local asymptotic normality for finite dimensional quantum systems previous results on local asymptotic normality lan for qubits 16 19 are extended to quantum systems of arbitrary finite dimension d lan means that the quantum statistical model consisting of n identically prepared d dimensional systems with joint state rho otimes n converges as n to a statistical model consisting of classical and quantum gaussian variables with fixed and known covariance matrix and unknown means related to the parameters of the density matrix remarkably the limit model splits into a product of a classical gaussian with mean equal to the diagonal parameters and independent harmonic oscillators prepared in thermal equilibrium states displaced by an amount proportional to the off diagonal elements as in the qubits case 16 lan is the main ingredient in devising a general two step adaptive procedure for the optimal estimation of completely unknown d dimensional quantum states this measurement strategy shall be described in a forthcoming paper 18

asymptotic performance of optimal state estimation in qubit system we derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus it is also proven that this bound can be achieved in any statistical model in the qubit system moreover we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models that is in such a statistical model the quantum correlation can improve the accuracy of the estimation in an asymptotic setting

numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

lagrangians of physics and the game of fisher information transfer the lagrangians of physics arise out of a mathematical game between a smart measurer and nature personified by a demon each contestant wants to maximize his level of fisher information i the game is zero sum by conservation of information in the closed system the payoff of the game introduces a variational principle extreme physical information epi which fixes both the lagrangian and the physical constant of each scenario the epi approach provides an understanding of the relationship between measurement and physical law epi also defines a prescription for constructing lagrangians the prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which i remains invariant as an example when applied to the smart measurement of the space time coordinate of a particle the symmetry used is that between position time space and momentum energy space then the unitary transformation is the fourier one and epi derives the following the equivalence of energy momentum and mass the constancy of planck s parameter h and the lagrangian that implies both the klein gordon equation and the dirac equation of quantum mechanics

generalization of the aoki yoshikawa sectoral productivity model based on extreme physical information principle abstract this paper presents a continuous variable generalization of the aoki yoshikawa sectoral productivity model information theoretical methods from the frieden soffer extreme physical information statistical estimation methodology were used to construct exact solutions both approaches coincide in first order approximation the approach proposed here can be successfully applied in other fields of research

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

a new approach to the cramer rao type bound of the pure state model this paper sheds light on non commutativity in quantum theory as regards theoretical estimation in it we calculate the quantum cramer rao type bound for many cases by use of a newly proposed powerful technique we also discuss the use of collective measurement in statistical estimation

finite element approximations of the lame system with penalized ideal contact boundary conditions we consider finite element approximations of the lame system of elasticity with ideal contact boundary conditions imposed with the penalty method for a polygonal or polyhedral boundary we prove convergence estimates in terms of both the penalty and discretization parameters in the case of a smooth curved boundary we show through a numerical two dimensional example that convergence may not hold due to a babuska s type paradox we also propose and test numerically several remedies

subjective modelling of supply and demand the minimum of fisher information solution abstract two of the present authors have put forward a projective geometry based model of rational trading that implies a model for subjective demand supply profiles if one considers closing of a position as a random process we would like to present the analysis of a subjectivity in such trading models in our model the trader gets the maximal profit intensity when the probability of transaction is 0 5853 we also present a comparison with the model based on the maximum of entropy principle to the best of our knowledge this is one of the first analyses that show a concrete situation in which trader profit optimal value is in the class of price negotiating algorithms strategies resulting in non monotonic demand supply curves of the rest of the world a collective opponent our model suggests that there might be a new class of rational trader strategies that almost neglects the supply demand profile of the market this class emerges when one tries to minimize the information that strategies reveal

a fibre bundle over manifolds of quantum channels and its application to quantum statistics a fibre bundle structure is introduced over manifolds of quantum channels this structure has a close connection with the problem of estimating an unknown quantum channel specified by a parameter it is shown that the quantum fisher information of the family of output states maximized over all input states which quantifies the ultimate statistical distinguishability of the parameter is expressed in terms of a geometrical quantity on the fibre bundle using this formula a criterion for the maximum quantum fisher information of the nth extended channel id n to be o n is derived this criterion further proves that for almost all quantum channels the maximum quantum fisher information increases in the order of o n

predictive inference for best linear combination of biomarkers subject to limits of detection measuring the accuracy of diagnostic tests is crucial in many application areas including medicine machine learning and credit scoring the receiver operating characteristic roc curve is a useful tool to assess the ability of a diagnostic test to discriminate between two classes or groups in practice multiple diagnostic tests or biomarkers are combined to improve diagnostic accuracy often biomarker measurements are undetectable either below or above the so called limits of detection lod in this paper nonparametric predictive inference npi for best linear combination of two or more biomarkers subject to limits of detection is presented npi is a frequentist statistical method that is explicitly aimed at using few modelling assumptions enabled through the use of lower and upper probabilities to quantify uncertainty the npi lower and upper bounds for the roc curve subject to limits of detection are derived where the objective function to maximize is the area under the roc curve in addition the paper discusses the effect of restriction on the linear combination s coefficients on the analysis examples are provided to illustrate the proposed method copyright 2017 john wiley sons ltd

optimal estimation of one parameter quantum channels we explore the task of optimal quantum channel identification and in particular the estimation of a general one parameter quantum process we derive new characterizations of optimality and apply the results to several examples including the qubit depolarizing channel and the harmonic oscillator damping channel we also discuss the geometry of the problem and illustrate the usefulness of using entanglement in process estimation

attaining the ultimate precision limit in quantum state estimation we derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non degenerate our results hold under an assumption called local asymptotic covariance which is weaker than unbiasedness or local unbiasedness the derivation is based on an analysis of the limiting distribution of the estimator s deviation from the true value of the parameter and takes advantage of quantum local asymptotic normality a useful asymptotic characterization of identically prepared states in terms of gaussian states we first prove our results for the mean square error of a special class of models called d invariant and then extend the results to arbitrary models generic cost functions and global state estimation where the unknown parameter is not restricted to a local neighbourhood of the true value the extension includes a treatment of nuisance parameters i e parameters that are not of interest to the experimenter but nevertheless affect the precision of the estimation as an illustration of the general approach we provide the optimal estimation strategies for the joint measurement of two qubit observables for the estimation of qubit states in the presence of amplitude damping noise and for noisy multiphase estimation

penalty finite element approximation of stokes equations with slip boundary conditions we consider the finite element approximation of the stationary stokes equations with slip boundary conditions on a domain with a smooth curved boundary the slip boundary condition is imposed weakly with the penalty method on polygonal domains approaching the smooth domain for taylor hood elements we derive error estimates which depend on the penalty parameter varepsilon the disctretization parameter h h and the approximation error of the normal to the boundary in particular if in the penalty term we use the normal to the polygonal boundary the best convergence order is 2 3 2 3 and it is obtained with varepsilon c h 2 3 c h 2 3 this convergence result shows that babuska s paradox associated to stokes equations with slip boundary conditions is circumvented a numerical example illustrates the theoretical results notably that regularized normal approximations give better approximations and convergence orders

optimal parameter estimation of pauli channels the optimal measurement configuration i e the optimal input quantum state and measurement in the form of a positive operator valued measure with two elements is investigated in this paper for qubit and generalized pauli channels the channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of pauli matrices in the qubit pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the fisher information of the estimated parameters consists of three von neumann measurements together with pure input states directed to the appropriate channel directions extensions of these results for generalized pauli channels are also given furthermore a computationally efficient estimation method is proposed for estimating the channel directions in the qubit pauli channel case the efficiency of this method is illustrated by simulations it is shown that its variance is of order 1 n where n is the number of used measurements

maximum likelihood ratio tests for comparing the discriminatory ability of biomarkers subject to limit of detection in this article we consider comparing the areas under correlated receiver operating characteristic roc curves of diagnostic biomarkers whose measurements are subject to a limit of detection lod a source of measurement error from instruments sensitivity in epidemiological studies we propose and examine the likelihood ratio tests with operating characteristics that are easily obtained by classical maximum likelihood methodology

n body extended channel estimation for low noise parameters the notion of low noise channels was recently proposed and analysed in detail in order to describe noise processes driven by the environment hotta m karasawa t and ozawa m 2005 phys rev a 72 052334 an estimation theory of low noise parameters of channels has also been developed in this paper we address the low noise parameter estimation problem for the n body extension of the dissipative low noise channels we perturbatively calculate the fisher information of the output states in order to evaluate the lower bound of the mean square error of the parameter estimation we show that the maximum of the fisher information over all input states can be attained by a factorized input state in the leading order of the low noise parameter thus to achieve optimal estimation it is not necessary for there to be entanglement of the n subsystems as long as the true low noise parameter is sufficiently small and the channel is properly dissipative

combining multiple biomarker models in logistic regression in medical research there is great interest in developing methods for combining biomarkers we argue that selection of markers should also be considered in the process traditional model variable selection procedures ignore the underlying uncertainty after model selection in this work we propose a novel model combining algorithm for classification in biomarker studies it works by considering weighted combinations of various logistic regression models five different weighting schemes are considered in the article the weights and algorithm are justified using decision theory and risk bound results simulation studies are performed to assess the finite sample properties of the proposed model combining method it is illustrated with an application to data from an immunohistochemical study in prostate cancer

increased efficiency of quantum state estimation using non separable measurements we address the major open problem of evaluating how much increased efficiency in estimation is possible using non separable as opposed to separable measurements of n copies of m level quantum systems first we study the six cases m 2 n 2 7 by computing the 3 3 fisher information matrices for the corresponding optimal measurements recently devised by vidal et al we obtain simple polynomial expressions for the gill massar traces of the products of the inverse of the quantum helstrom information matrix and these fisher information matrices the six traces all have minima of 2n 1 in the pure state limit while for separable measurements the traces can equal n but not exceed it then the result of an analysis for m 3 n 2 leads us to conjecture that for optimal measurements for all m and n the gill massar trace achieves a minimum of 2n 1 m 1 in the pure state limit

linear combinations of multiple diagnostic markers abstract the receiver operating characteristic roc curve is a simple and meaningful measure to assess the usefulness of diagnostic markers to use the information carried by multiple markers we note that fisher s linear discriminant function provides a linear combination of markers to maximize the sensitivity over the entire specificity range uniformly under the multivariate normal distribution model with proportional covariance matrices with no restriction on covariance matrices we also provide a solution of the best linear combination of markers in the sense that the area under the roc curve of this combination is maximized among all possible linear combinations we illustrate both situations discussed in the article with a cancer clinical trial data

how continuous quantum measurements in finite dimensions are actually discrete we show that in finite dimensions a quantum measurement with a continuous set of outcomes can be always realized as a continuous random choice of measurements with a finite number of outcomes

towards optimal finite element error estimates for the penalized dirichlet problem in a domain with curved boundary we consider the finite element approximation of an elliptic problem with homogeneous dirichlet boundary conditions on a curved boundary and imposed using the penalty method we establish optimal h 1 error estimates with suitable assumptions on the penalty parameter e as a function of the elements size h our focus is on establishing these results with least restrictive assumptions possible on this dependency

quantum parameter estimation of a generalized pauli channel we present a quantum parameter estimation theory for a generalized pauli channel d d where the parameter is regarded as a coordinate system of the probability simplex d2 1 we show that for each degree n of extension id n d d n d d n the sld fisher information matrix for the output states takes the maximum when the input state is an n tensor product of a maximally entangled state me d d we further prove that for the corresponding quantum cramer rao inequality there is an efficient estimator if and only if the parameter is m affine in d2 1 these results rely on the fact that the family id me of output states can be identified with d2 1 in the sense of quantum information geometry this fact further allows us to investigate submodels of generalized pauli channels in a unified manner

fidelity susceptibility and quantum fisher information for density operators with arbitrary ranks abstract taking into account the density matrices with non full ranks we show that the fidelity susceptibility is determined by the support of the density matrix combining with the corresponding expression of the quantum fisher information we rigorously prove that the fidelity susceptibility is proportional to the quantum fisher information as this proof can be naturally extended to the full rank case this proportional relation is generally established for density matrices with arbitrary ranks furthermore we give an analytical expression of the quantum fisher information matrix and show that it can also be represented in the density matrix s support

geometry of optimal estimation scheme for su d channels the problem of estimating an unknown su d channel u u u is studied based on the quantum cramer rao inequality it is shown that the minimum estimation error is of o 1 n2 where n is the degree of extension of the channel the mechanism behind this asymptotic behaviour is investigated from a differential geometrical point of view

enhanced functional evaluation for the finite element penalty method abstract we analyze the impact of using the penalty method on the estimation of a prescribed solution functional or quantity of interest qoi specifically we consider the use of penalty methods to enforce dirichlet boundary constraints focusing our attention on boundary fluxes as qois we propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter we show that the new estimator reduces the error arising from the use of the penalty method from o to o a well posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method errors in the enhanced flux estimator are then controlled using adjoint based techniques several numerical experiments are presented to demonstrate that the enhanced estimator in combination with adjoint error estimation allows one to efficiently control both the discretization and penalty errors in target qois

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

optimal feedback scheme and universal time scaling for hamiltonian parameter estimation time is a valuable resource and it is expected that a longer time period should lead to better precision in hamiltonian parameter estimation however recent studies in quantum metrology have shown that in certain cases more time may even lead to worse estimations which puts this intuition into question in this letter we show that by including feedback controls this intuition can be restored by deriving asymptotically optimal feedback controls we quantify the maximal improvement feedback controls can provide in hamiltonian parameter estimation and show a universal time scaling for the precision limit under the optimal feedback scheme our study reveals an intriguing connection between noncommutativity in the dynamics and the gain of feedback controls in hamiltonian parameter estimation

frieden wave function representations via an einstein podolsky rosen bohm experiment the appearance of the spin 1 2 and spin 1 representations in the frieden soffer extreme physical information epi statistical approach to the einstein podolsky rosen bohm epr bohm experiment is shown in order to obtain the epr bohm result in addition to the observed structural and variational information principles of the epi method the condition of the regularity of the probability distribution is used the observed structural information principle is obtained from the analyticity of the logarithm of the likelihood function it is suggested that due to the self consistent analysis of both information principles quantum mechanics is covered by the statistical information theory the estimation of the angle between the analyzers in the epr bohm experiment is discussed

combining markers with and without the limit of detection in this paper we consider the combination of markers with and without the limit of detection lod lod is often encountered when measuring proteomic markers because of the limited detecting ability of an equipment or instrument it is difficult to measure markers at a relatively low level suppose that after some monotonic transformation the marker values approximately follow multivariate normal distributions we propose to estimate distribution parameters while taking the lod into account and then combine markers using the results from the linear discriminant analysis our simulation results show that the roc curve parameter estimates generated from the proposed method are much closer to the truth than simply using the linear discriminant analysis to combine markers without considering the lod in addition we propose a procedure to select and combine a subset of markers when many candidate markers are available the procedure based on the correlation among markers is different from a common understanding that a subset of the most accurate markers should be selected for the combination the simulation studies show that the accuracy of a combined marker can be largely impacted by the correlation of marker measurements our methods are applied to a protein pathway dataset to combine proteomic biomarkers to distinguish cancer patients from non cancer patients

an upper bound on the number of compatible parameters in simultaneous quantum estimation simultaneous estimation of multiple parameters is required in many practical applications a lower bound on the variance of simultaneous estimation is given by the quantum fisher information matrix this lower bound is however not necessarily achievable there exists a necessary and sufficient condition for its achievability it is unknown how many parameters can be estimated while satisfying this condition in this paper we analyse an upper bound on the number of such parameters through linear algebraic techniques this upper bound depends on the algebraic structure of the quantum system used as a probe we explicitly calculate this bound for two quantum systems single qubit and two qubit x states

convex optimization based parameter estimation and experiment design for pauli channels in this technical note we study the problem of parameter estimation for quantum channels which are completely positive maps acting on quantum states we focus on an important subclass of channels called pauli channels which are characterized by a certain set of vectors directions and a number of scalar parameters for the case where the directions are known a special parametrization turns the parameter estimation problem into a convex optimization problem for the case of unknown directions we give a simple algorithm to estimate the directions for qubit pauli channels these results assume that the identification experiment configuration is given namely a set of quantum states are given on which the channel acts and a set of positive measurement operator s are fixed from which information is gathered in the second part of the technical note we consider the problem of determining the experiment design namely determining states and measurements for optimal parameter identification of the channels we formalize this problem as a maximization problem for fisher information and assuming known channel directions prove that this problem is convex we also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal for qubit pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions we illustrate the results of the technical note with numerical examples and simulations

bloch vectors for qudits we present three different matrix bases that can be used to decompose density matrices of d dimensional quantum systems so called qudits the generalized gell mann matrix basis the polarization operator basis and the weyl operator basis such a decomposition can be identified with a vector the bloch vector i e a generalization of the well known qubit case and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices we present a new method to decompose density matrices via so called standard matrices consider the important case of an isotropic two qudit state and decompose it according to each basis in the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin 1 measurements which is appropriate for an experimental realization

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

comparison between the cramer rao and the mini max approaches in quantum channel estimation in a unified viewpoint in quantum channel estimation we compare the cramer rao and the mini max approaches which gives the bayesian bound in the group covariant model for this purpose we introduce the local asymptotic mini max bound whose maximum is shown to be equal to the asymptotic limit of the mini max bound it is shown that the local asymptotic mini max bound is strictly larger than the cramer rao bound in the phase estimation case while both bounds coincide when the minimum mean square error decreases with the order o frac 1 n we also derive a sufficient condition so that the minimum mean square error decreases with the order o frac 1 n

finite element approximation of the dirichlet problem using the boundary penalty method summary this paper considers a finite element approximation of the dirichlet problem for a second order self adjoint elliptic equation a u f in a region f2cn n 2 or 3 by the boundary penalty method if the finite element space defined over d h a union of elements has approximation power h in the l 2 norm then i for f2 d h convex polyhedral we show that choosing the penalty parameter e h x with 2 k yields optimal h 1 and l 2 error bounds if u hk i q

estimating the spectrum of a density matrix with locc the problem of estimating the spectrum of a density matrix is considered other problems such as bipartite pure state entanglement can be reduced to spectrum estimation a local operations and classical communication locc measurement strategy is shown which is asymptotically optimal this means that for a very large number of copies it becomes unnecessary to perform collective measurements which should be more difficult to implement in practice

explicit formula for the holevo bound for two parameter qubit state estimation problem the main contribution of this paper is to derive an explicit expression for the fundamental precision bound the holevo bound for estimating any two parameter family of qubit mixed states in terms of quantum versions of fisher information the obtained formula depends solely on the symmetric logarithmic derivative sld the right logarithmic derivative rld fisher information and a given weight matrix this result immediately provides necessary and sufficient conditions for the following two important classes of quantum statistical models the holevo bound coincides with the sld cramer rao bound and it does with the rld cramer rao bound one of the important results of this paper is that a general model other than these two special cases exhibits an unexpected property the structure of the holevo bound changes smoothly when the weight matrix varies in particular it always coincides with the rld cramer rao bound for a certain choice of the weight matrix several examples illustrate these findings

probing the qudit depolarizing channel for the quantum depolarizing channel with any finite dimension we compare three schemes for channel identification unentangled probes probes maximally entangled with an external ancilla and maximally entangled probe pairs this comparison includes cases where the ancilla is itself depolarizing and where the probe is circulated back through the channel before measurement compared on the basis of quantum fisher information gained per channel use we find broadly that entanglement with an ancilla dominates the other two schemes but only if entanglement is cheap relative to the cost per channel use and only if the external ancilla is well shielded from depolarization we arrive at these results by a relatively simple analytical means a separate more complicated analysis for partially entangled probes shows for the qudit depolarizing channel that any amount of probe entanglement is advantageous and that the greatest advantage comes with maximal entanglement

an estimation theoretical characterization of coherent states we introduce a class of quantum pure state models called the coherent models a coherent model is an even dimensional manifold of pure states whose tangent space is characterized by a symplectic structure in a rigorous framework of noncommutative statistics it is shown that a coherent model inherits and expands the original spirit of the minimum uncertainty property of coherent states

evaluating the holevo cramer rao bound for multiparameter quantum metrology only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed this is due to the incompatibility of observables the fundamental bound for multiparameter quantum estimation is the holevo cram er rao bound hcrb whose evaluation has so far remained elusive for finite dimensional systems we recast its evaluation as a semidefinite program with reduced size for rank deficient states we show that it also satisfies strong duality we use this result to study phase and loss estimation in optical interferometry and three dimensional magnetometry with noisy multiqubit systems for the former we show that in some regimes it is possible to attain the hcrb with the optimal single copy measurement for phase estimation for the latter we show a nontrivial interplay between the hcrb and incompatibility and provide numerical evidence that projective single copy measurements attain the hcrb in the noiseless 2 qubit case

analysis of a convenient information bound for general quantum channels open questions from sarovar and milburn 2006 j phys a math gen 39 8487 are answered sarovar and milburn derived a convenient upper bound for the fisher information of a one parameter quantum channel they showed that for quasi classical models their bound is achievable and they gave a necessary and sufficient condition for positive operator valued measures povms attaining this bound they asked i whether their bound is attainable more generally ii whether explicit expressions for optimal povms can be derived from the attainability condition we show that the symmetric logarithmic derivative sld quantum information is less than or equal to the sm bound i e h c and we find conditions for equality as the fisher information is less than or equal to the sld quantum information i e fm h we can deduce when equality holds in fm c equality does not hold for all channels as a consequence the attainability condition cannot be used to test for optimal povms for all channels these results are extended to multi parameter channels

duality versus dual flatness in quantum information geometry we investigate questions in quantum informationgeometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finite dimensional hilbert spaces we show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal we prove moreover that there is never a dually flat structure on the set of pure states thus any general theory of quantum informationgeometry that involves duality concepts must inevitably be based on dual structures which are nonflat

on linear combinations of biomarkers to improve diagnostic accuracy we consider combining multiple biomarkers to improve diagnostic accuracy su and liu derived the linear combinations that maximize the area under the receiver operating characteristic roc curves these linear combinations however may have unsatisfactory low sensitivity over a certain range of desired specificity in this paper we consider maximizing sensitivity over a range of specificity we first present a simpler proof for su and liu s main theorem and further investigate some other optimal properties of their linear combinations we then derive alternative linear combinations that have higher sensitivity over a range of high or low specificity the methods are illustrated using data from a study evaluating biomarkers for coronary heart disease