quantum homodyne tomography as an informationally complete positive operator valued measure we define a positive operator valued measure e on describing the measurement of randomly sampled quadratures in quantum homodyne tomography and we study its probabilistic properties moreover we give a mathematical analysis of the relation between the description of a state in terms of e and the description provided by its wigner transform

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

quantum key distribution based on private states unconditional security over untrusted channels with zero quantum capacity in this paper we prove unconditional security for a quantum key distribution qkd protocol based on distilling pbits twisted ebits from an arbitrary untrusted state that is claimed to contain distillable key our main result is that we can verify security using only public communication via parameter estimation of the given untrusted state the technique applies even to bound entangled states thus extending qkd to the regime where the available quantum channel has zero quantum capacity we also show how to convert our purification based qkd schemes to prepare measure schemes

generalized coherent states and the diagonal representation for operators we consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with a unitary irreducible representation of a compact lie group we show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining clebsch gordan theory and the reciprocity theorem associated with induced unitary group representations applications to several examples involving su 2 su 3 and the heisenberg weyl group are presented showing that there are simple examples of generalized coherent states which do not meet these conditions our results are relevant for phase space description of quantum mechanics and quantum state reconstruction problems

convolutions for berezin quantization and berezin lieb inequalities concepts and results from quantum harmonic analysis such as the convolution between functions and operators or between two operators are identified as the appropriate setting for berezin quantization and berezin lieb inequalities based on this insight we provide a rigorous approach to the generalized phase space representation introduced by klauder skagerstam and their variants of berezin lieb inequalities in this setting hence our presentation of the results of klauder skagerstam gives a more conceptual framework which yields as a byproduct an interesting perspective on the connection between the berezin quantization and weyl quantization concepts and results from quantum harmonic analysis such as the convolution between functions and operators or between two operators are identified as the appropriate setting for berezin quantization and berezin lieb inequalities based on this insight we provide a rigorous approach to the generalized phase space representation introduced by klauder skagerstam and their variants of berezin lieb inequalities in this setting hence our presentation of the results of klauder skagerstam gives a more conceptual framework which yields as a byproduct an interesting perspective on the connection between the berezin quantization and weyl quantization

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

frames from imprimitivity systems let p v be an irreducible imprimitivity system for a group h based on a dual group of an abelian group a and acting on a hilbert space h given h we find necessary and sufficient conditions in order that the set of vectors a vh 1 dp a a h h be a frame in h moreover we apply these results to some examples that are considered in the literature in the context of square integrability modulo a coset space

the schwinger su 3 construction i multiplicity problem and relation to induced representations the schwinger oscillator operator representation of su 3 is analyzed with particular reference to the problem of multiplicity of irreducible representations it is shown that with the use of an sp 2 r unitary representation commuting with the su 3 representation the infinity of occurrences of each su 3 irreducible representation can be handled in complete detail a natural generating representation for su 3 containing each irreducible representation exactly once is identified within a subspace of the schwinger construction and this is shown to be equivalent to an induced representation of su 3

group theoretical quantum tomography the paper is devoted to the mathematical foundation of quantum tomography using the theory of square integrable representations of unimodular lie groups

quantum harmonic analysis on phase space relative to an irreducible representation of the canonical commutation relations convolutions between quantum mechanical operators and between functions and operators are defined for which the usual weyl transform acts as a fourier transform basic properties of these operations are developed in close analogy to harmonic analysis on r2n using the quantum version of wiener s approximation theorem a natural one to one correspondence between the closed phase space translation invariant subspaces of classical and quantum observables is established

su n coherent states and irreducible schwinger bosons we exploit the su n irreducible schwinger boson to construct su n coherent states this construction of an su n coherent state is analogous to the construction of the simplest heisenberg weyl coherent states the coherent states belonging to irreducible representations of su n are labeled by the eigenvalues of n 1 su n casimir operators and characterized by n 1 complex orthonormal vectors describing the su n group manifold

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

compressive wideband power spectrum estimation in several applications such as wideband spectrum sensing for cognitive radio only the power spectrum a k a the power spectral density is of interest and there is no need to recover the original signal itself in addition high rate analog to digital converters adcs are too power hungry for direct wideband spectrum sensing these two facts have motivated us to investigate compressive wideband power spectrum sensing which consists of a compressive sampling procedure and a reconstruction method that is able to recover the unknown power spectrum of a wide sense stationary signal from the obtained sub nyquist rate samples it is different from spectrum blind sampling sbs which aims at reconstructing the original signal instead of the power spectrum in this paper a solution is first presented based on a periodic sampling procedure and a simple least squares reconstruction method we evaluate the reconstruction process both in the time and frequency domain then we examine two possible implementations for the compressive sampling procedure namely complex gaussian sampling and multicoset sampling although we mainly focus on the latter a new type of multicoset sampling is introduced based on the so called minimal sparse ruler problem next we analyze the statistical properties of the estimated power spectrum the computation of the mean and the covariance of the estimates allows us to calculate the analytical normalized mean squared error nmse of the estimated power spectrum further when the received signal is assumed to contain only circular complex zero mean gaussian i i d noise the computed mean and covariance can be used to derive a suitable detection threshold simulation results underline the promising performance of our proposed approach note that all benefits of our method arise without putting any sparsity constraints on the power spectrum

wigner distributions and quantum mechanics on lie groups the case of the regular representation we consider the problem of setting up the wigner distribution for states of a quantum system whose configuration space is a lie group the basic properties of wigner distributions in the familiar cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from n dimensional euclidean space rn to a lie group g the notion of canonical momentum is carefully analyzed and the meanings of marginal probability distributions and their recovery from the wigner distribution are clarified for the case of compact g an explicit definition of the wigner distribution is proposed possessing all the required properties geodesic curves in g which help introduce a notion of the midpoint of two group elements play a central role in the construction

a fermionic de finetti theorem quantum versions of de finetti s theorem are powerful tools yielding conceptually important insights into the security of key distribution protocols or tomography schemes and allowing one to bound the error made by mean field approaches such theorems link the symmetry of a quantum state under the exchange of subsystems to negligible quantum correlations and are well understood and established in the context of distinguishable particles in this work we derive a de finetti theorem for finite sized majorana fermionic systems it is shown much reflecting the spirit of other quantum de finetti theorems that a state which is invariant under certain permutations of modes loses most of its anti symmetric character and is locally well described by a mode separable state we discuss the structure of the resulting mode separable states and establish in specific instances a quantitative link to the quality of the hartree fock approximation of quantum systems we hint at a link to generalized pauli principles fo

operator integrals and phase space observables the operator integral f de of a complex valued measurable function f with respect to a positive operator measure e is considered if f is a neumark dilation of e into a projection measure then the projected operator integral pr fdf is a restriction of the operator f de necessary and sufficient conditions for the equality pr f df f de are obtained the results are applied to determine the moment operators of the phase space observables generated by the number states

on exchangeable continuous variable systems we investigate permutation invariant continuous variable quantum states and their covariance matrices we provide a complete characterization of the latter with respect to permutation invariance and exchangeability and representing convex combinations of tensor power states on the level of the respective density operators this leads to necessary criteria for all these properties which become necessary and sufficient for gaussian states for these we use the derived results to provide de finetti type theorems for various distance measures

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

the cumulant theory of cyclostationary time series ii development and applications for pt i see ibid vol 42 no 12 p 3387 3408 1994 the development of the theory of nonlinear processing of cyclostationary time series that is initiated in part i is continued a new type of cumulant for complex valued variables is introduced and used to generalize the temporal and spectral moments and cumulants for cyclostationary time series from real valued to complex valued time series the relations between the temporal and spectral moments and cumulants at the inputs and outputs of several signal processing operations are determined formulas for the temporal and spectral cumulants of complex valued pulse amplitude modulated time series are derived estimators for the temporal moments and cumulants and for the cyclic polyspectra are presented and their properties are discussed the performance of these estimators is illustrated by several computer simulation examples for pulse amplitude modulated time series the theory is applied to the problems of weak signal detection and interference tolerant time delay estimation

covariant observables on a nonunimodular group it is shown that the characterization of covariant positive operator measures on nonunimodular locally compact groups can be obtained by using vector measure theoretic methods without an application of mackey s imprimitivity theorem

an invitation to quantum tomography we describe quantum tomography as an inverse statistical problem in which the quantum state of a light beam is the unknown parameter and the data are given by results of measurements performed on identical quantum systems the state can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the wigner function we present consistency results for pattern function projection estimators and for sieve maximum likelihood estimators for both the density matrix of the quantum state and its wigner function we illustrate the performance of the estimators on simulated data an em algorithm is proposed for practical implementation there remain many open problems e g rates of convergence adaptation and studying other estimators a main purpose of the paper is to bring these to the attention of the statistical community

positive operator valued measures covariant with respect to an irreducible representation given an irreducible representation of a group g we show that all the covariant positive operator valued measures based on g z where z is a central subgroup are described by trace class trace one positive operators

the sum of squares hierarchy on the sphere and applications in quantum information theory we consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum of squares relaxations exploiting the polynomial kernel technique we obtain a quadratic improvement of the known convergence rate by reznick and doherty and wehner specifically we show that the rate of convergence is no worse than o d 2 ell 2 in the regime ell omega d where ell is the level of the hierarchy and d the dimension solving a problem left open in the recent paper by de klerk and laurent arxiv 1904 08828 importantly our analysis also works for matrix valued polynomials on the sphere which has applications in quantum information for the best separable state problem by exploiting the duality relation between sums of squares and the doherty parrilo spedalieri hierarchy in quantum information theory we show that our result generalizes to nonquadratic polynomials the convergence rates of navascues owari and plenio

minimax and adaptive estimation of the wigner function in quantum homodyne tomography with noisy data we estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems the state is represented through the wigner function a density on r 2 which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics the effect of the losses due to detection inefficiencies which are always present in a real experiment is the addition to the tomographic data of independent gaussian noise we construct a kernel estimator for the wigner function and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions for the l2 risk we compute the upper bounds of a truncated kernel estimator over the same classes restricted to functions with sub gaussian asymptotic behaviour we construct adaptive estimators i e which do not depend on the smoothness parameters and prove that in some set ups they attain the minimax rates for the corresponding smoothness class

withdrawn sparse representation of higher order statistics this article has been withdrawn please see elsevier policy on article withdrawal http www elsevier com locate withdrawalpolicy this article has been withdrawn at the request of the editor and publisher the publisher regrets that an error occurred which led to the premature publication of this paper this error bears no reflection on the article or its authors the publisher apologizes to the authors and the readers for this unfortunate error

finite size mean field models we characterize the two site marginals of exchangeable states of a system of quantum spins in terms of a simple positivity condition this result is used in two applications we first show that the distance between two site marginals of permutation invariant states on n spins and exchangeable states is of order 1 n the second application relates the mean ground state energy of a mean field model of composite spins interacting through a product pair interaction with the mean ground state energies of the components

coherent states and the classical limit on irreducible representations we give an explicit construction of the coherent states for an arbitrary irreducible representation we also construct the symplectic structure on the manifold of coherent states find canonical variables and discuss various classical limits of quantum mechanical systems with relevant observables that obey commutation relations

normal covariant quantization maps we consider questions related to quantizing complex valued functions defined on a locally compact topological group in the case of bounded functions we generalize r werner s approach to prove the characterization of the associated normal covariant quantization maps

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

coding theorem and strong converse for quantum channels we present a new proof of holevo s 1973 1977 coding theorem for transmitting classical information through quantum channels and its strong converse the technique is largely inspired by wolfwitz s 1964 combinatorial approach using types of sequences as a byproduct of our approach which is independent of previous ones both in the coding theorem and the converse we can give a new proof of holevo s information bound

square integrable imprimitivity systems we give a definition of square integrability for imprimitivity systems and we prove that the square integrable ones share most of the properties of square integrable representations of groups

de finetti theorem on the car algebra the symmetric states on a quasi local c algebra on the infinite set of indices j are those invariant under the action of the group of the permutations moving only a finite but arbitrary number of elements of j the celebrated de finetti theorem describes the structure of the symmetric states i e exchangeable probability measures in classical probability in the present paper we extend the de finetti theorem to the case of the car algebra that is for physical systems describing fermions namely after showing that a symmetric state is automatically even under the natural action of the parity automorphism we prove that the compact convex set of such states is a choquet simplex whose extremal i e ergodic w r t the action of the group of permutations previously described are precisely the product states in the sense of araki moriya in order to do that we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self containing interest

the schwinger su 3 construction ii relations between heisenberg weyl and su 3 coherent states the schwinger oscillator operator representation of su 3 studied in a previous paper from the representation theory point of view is analyzed to discuss the intimate relationships between standard oscillator coherent state systems and systems of su 3 coherent states both su 3 standard coherent states based on choice of highest weight vector as fiducial vector and certain other specific systems of generalized coherent states are found to be relevant a complete analysis is presented covering all the oscillator coherent states without exception and amounting to su 3 harmonic analysis of these states

a general theorem on square integrability vector coherent states we derive a generalization of the well known theorem for the square integrability of a unitary irreducible representation of a locally compact group the generalization covers the case of representations admitting vector coherent states the result is illustrated by an example drawn from the isochronous galilei group the construction yields a wide variety of coherent states labeled by phase space points which satisfy a resolution of the identity condition and incorporate spin degrees of freedom

coherent states for su 3 we define coherent states for su 3 using six bosonic creation and annihilation operators these coherent states are explicitly characterized by six complex numbers with constraints for the completely symmetric representations n 0 and 0 m only three of the bosonic operators are required for mixed representations n m all six operators are required the coherent states provide a resolution of identity satisfy the continuity property and possess a variety of group theoretic properties we introduce an explicit parametrization of the group su 3 and the corresponding integration measure finally we discuss the path integral formalism for a problem in which the hamiltonian is a function of su 3 operators at each site

quantum tomography phase space observables and generalized markov kernels we construct a generalized markov kernel which transforms the observable associated with the homodyne tomography into a covariant phase space observable with a regular kernel state illustrative examples are given in the cases of a schrodinger cat kernel state and the cahill glauber s parametrized distributions also we consider an example of a kernel state when the generalized markov kernel cannot be constructed

informationally complete sets of gaussian measurements we prove the necessary and sufficient conditions for the informational completeness of an arbitrary set of gaussian observables on continuous variable systems with a finite number of degrees of freedom in particular we show that an informationally complete set either contains a single informationally complete observable or includes infinitely many observables we show that for a single informationally complete observable the minimal outcome space is the phase space and the corresponding probability distribution can always be obtained from the quantum optical q function by linear postprocessing and gaussian convolution in a suitable symplectic coordinatization of the phase space in the case of projection valued gaussian observables e g generalized field quadratures we show that an informationally complete set of observables is necessarily infinite finally we generalize the treatment to the case where the measurement coupling is given by a general linear bosonic channel and characterize informational completeness for an arbitrary set of the associated observables

a proof for the informational completeness of the rotated quadrature observables we give a new mathematically rigorous proof for the fact that when s is a dense subset of 0 2 the rotated quadrature operators q s of a single mode electromagnetic field constitute an informationally complete set of observables

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