian of the bulk 2d peps has a constant gap in the thermodynamic limit second we then show that gibbs state of a local finite range hamiltonian satisfy such condition the proof applies to the case of injective and mpo injective peps employs the martingale method of nearly commuting projectors and exploits a result of araki commun math phys 14 2 120 157 1969 on the robustness of one dimensional gibbs states our result provides one of the first rigorous connections between boundary theories and dynamical properties in an interacting many body system

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

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

how not to renyi generalize the quantum conditional mutual information we study the relation between the quantum conditional mutual information and the quantum renyi divergences considering the totally antisymmetric state we show that it is not possible to attain a proper generalization of the quantum conditional mutual information by optimizing the distance in terms of quantum renyi divergences over the set of all markov states the failure of the approach considered arises from the observation that a small quantum conditional mutual information does not imply that the state is close to a quantum markov state

entanglement of the antisymmetric state we analyse the entanglement of the antisymmetric state in dimension d d and present two main results first we show that the amount of secrecy that can be extracted from the state is low more precisely the distillable key is bounded by o 1 d second we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state entanglement cost is larger than a constant independent of d the second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant finally we note that the regularised relative entropy of entanglement is asymptotically continuous in the state elementary and advanced facts from the representation theory of the unitary group including the concept of plethysm play a central role in the proofs of the main results

squashed entanglement an additive entanglement measure in this paper we present a new entanglement monotone for bipartite quantum states its definition is inspired by the so called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions we derive certain properties of the new measure which we call squashed entanglement it is a lower bound on entanglement of formation and an upper bound on distillable entanglement furthermore it is convex additive on tensor products and superadditive in general continuity in the state is the only property of our entanglement measure which we cannot provide a proof for we present some evidence however that our quantity has this property the strongest indication being a conjectured fannes type inequality for the conditional von neumann entropy this inequality is proved in the classical case

adaptive cluster sampling abstract in many real world sampling situations researchers would like to be able to adaptively increase sampling effort in the vicinity of observed values that are high or otherwise interesting this article describes sampling designs in which whenever an observed value of a selected unit satisfies a condition of interest additional units are added to the sample from the neighborhood of that unit if any of these additional units satisfies the condition still more units may be added sampling designs such as these in which the selection procedure is allowed to depend on observed values of the variable of interest are in contrast to conventional designs in which the entire selection of units to be included in the sample may be determined prior to making any observations because the adaptive selection procedure introduces biases into conventional estimators several estimators are given that are design unbiased for the population mean with the adaptive cluster designs of this article that is the

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

strip adaptive cluster sampling probability proportional to size selection of primary units summary strip adaptive cluster sampling is extended to situations where the primary units have unequal numbers of secondary units primary units are selected with replacement and with probabilities proportional to the number of secondary units the horvitz thompson estimator of the parametric total is used as the basis for estimation two cases are considered basic strip adaptive cluster sampling and strip adaptive cluster sampling under stratification of primary units a numerical example is presented in which the goal is to estimate the number of objects on a site the sampling strategy is useful for ecological studies where a rectangular grid of secondary units is not adequate to cover the study site

min and max entropy in infinite dimensions we consider an extension of the conditional min and max entropies to infinite dimensional separable hilbert spaces we show that these satisfy characterizing properties known from the finite dimensional case and retain information theoretic operational interpretations e g the min entropy as maximum achievable quantum correlation and the max entropy as decoupling accuracy we furthermore generalize the smoothed versions of these entropies and prove an infinite dimensional quantum asymptotic equipartition property to facilitate these generalizations we show that the min and max entropy can be expressed in terms of convergent sequences of finite dimensional min and max entropies which provides a convenient technique to extend proofs from the finite to the infinite dimensional setting

minimum rates of approximate sufficient statistics given a sufficient statistic for a parametric family of distributions one can estimate the parameter without access to the data however the memory or code size for storing the sufficient statistic may nonetheless still be prohibitive indeed for n independent samples drawn from a k nomial distribution with d k 1 degrees of freedom the length of the code scales as d log n o 1 in many applications we may not have a useful notion of sufficient statistics e g when the parametric family is not an exponential family and we may also not need to reconstruct the generating distribution exactly by adopting a shannon theoretic approach in which we allow a small error in estimating the generating distribution we construct various approximate sufficient statistics and show that the code length can be reduced to d 2 log n o 1 we consider errors measured according to the relative entropy and variational distance criteria for the code constructions we leverage rissanen s minimum description length principle which yields a non vanishing error measured according to the relative entropy for the converse parts we use clarke and barron s formula for the relative entropy of a parameterized distribution and the corresponding mixture distribution however this method only yields a weak converse for the variational distance we develop new techniques to achieve vanishing errors and we also prove strong converses the latter means that even if the code is allowed to have a non vanishing error its length must still be at least d 2 log n

quantum conditional relative entropy and quasi factorization of the relative entropy the existence of a positive log sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the markov approximation for classical spin systems such constant was proven to exist under the assumption of a mixing condition in the gibbs measure associated to their dynamics via a quasi factorization of the entropy in terms of the conditional entropy in some sub algebras in this work we analyze analogous quasi factorization results in the quantum case for that we define the quantum conditional relative entropy and prove several quasi factorization results for it as an illustration of their potential we use one of them to obtain a positive log sobolev constant for the heat bath dynamics with product fixed point

cryptographic distinguishability measures for quantum mechanical states this paper mostly expository in nature surveys four measures of distinguishability for quantum mechanical states this is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions namely the probability of an identification error the kolmogorov distance the bhattacharyya coefficient and the shannon 1948 distinguishability as defined through mutual information these measures have a long history of use in statistical pattern recognition and classical cryptography we obtain several inequalities that relate the quantum distinguishability measures to each other one of which may be crucial for proving the security of quantum cryptographic key distribution in another vein these measures and their connecting inequalities are used to define a single notion of cryptographic exponential indistinguishability for two families of quantum states this is a tool that may prove useful in the analysis of various quantum cryptographic protocols

robustness of quantum markov chains if the conditional information of a classical probability distribution of three random variables is zero then it obeys a markov chain condition if the conditional information is close to zero then it is known that the distance minimum relative entropy of the distribution to the nearest markov chain distribution is precisely the conditional information we prove here that this simple situation does not obtain for quantum conditional information we show that for tri partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum markov chain state but the distance can be much greater indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor

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

a lower bound of quantum conditional mutual information in this paper a lower bound of quantum conditional mutual information is obtained by employing the peierls bogoliubov inequality and the golden thompson inequality comparison with the bounds obtained by other researchers indicates that our result is independent of any measurements it may give some new insights into squashed entanglement and perturbations of markov chain states

strong converse for the classical capacity of entanglement breaking and hadamard channels via a sandwiched renyi relative entropy a strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel along with a corresponding achievability statement for rates below the capacity such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication here we show that such a strong converse theorem holds for the classical capacity of all entanglement breaking channels and all hadamard channels the complementary channels of the former these results follow by bounding the success probability in terms of a sandwiched renyi relative entropy by showing that this quantity is subadditive for all entanglement breaking and hadamard channels and by relating this quantity to the holevo capacity prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results

dimensional reduction and quiver bundles the so called hitchin kobayashi correspondence proved by donaldson uhlenbeck and yau establishes that an indecomposable holomorphic vector bundle over a compact kahler manifold admits a hermitian einstein metric if and only if the bundle satisfies the mumford takemoto stability condition in this paper we consider a variant of this correspondence for g equivariant vector bundles on the product of a compact kahler manifold x by a flag manifold g p where g is a complex semisimple lie group and p is a parabolic subgroup the modification that we consider is determined by a filtration of the vector bundle which is naturally defined by the equivariance of the bundle the study of invariant solutions to the modified hermitian einstein equation over xg p leads via dimensional reduction techniques to gauge theoretic equations on x these are equations for hermitian metrics on a set of holomorphic bundles on x linked by morphisms defin ing what we call a quiver bundle for a quiver with relations whose structure is entirely determined by the parabolic subgroup p similarly the corresponding stability condition for the invariant filtration over xg p gives rise to a stability condition for the quiver bundle on x and hence to a hitchin kobayashi correspondence in the simplest case when the flag manifold is the complex projective line one recovers the theory of vortices stable triples and stable chains as studied by bradlow the authors and others

stratified adaptive cluster sampling summary stratified adaptive cluster sampling refers to designs in which following an initial stratified sample additional units are added to the sample from the neighbourhood of any selected unit with an observed value that satisfies a condition of interest if any of the added units in turn satisfies the condition still more units are added to the sample estimation of the population mean or total with the stratified adaptive cluster designs is complicated by the possibility that a selection in one stratum may result in the addition of units from other strata to the sample so that observations in separate strata are not independent since conventional estimators such as the stratified sample mean are biased with the adaptive designs of this paper several types of estimators are developed which are unbiased for the population mean or total with stratified adaptive cluster sampling