on the regular hilbert space representation of a moyal quantization it is shown that in the regular viz phase space representation of a moyal quantization real polynomials in the phase space variables become essentially self adjoint operators and that functions in the positive cone of monomials of even power in the phase space variables yield positive operators conditions on the boundedness of moyal operators are given the classical limit of a moyal operator is defined and used to relate classical with quantum entropy

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

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

brunn minkowski and zhang inequalities for convolution bodies abstract a quantitative version of minkowski sum extending the definition of convolution of convex bodies is studied to obtain extensions of the brunn minkowski and zhang inequalities as well as other interesting properties on convex geometry involving convolution bodies or polar projection bodies the extension of this new version to more than two sets is also given

the conditional entropy power inequality for quantum additive noise channels we prove the quantum conditional entropy power inequality for quantum additive noise channels this inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory we also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of gaussian input states we apply the conditional entropy power inequality to find an array of information theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting furthermore we give a simple proof of the convergence rate of the quantum ornstein uhlenbeck semigroup based on entropy power inequalities we prove the quantum conditional entropy power inequality for quantum additive noise channels this inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory we also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of gaussian input states we apply the conditional entropy power inequality to find an array of information theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting furthermore we give a simple proof of the convergence rate of the quantum ornstein uhlenbeck semigroup based on entropy power inequalities

a generalization of the entropy power inequality with applications the authors prove the following generalization of the entropy power inequality h ax or h ax where h denotes joint differential entropy x x sub 1 x sub n is a random vector with independent components x x x sub n is a gaussian vector with independent components such that h x sub i h x sub i i 1 n and a is any matrix this generalization of the entropy power inequality is applied to show that a non gaussian vector with independent components becomes closer to gaussianity after a linear transformation where the distance to gaussianity is measured by the information divergence another application is a lower bound greater than zero for the mutual information between nonoverlapping spectral components of a non gaussian white process they also describe a dual generalization of the fisher information inequality

symplectic coarse grained classical and semclassical evolution of subsystems new theoretical approach we study the classical and semiclassical time evolutions of subsystems of a hamiltonian system this is done using a generalization of heller s thawed gaussian approximation introduced by littlejohn the key tool in our study is an extension of gromov s principle of the symplectic camel obtained in collaboration with dias de gosson and prata arxiv 1911 03763v1 math sg 2019 this extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius in the quantum case the radii of these symplectic balls are taken equal to and represent the ellipsoids of minimum uncertainty which we called quantum blobs in previous work

bell s inequalities and quantum field theory ii bell s inequalities are maximally violated in the vacuum in the context of the study of bell s inequalities carried out in paper i j math phys 28 2440 1987 it is proven that bell s inequalities are maximally violated in the vacuum state by suitable spacelike separated observables for both bose and fermi free quantum field theories

the algebra of grassmann canonical anticommutation relations and its applications to fermionic systems we present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti commutation relations car in analogy to quasifree states on the algebra of canonical commutation relations ccr the used mathematical tools are based on a new algebraic structure the grassmann algebra of canonical anticommutation relations gar algebra which is given by the twisted tensor product of a grassmann and a car algebra as a new application the corresponding theory provides an elegant tool for calculating the fidelity of two quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems

geometric inequalities from phase space translations we establish a quantum version of the classical isoperimetric inequality relating the fisher information and the entropy power of a quantum state the key tool is a fisher information inequality for a state which results from a certain convolution operation the latter maps a classical probability distribution on phase space and a quantum state to a quantum state we show that this inequality also gives rise to several related inequalities whose counterparts are well known in the classical setting in particular it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup as an application we derive a log sobolev inequality for the quantum ornstein uhlenbeck semigroup and argue that it implies fast convergence towards the fixed point for a large class of initial states

corrections to the entropy power inequality for quantum systems we correct an intermediate step in the derivation of our main statements in 1 specifically inequality 63 in the mentioned paper intended to give an upper bound on the entropy of certain gaussian states is incorrect in that paper we used inequality 63 to derive the asymptotic large time scaling of the entropy under the quantum version of the heat equation we provide alternative derivations of this result which sidestep the bound 63 the main conclusions of the paper therefore remain unaffected we thank giacomo de palma andrea mari and vittorio giovannetti for pointing out this issue and simultaneously providing us with a resolution in a very detailed communication we also thank them for their permission to include their discussion in this erratum we note that one of the alternative derivations of the asymptotic scaling presented here has previously appeared in their publication

noncommutative martingale deviation and poincare type inequalities with applications we prove a deviation inequality for noncommutative martingales by extending oliveira s argument for random matrices by integration we obtain a burkholder type inequality with satisfactory constant using continuous time we establish noncommutative poincare type inequalities for nice semigroups with a positive curvature condition these results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to bobkov and gotze in the commutative case to demonstrate our setting is general enough we give various examples including certain group von neumann algebras random matrices and classical diffusion processes among others

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

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

rogers shephard inequality for log concave functions abstract in this paper we prove different functional inequalities extending the classical rogers shephard inequalities for convex bodies the original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body such as its difference body we characterize the equality cases in all these inequalities our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log concave functions and the study of some geometrical properties of these new sets

generators of kms symmetric markov semigroups on mathcal b rm h symmetry and quantum detailed balance we find the structure of generators of norm continuous quantum markov semigroups on mathcal b rm h that are symmetric with respect to the scalar product tr 1 2x 1 2y induced by a faithful normal invariant state and satisfy two quantum generalisations of the classical detailed balance condition related with this non commutative notion of symmetry the so called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal

dirichlet forms and markovian semigroups on standard forms of von neumann algebras we characterize w continuous markovian semigroups on a von neumann algebra m which are 0 symmetric w r t a faithful normal state 0in m in terms of quadratic forms on the hilbert space h of a standard form m h p j we characterize also symmetric strongly continuous contraction semigroups on a real hilbert space h which leave invariant a closed convex set in h in terms of a contraction property of the associated quadratic forms we apply the results to give criteria of essential selfadjointness for quadratic form sums and to give a characterization of w continuous markovian semigroups on m which commute with the modular automorphism group 0t

semispectral measures as convolutions and their moment operators the moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied paying attention to the natural domains of these unbounded operators the results are then applied to conveniently determine the moment operators of the cartesian margins of the phase space observables

moment operators of the cartesian margins of the phase space observables the theory of operator integrals is used to determine the moment operators of the cartesian margins of the phase space observables generated by the mixtures of the number states the moments of the x margin are polynomials of the position operator and those of the y margin are polynomials of the momentum operator

gaussian optimizers for entropic inequalities in quantum information we survey the state of the art for the proof of the quantum gaussian optimizer conjectures of quantum information theory these fundamental conjectures state that quantum gaussian input states are the solution to several optimization problems involving quantum gaussian channels these problems are the quantum counterpart of three fundamental results of functional analysis and probability the entropy power inequality the sharp young s inequality for convolutions and the theorem gaussian kernels have only gaussian maximizers quantum gaussian channels play a key role in quantum communication theory they are the quantum counterpart of gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime the quantum gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space the restriction of the quantum limited gaussian attenuator to input states diagonal in the fock basis coincide

ruled lagrangian submanifolds of the 6 sphere this article sets out to serve a dual purpose on the one hand we give an explicit description of the lagrangians in the nearly kahler 6 sphere s 6 which are ruled by circles of constant radius using weierstrass formulae on the other we recognise all known examples of these lagrangians as being ruled by such circles therefore we describe all families of lagrangians in s 6 whose second fundamental form satisfies natural pointwise conditions so called second order families

a solution of gaussian optimizer conjecture for quantum channels the long standing conjectures of the optimality of gaussian inputs and additivity are solved for a broad class of gauge covariant or contravariant bosonic gaussian channels which includes in particular thermal additive classical noise and amplifier channels restricting to the class of states with finite second moments we show that the vacuum is the input state which minimizes the entropy at the output of such channels this allows us to show also that the classical capacity of these channels under the input energy constraint is additive and is achieved by gaussian encodings

on the similarity of the entropy power inequality and the brunn minkowski inequality corresp the entropy power inequality states that the effective variance entropy power of the sum of two independent random variables is greater than the sum of their effective variances the brunn minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii both these inequalities are recast in a form that enhances their similarity in spite of this similarity there is as yet no common proof of the inequalities nevertheless their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found two applications of this reasoning are presented first an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the fisher information matrix given an entropy constraint just as a sphere minimizes the surface area given a volume constraint second a theorem involving the effective radii of growing convex sets is proved

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

correspondence theory on p fock spaces with applications to toeplitz algebras abstract we prove several results concerning the theory of toeplitz algebras over p fock spaces using a correspondence theory of translation invariant symbol and operator spaces the most notable results are the full toeplitz algebra is the norm closure of all toeplitz operators with bounded uniformly continuous symbols this generalizes a result obtained by j xia 25 in the case p 2 which was proven by different methods further we prove that every toeplitz algebra which has a translation invariant c subalgebra of the bounded uniformly continuous functions as its set of symbols is linearly generated by toeplitz operators with the same space of symbols

universal joint measurement uncertainty relation for error bars we formulate and prove a new universally valid uncertainty relation for the necessary error bar widths in any approximate joint measurement of position and momentum

wigner quantisation of arrival time and oscillator phase by means of the wigner distribution function an expression is derived for the quasiprobability that a free particle in one dimension arrives at the origin during a given time interval and also for the quasiprobability that the phase of a one dimensional harmonic oscillator lies in a given segment of the circle it is shown that the best bounds on these quasiprobabilities p are 0 1559 or p or 1 0077 independent of the size of the interval it is shown that on disjoint unions of infinitely many intervals the quasiprobabilities may add up to any positive or negative real value it is briefly indicated how negative probabilities can be avoided altogether in an alternative approach to the quantisation of arrival time and oscillator phase

on a quantum entropy power inequality of audenaert datta and ozols we give a short proof of a recent inequality of audenaert datta and ozols and determine cases of equality

volume of mixed bodies by using inequalities obtained for the volume of mixed bodies and the petty projection inequality sharp isoperimetric inequalities are derived for the projection measures quermassintegrale of a convex body these projection measure inequalities which involve mixed projection bodies zonoids are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body the inequality obtained for the volume of mixed bodies is also used to derive a form of the brunn minkowski inequality involving mixed bodies as an application inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies o introduction in 24 it was shown that the petty projection inequality 26 will yield extensions of the known isoperimetric inequalities between the volume of a convex body and its projection measures quermassintegrale in this article we show that the petty projection inequality can be used to obtain sharp isoperimetric inequalities corollary 5 20 which are strengthened forms of the classical inequalities between the projection measures of a convex body this is accomplished by using the so called mixed convex bodies introduced by firey 17 geometric inequalities are derived for the volume of mixed bodies theorems 4 1 and 4 3 which when combined with the petty projection inequality yield inequalities involving projection bodies zonoids which are related to the inequalities between the projection measures of a convex body the inequalities for mixed bodies are also used to obtain a version of the brunn minkowski inequality theorem 4 2 involving mixed bodies also derived are inequalities between the volume of convex bodies and their associated mixed projection integral theorems 6 4 and 6 4 that are extensions and improvements of some previously established inequalities since some of our results concern projection bodies a topic of interest to workers outside the area of geometric convexity we have restated a few basic definitions and standard results from the field of geometric inequalities in 1 we collect for quick reference some well known results concerning mixed volumes and mixed area measures which will be required later references for material presented in this section are bonnesen and fenchel 7 for mixed volumes busemann 9 and leichtweiss 20 the original works of aleksandrov 1 4 and fenchel and jessen 15 supplemented by the survey of schneider 28 are recommended references for mixed area measures 2 contains introductory material regarding blaschke addition received by the editors january 8 1985 1980 mcithematics subject classihecition primary 52a40 secondary 53a15 46b20

phase space quantization and the operator moment problem we consider questions related to a quantization scheme in which a classical variable f r on a phase space is associated with a preferably unique semispectral measure ef such that the moment operators of ef are required to be of the form fk with a suitable mapping from the set of classical variables to the set of not necessarily bounded operators in the hilbert space of the quantum system in particular we investigate the situation where the map is implemented by the operator integral with respect to some fixed positive operator measure the phase space is first taken to be an abstract measurable space then a locally compact unimodular group and finally r2 where we determine explicitly the relevant operators fk for certain variables f in the case where the quantization map is implemented by a translation covariant positive operator measure in addition we consider the question under what conditions a positive operator measure is projection valued

characterization of informational completeness for covariant phase space observables in the nonrelativistic setting with finitely many canonical degrees of freedom a shift covariant phase space observable is uniquely characterized by a positive operator of trace one and in turn by the fourier weyl transform of this operator we study three properties of such observables and characterize them in terms of the zero set of this transform the first is informational completeness for which it is necessary and sufficient that the zero set has dense complement the second is a version of informational completeness for the hilbert schmidt class equivalent to the zero set being of measure zero and the third known as regularity is equivalent to the zero set being empty we give examples demonstrating that all three conditions are distinct the three conditions are the special cases for p 1 2 of a more general notion of p regularity defined as the norm density of the span of translates of the operator in the schatten p class we show that the relation between zero sets and p regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis

first order sensitivity analysis of symplectic eigenvalues abstract for every 2 n 2 n positive definite matrix a there are n positive numbers d 1 a d n a associated with a called the symplectic eigenvalues of a it is known that d m are continuous functions of a but are not differentiable in general in this paper we show that the directional derivative of d m exists and derive its expression we also discuss various subdifferential properties of d m such as clarke and michel penot subdifferentials

volume inequalities for the i th convolution bodies abstract we obtain a new extension of rogers shephard inequality providing an upper bound for the volume of the sum of two convex bodies k and l we also give lower bounds for the volume of the k th limiting convolution body of two convex bodies k and l special attention is paid to the n 1 th limiting convolution body for which a sharp inequality which is equality only when k l is a simplex is given since the n th limiting convolution body of k and k is the polar projection body of k these inequalities can be viewed as an extension of zhang s inequality

the conditional entropy power inequality for bosonic quantum systems we prove the conditional entropy power inequality for gaussian quantum systems this fundamental inequality determines the minimum quantum conditional von neumann entropy of the output of the beam splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies we also prove that for any couple of values of the quantum conditional entropies of the two inputs the minimum of the quantum conditional entropy of the output given by the conditional entropy power inequality is asymptotically achieved by a suitable sequence of quantum gaussian input states our proof of the conditional entropy power inequality is based on a new stam inequality for the quantum conditional fisher information and on the determination of the universal asymptotic behaviour of the quantum conditional entropy under the heat semigroup evolution the beam splitter and the squeezing are the central elements of quantum optics and can model the attenuation the amplification and the noise of electromagnetic signals this conditional entropy power inequality will have a strong impact in quantum information and quantum cryptography among its many possible applications there is the proof of a new uncertainty relation for the conditional wehrl entropy

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

measurement uncertainty relations measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables they can be seen as a generalization of the error disturbance tradeoff first discussed heuristically by heisenberg here we prove such relations for the case of two canonically conjugate observables like position and momentum and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state both sets of relations are generalized to means of order rather than the usual quadratic means and we show that the optimal constants are the same for preparation and for measurement uncertainty the constants are determined numerically and compared with some bounds in the literature in both cases the near saturation of the inequalities entails that the state resp observable is uniformly close to a minimizing one

spectral analysis and feller property for quantum ornstein uhlenbeck semigroups a class of dynamical semigroups arising in quantum optics models of masers and lasers is investigated the semigroups are constructed by means of noncommutative dirichlet forms on the full algebra of bounded operators on a separable hilbert space the explicit action of their generators on a core in the domain is used to demonstrate the feller property of the semigroups with respect to the c subalgebra of compact operators the dirichlet forms are analysed and the c 2 spectrum together with eigenspaces are found when reduced to certain maximal abelian subalgebras the semigroups give rise to the markov semigroups of classical ornstein uhlenbeck processes on the one hand and of classical birth and death processes on the other

inequalities for mixed projection bodies mixed projection bodies are related to ordinary projection bodies zonoids in the same way that mixed volumes are related to ordinary volume analogs of the classical inequalities from the brunn minkowski theory such as the minkowski brunn minkowski and aleksandrov fenchel inequalities are developed for projection and mixed projection bodies

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

the rate distortion function for the quadratic gaussian ceo problem a new multiterminal source coding problem called the ceo problem was presented and investigated by berger zhang and viswanathan recently viswanathan and berger have investigated an extension of the ceo problem to gaussian sources and call it the quadratic gaussian ceo problem they considered this problem from a statistical viewpoint deriving some interesting results in this paper we consider the quadratic gaussian ceo problem from a standpoint of multiterminal rate distortion theory we regard the ceo problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given this viewpoint leads us to a complete solution to the problem we determine the tradeoff between the total amount of rate and squared distortion deriving an explicit formula of the rate distortion function the derived function has the form of a sum of two nonnegative functions one is a classical rate distortion function for single gaussian source and the other is a new rate distortion function which dominates the performance of the system for a relatively small distortion it follows immediately from our result that the conjecture of viswanathan and berger on the asymptotic behavior of minimum squared distortion for large rates is true

on the volume of the minkowski sum of line sets and the entropy power inequality we derive a version of the brunn minkowski inequality which gives a nontrivial lower bound on the volume of the minkowski sum of degenerate sets namely line sets this inequality parallels a recently obtained matrix generalization of the entropy power inequality

entanglement assisted capacity of a quantum channel and the reverse shannon theorem the entanglement assisted classical capacity of a noisy quantum channel c sub e is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel assuming that the sender and receiver have access to the resource of shared quantum entanglement which may be used up by the communication protocol we show that the capacity c sub e is given by an expression parallel to that for the capacity of a purely classical channel i e the maximum over channel inputs spl rho of the entropy of the channel input plus the entropy of the channel output minus their joint entropy the latter being defined as the entropy of an entangled purification of spl rho after half of it has passed through the channel we calculate entanglement assisted capacities for two interesting quantum channels the qubit amplitude damping channel and the bosonic channel with amplification attenuation and gaussian noise we discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel alone or in the presence of ancillary resources such as prior entanglement in the classical analog of entanglement assisted communication communication over a discrete memoryless channel dmc between parties who share prior random information we show that one parameter is sufficient i e that in the presence of prior shared random information all dmcs of equal capacity can simulate one another with unit asymptotic efficiency

entanglement assisted capacity of quantum multiple access channels we find a regularized formula for the entanglement assisted ea capacity region for quantum multiple access channels qmac we illustrate the capacity region calculation with the example of the collective phase flip channel which admits a single letter characterization on the way we provide a first principles proof of the ea coding theorem based on a packing argument we observe that the holevo schumacher westmoreland theorem may be obtained from a modification of our ea protocol we remark on the existence of a family hierarchy of protocols for multiparty scenarios with a single receiver in analogy to the two party case in this way we relate several previous results regarding qmacs

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

the conditional entropy power inequality for gaussian quantum states we propose a generalization of the quantum entropy power inequality involving conditional entropies for the special case of gaussian states we give a proof based on perturbation theory for symplectic spectra we discuss some implications for entanglement assisted classical communication over additive bosonic noise channels

a simple proof of the entropy power inequality this correspondence gives a simple proof of shannon s entropy power inequality epi using the relationship between mutual information and minimum mean square error mmse in gaussian channels

an extremal inequality motivated by multiterminal information theoretic problems we prove a new extremal inequality motivated by the vector gaussian broadcast channel and the distributed source coding with a single quadratic distortion constraint problems as a corollary this inequality yields a generalization of the classical entropy power inequality epi as another corollary this inequality sheds insight into maximizing the differential entropy of the sum of two dependent random variables

contractivity properties of a quantum diffusion semigroup we consider a quantum generalization of the classical heat equation and study contractivity properties of its associated semigroup we prove a nash inequality and a logarithmic sobolev inequality the former leads to an ultracontractivity result this in turn implies that the largest eigenvalue and the purity of a state with positive wigner function evolving under the action of the semigroup decrease at least inverse polynomially in time while its entropy increases at least logarithmically in time

entropy power inequalities for qudits shannon s entropy power inequality epi can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule f a x 1 a y af x 1 a f y a 0 1 here x and y are continuous random variables and the function f is either the differential entropy or the entropy power konig and smith ieee trans inf theory 60 3 1536 1548 2014 and de palma mari and giovannetti nat photonics 8 12 958 964 2014 obtained quantum analogues of these inequalities for continuous variable quantum systems where x and y are replaced by bosonic fields and the addition rule is the action of a beam splitter with transmissivity a on those fields in this paper we similarly establish a class of epi analogues for d level quantum systems i e qudits the underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter a 0 1 and acts like a finite dimensional analogue of a beam splitter with transmissiv

densities of covariant observables positive operator valued observables in a complex hilbert space h based on a topological homogeneous space g h and g covariant with respect to a strongly continuous unitary representation on h of a locally compact group g are investigated the goal is to determine whether they are the weak integrals of some positive operator valued densities relative to a g quasi invariant measure on g h to this end the notion of a kernel hilbert space is generalized to spaces of equivalence classes with respect to of mappings of g h into a complex hilbert space k

perturbation bounds for williamson s symplectic normal form abstract given a real valued positive semidefinite matrix williamson proved that it can be diagonalised using symplectic matrices the corresponding diagonal values are known as the symplectic spectrum this paper is concerned with the stability of williamson s decomposition under perturbations we provide norm bounds for the stability of the symplectic eigenvalues and prove that if s diagonalises a given matrix m to williamson form then s is stable if the symplectic spectrum is nondegenerate and s t s is always stable finally we sketch a few applications of the results in quantum information theory

sharp uncertainty relations for number and angle we study uncertainty relations for pairs of conjugate variables like number and angle of which one takes integer values and the other takes values on the unit circle the translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively and the bounds depend only on the choice of two metrics used to quantify the difference of number and angle outputs respectively for each type of observable we discuss two natural choices of metric and discuss the resulting optimal bounds with both numerical and analytical methods we also develop some simple and explicit albeit not sharp lower bounds using an apparently new method for obtaining certified lower bounds to ground state problems

the classical limit of a state on the weyl algebra this paper considers states on the weyl algebra of the canonical commutation relations over the phase space r 2n we show that a state is regular iff its classical limit is a countably additive borel probability measure on r 2n it follows that one can reduce the state space of the weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical limit is physical

gradient flows of the entropy for finite markov chains let k be an irreducible and reversible markov kernel on a finite set x we construct a metric w on the set of probability measures on x and show that with respect to this metric the law of the continuous time markov chain evolves as the gradient flow of the entropy this result is a discrete counterpart of the wasserstein gradient flow interpretation of the heat flow in rn by jordan kinderlehrer and otto 1998 the metric w is similar to but different from the l2 wasserstein metric and is defined via a discrete variant of the benamou brenier formula

special lagrangian submanifolds with isolated conical singularities v survey and applications this is the last in a series of five papers studying compact special lagrangian submanifolds sl m folds x in almost calabi yau m folds m with singularities x1 x n locally modelled on special lagrangian cones c1 c n in c m with isolated singularities at 0 readers are advised to begin with this paper we survey the major results of the previous four papers giving brief explanations of the proofs we apply the results to describe the boundary of a moduli space of compact nonsingular sl m folds n in m we prove the existence of special lagrangian connected sums n1 nk of sl mfolds n1 n k in m we also study sl 3 folds with t 2 cone singularities proving results related to ideas of the author on invariants of calabi yau 3 folds and the syz conjecture let x be a compact sl m fold with isolated conical singularities xi and cones ci for i 1 n the first paper studied the regularity of x near its singular points and the the second the moduli space of deformations of x the third and fourth papers construct desingularizations of x realizing x as a limit of a family of compact nonsingular sl m folds n t in m for small t 0 let li be an asymptotically conical sl m fold in cm asymptotic to ci at infinity we make n t by gluing tli into x at xi for i 1 n

bounds on information combining with quantum side information bounds on information combining are entropic inequalities that determine how the information entropy of a set of random variables can change when these are combined in certain prescribed ways such bounds play an important role in classical information theory particularly in coding and shannon theory entropy power inequalities are special instances of them the arguably most elementary kind of information combining is the addition of two binary random variables a cnot gate and the resulting quantities play an important role in belief propagation and polar coding we investigate this problem in the setting where quantum side information is available which has been recognized as a hard setting for entropy power inequalities our main technical result is a non trivial and close to optimal lower bound on the combined entropy which can be seen as an almost optimal quantum mrs gerber s lemma our proof uses three main ingredients 1 a new bound on the concavity of von neumann entropy which is tight in the regime of low pairwise state fidelities 2 the quantitative improvement of strong subadditivity due to fawzi renner in which we manage to handle the minimization over recovery maps and 3 recent duality results on classical quantum channels due to renes et al we furthermore present conjectures on the optimal lower and upper bounds under quantum side information supported by interesting analytical observations and strong numerical evidence we finally apply our bounds to polar coding for binary input classical quantum channels and show the following three results 1 even non stationary channels polarize under the polar transform 2 the blocklength required to approach the symmetric capacity scales at most sub exponentially in the gap to capacity and 3 under the aforementioned lower bound conjecture a blocklength polynomial in the gap suffices

a proof of the fisher information inequality via a data processing argument the fisher information j x of a random variable x under a translation parameter appears in information theory in the classical proof of the entropy power inequality epi it enters the proof of the epi via the de bruijn identity where it measures the variation of the differential entropy under a gaussian perturbation and via the convolution inequality j x y sup 1 spl ges j x sup 1 j y sup 1 for independent x and y known as the fisher information inequality fii the fii is proved in the literature directly in a rather involved way we give an alternative derivation of the fii as a simple consequence of a data processing inequality for the cramer rao lower bound on parameter estimation

mixed states with positive wigner functions the wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent according to a result of hudson the characterization of mixed states with a positive wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters i e functions with given wigner spectrum in the sense of narcowich if a state is a convex combination of coherent states it has the property that the wigner spectrum contains the unit interval which in turn implies that the wigner function is positive it is shown by explicit examples that the converses of both implications are false the examples are taken from a low dimensional section of the state space in which all wigner spectra can be computed in this set counterexamples to a conjecture by narcowich concerning the wigner spectrum of products are also found as well as a state whose wigner spectrum is a convergent sequence of discrete points

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

generalized minimal output entropy conjecture for one mode gaussian channels definitions and some exact results a formulation of the generalized minimal output entropy conjecture for gaussian channels is presented it asserts that for states with fixed input entropy the minimal value of the output entropy of the channel i e the minimal output entropy increment for fixed input entropy is achieved by gaussian states in the case of centered channels i e channels which do not add squeezing to the input state this implies that the minimum is obtained by thermal gibbs inputs the conjecture is proved to be valid in some special cases

the wigner distribution of n arbitrary observables we study a generalization of the wigner function to arbitrary tuples of hermitian operators we show that for any collection of hermitian operators a1 an and any quantum state there is a unique joint distribution on rn with the property that the marginals of all linear combinations of the ak coincide with their quantum counterparts in other words we consider the inverse radon transform of the exact quantum probability distributions of all linear combinations we call it the wigner distribution because for position and momentum this property defines the standard wigner function we discuss the application to finite dimensional systems establish many basic properties and illustrate these by examples the properties include the support the location of singularities positivity the behavior under symmetry groups and informational completeness

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

volumes of restricted minkowski sums and the free analogue of the entropy power inequality in noncommutative probability theory independence can be based on free products instead of tensor products this yields a highly noncommutative theory free probability theory for an introduction see 9 the analogue of entropy in the free context was introduced by the second named author in 8 here we show that shannon s entropy power inequality 6 1 has an analogue for the free entropy x theorem 2 1

rate distortion performance in coding bandlimited sources by sampling and dithered quantization the rate distortion characteristics of a scheme for encoding continuous time band limited stationary sources with a prescribed band is considered in this coding procedure the input is sampled at nyquist s rate or faster the samples undergo dithered uniform or lattice quantization using subtractive dither and the quantizer output is entropy coded the rate distortion performance and the tradeoff between the sampling rate and the quantization accuracy is investigated utilizing the observation that the coding scheme is equivalent to an additive noise channel it is shown that the mean square error of the scheme is fixed as long as the product of the sampling period and the quantizer second moment is kept constant while for a fixed distortion the coding rate generally increases when the sampling rate exceeds the nyquist rate finally as the lattice quantizer dimension becomes large the equivalent additive noise channel of the scheme tends to be white gaussian and both the rate and the distortion performance become invariant to the sampling rate

the entropy power inequality for quantum systems when two independent analog signals x and y are added together giving z x y the entropy of z h z is not a simple function of the entropies h x and h y but rather depends on the details of x and y s distributions nevertheless the entropy power inequality epi which states that e2h z e2h x e2h y gives a very tight restriction on the entropy of z this inequality has found many applications in information theory and statistics the quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter the purpose of this paper is to give a detailed outline of the proof of two separate generalizations of the epi to the quantum regime our proofs are similar in spirit to the standard classical proofs of the epi but some new quantities and ideas are needed in the quantum setting in particular we find a new quantum de bruijin identity relating entropy production under diffusion to a divergence based quantum fisher information furthermore this fisher information exhibits certain convexity properties in the context of beam splitters

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power

constructing special lagrangian m folds in c m by evolving quadrics this is the second in a series of papers constructing explicit examples of special lagrangian submanifolds sl m folds in c the first paper of the series 7 studied sl m folds with large symmetry groups and subsequent papers 8 9 10 11 construct examples of sl 3 folds in c using evolution equations symmetries ruled submanifolds and integrable systems the principal motivation for these papers is to lay the foundations for a study of the singularities of compact special lagrangian m folds in calabi yau m folds particularly in low dimensions such as m 3 special lagrangian m folds in c and especially special lagrangian cones should provide local models for singularities of sl m folds in calabi yau m folds understanding such singularities will be essential in making rigorous the explanation of mirror symmetry of calabi yau 3 folds x x proposed by strominger yau and zaslow 13 which involves dual fibrations of x x by special lagrangian 3 tori with some singular fibres it will also be important in resolving conjectures made by the author 6 which attempt to define an invariant of calabi yau 3 folds by counting special lagrangian homology 3 spheres the paper falls into three parts the first this section and 2 is introductory the second part 3 4 describes a general construction of special lagrangianm folds n in c depending on a set of evolution data p where p is an m 1 submanifold in r then n is the subset of c swept out by the image of p under a 1 parameter family of linear or affine maps t r n c which satisfy a first order nonlinear o d e in t examples of sets of evolution data will be given in 4 together with some progress towards a classification of such data the simplest interesting sets of evolution data occur when n m and p is a nondegenerate quadric in r in the third part 5 7 we apply the construction to these examples in this case t r must be a lagrangian plane in c for each t thus n is fibred by quadrics in lagrangian planes r in c the construction of 3 4 will also be used in the sequel to this paper 8 with different evolution data to construct families of sl 3 folds in c the construction has both a linear and an affine version in the linear version

quantization and the resolvent algebra abstract we introduce a novel commutative c algebra c r x of functions on a symplectic vector space x admitting a complex structure along with a strict deformation quantization that maps a dense subalgebra of c r x to the resolvent algebra introduced by buchholz and grundling 2 the associated quantization map is a field theoretical weyl quantization compatible with the work of binz honegger and rieckers 1 we also define a berezin type quantization map on all of c r x which continuously and bijectively maps it onto the resolvent algebra the commutative resolvent algebra c r x generally defined on a real inner product space x intimately depends on the finite dimensional subspaces of x we thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its gelfand spectrum