reversible arithmetic coding for quantum data compression we study the problem of compressing a block of symbols a block quantum state emitted by a memoryless quantum bernoulli source we present a simple to implement quantum algorithm for projecting with high probability the block quantum state onto the typical subspace spanned by the lending eigenstates of its density matrix we propose a fixed rate quantum shannon fano code to compress the projected block quantum state using a per symbol code rate that is slightly higher than the von neumann 1955 entropy limit finally we propose quantum arithmetic codes to efficiently implement quantum shannon fano 1948 codes our arithmetic encoder and decoder have a cubic circuit and a cubic computational complexity in the block size both the encoder and decoder are quantum mechanical inverses of each other and constitute an elegant example of reversible quantum computation

quantum rate distortion coding with auxiliary resources we extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression the first setting we consider is that of quantum rate distortion coding with the help of a classical side channel our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function extending earlier work of devetak and berger we also combine this bound with the entanglement assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source the second setting we consider is that of quantum rate distortion coding with quantum side information qsi available to the receiver in order to prove results in this setting we first state and prove a quantum reverse shannon theorem with qsi for tensor power states which extends the known tensor power quantum reverse shannon theorem the achievability part of this theorem relies on the quantum state redistribution protocol while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver s qsi this quantum reverse shannon theorem with qsi naturally leads to quantum rate distortion theorems with qsi with or without entanglement assistance

scattering theory and the trace of the wave group in 9 lax and phillips discuss the analogue for classical scattering theories in odd space dimensions of the poisson summation formula as generalized to compact manifolds by chazarain 3 colin de verdi e 141 and duistermaat and guillemin 5 and further generalized to manifolds with boundary in 1 71 however in 9 this trace formula is only proved for comparatively large values of the time variable see also ralston 14 1 bardos el al 2 this note is devoted to the proof of the same formula for all positive time for the case of scattering by a smooth compactly supported potential if n a is the number of poles a 3 then for every e 0 there exists c 0 such that

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

uniformity and the taylor expansion of ordinary lambda terms we define the complete taylor expansion of an ordinary lambda term as an infinite linear combination with rational coefficients of terms of a resource calculus similar to boudol s lambda calculus with multiplicities or with resources in our resource calculus all applications are multi linear in the algebraic sense i e commute with linear combinations of the function or the argument we study the collective behaviour of the beta reducts of the terms occurring in the taylor expansion of any ordinary lambda term using in a surprisingly crucial way a uniformity property that they enjoy as a corollary we obtain the main part of a proof that this taylor expansion commutes with bohm tree computation syntactically

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

a semantic measure of the execution time in linear logic we give a semantic account of the execution time i e the number of cut elimination steps leading to the normal form of an untyped mell net we first prove that 1 a net is head normalizable i e normalizable at depth 0 if and only if its interpretation in the multiset based relational semantics is not empty and 2 a net is normalizable if and only if its exhaustive interpretation a suitable restriction of its interpretation is not empty we then give a semantic measure of execution time we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut link from the interpretations of the two cut free nets these results are inspired by similar ones obtained by the first author for the untyped lambda calculus

intuitionistic differential nets and lambda calculus we define pure intuitionistic differential proof nets extending ehrhard and regnier s differential interaction nets with the exponential box of linear logic normalization of the exponential reduction and confluence of the full one is proved these results are directed and adjusted to give a translation of boudol s untyped calculus with resources extended with a linear nonlinear reduction a la ehrhard and regnier s differential calculus such reduction comes in two flavours baby step and giant step s reduction the translation based on girard s encoding a b a b and as such extending the usual one for calculus into proof nets enjoys bisimulation for giant step s reduction from this result we also derive confluence of both reductions

sharp polynomial bounds on the number of scattering poles of radial potentials abstract it is shown that for scattering by a radially symmetric potential in r n n odd the number of poles in a disk of radius r satisfies an estimate n r c n r 1 n this bound is sharp as shown by the special case of potentials nonvanishing at the boundary where n r k n a n r n 1 o 1 a being the diameter of the support

lower bounds for resonance counting functions for obstacle scattering in even dimensions in even dimensional euclidean scattering the resonances lie on the logarithmic cover of the complex plane this paper studies resonances for obstacle scattering in bbb r d with dirichlet neumann or admissible robin boundary conditions when d is even set n m r to be the number of resonances with norm at most r and argument between m pi and m 1 pi then lim sup r rightarrow infty log n m r over log r d if m in bbb z setminus 0

quantum rate distortion theory for memoryless sources we formulate quantum rate distortion theory in the most general setting where classical side information is included in the tradeoff using a natural distortion measure based on entanglement fidelity and specializing to the case of an unrestricted classical side channel we find the exact quantum rate distortion function for a source of isotropic qubits an upper bound we believe to be exact is found in the case of biased sources we establish that in this scenario optimal rate distortion codes produce no entropy exchange with the environment of any individual qubit

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

the differential lambda calculus we present an extension of the lambda calculus with differential constructions we state and prove some basic results confluence strong normalization in the typed case and also a theorem relating the usual taylor series of analysis to the linear head reduction of lambda calculus

reversible irreversible and optimal l machines abstract lambda calculus is the core of functional programming and many different ways to evaluate lambda terms have been considered one of the nicest from the theoretical point of view is head linear reduction we compare two ways of implementing that specific evaluation strategy krivine s abstract machine and the interaction abstract machine runs on those machines stand in a relation which can be accurately described using the call return symmetry discovered by asperti and laneve

on the born oppenheimer expansion for polyatomic molecules we consider the schrodinger operatorp h for a polyatomic molecule in the semiclassical limit where the mass ratioh 2 of electronic to nuclear mass tends to zero we obtain wkb type expansions of eigenvalues and eigenfunctions ofp h to all orders inh this allows to treat the splitting of the ground state energy of a non planar molecule our class of potentials covers the physical case of the coulomb interaction we use methods ofh pseudodifferential operators with operator valued symbols which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials

a semantic account of strong normalization in linear logic abstract we prove that given two cut free nets of linear logic by means of their relational interpretations one can determine 1 whether or not the net obtained by cutting the two nets is strongly normalizable 2 in case it is strongly normalizable the maximum length of the reduction sequences starting from that net as a by product of our semantic approach we obtain a new proof of the conservation theorem for multiplicative exponential linear logic mell which does not rely on confluence this yields an alternative proof of strong normalization for mell

quantum rate distortion reverse shannon theorems and source channel separation we derive quantum counterparts of two key theorems of classical information theory namely the rate distortion theorem and the source channel separation theorem the rate distortion theorem gives the ultimate limits on lossy data compression and the source channel separation theorem implies that a two stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel in spite of their importance in the classical domain there has been surprisingly little work in these areas for quantum information theory in this paper we prove that the quantum rate distortion function is given in terms of the regularized entanglement of purification we also determine a single letter expression for the entanglement assisted quantum rate distortion function and we prove that it serves as a lower bound on the unassisted quantum rate distortion function this implies that the unassisted quantum rate distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output in spite of barnum s conjecture that the coherent information would be relevant here moreover we prove several quantum source channel separation theorems the strongest of these are in the entanglement assisted setting in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion

the system f of variable types fifteen years later the semantic study of system f stumbles on the problem of variable types for which there was no convincing interpretation we develop here a semantics based on the category theoretic idea of direct limit so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones thus getting rid of the circularity of variable types to do so one has first to simplify somehow the extant semantic ideas replacing scott domains by the simpler and more finitary qualitative domains the interpretation obtained is extremely compact as shown on simple examples the paper also contains the definitions of a very small universal model of lambda calculus and investigates the concept totality

one shot lossy quantum data compression we provide a framework for one shot quantum rate distortion coding in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level we obtain a one shot characterization of the minimum qubit compression size for an entanglement assisted quantum rate distortion code in terms of the smooth max information a quantity previously employed in the one shot quantum reverse shannon theorem next we show how this characterization converges to the known expression for the entanglement assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source finally we give a tight finite blocklength characterization for the entanglement assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbolwise distortion constraint

the differential λμ calculus we define a differential l m calculus which is an extension of both parigot s l m calculus and ehrhard regnier s differential l calculus we prove some basic properties of the system reduction enjoys church rosser and simply typed terms are strongly normalizing

constructing differential categories and deconstructing categories of games differential categories were introduced by blute cockett and seely to axiomatize categorically ehrhard and regnier s syntactic differential operator we present an abstract construction that takes a symmetric monoidal category and yields a differential category and show how this construction may be applied to categories of games in one instance we recover the category previously used to give a fully abstract model of a nondeterministic imperative language the construction exposes the differential structure already present in this model and shows how the differential combinator may be encoded in the imperative language the second instance corresponds to a new cartesian differential category of games we give a model of a simply typed resource calculus resource pcf in this category and show that it possesses the finite definability property comparison with a semantics based on bucciarelli ehrhard and manzonetto s relational model reveals that the latter also possesses this property and is fully abstract

on the reversible extraction of classical information from a quantum source consider a source of pure quantum states with von neumann entropy s by the quantum source coding theorem arbitrarily long strings of signals may be encoded asymptotically into s qubits per signal the schumacher limit in such a way that entire strings may be recovered with arbitrarily high fidelity suppose that classical storage is free while quantum storage is expensive and suppose that the states of do not fall into two or more orthogonal subspaces we show that if can be compressed with arbitrarily high fidelity into a qubits per signal plus any amount of auxiliary classical storage then a must still be at least as large as the schumacher limit s of thus no part of the quantum information content of can be faithfully replaced by classical information if the states do fall into orthogonal subspaces then a may be less than s but only by an amount not exceeding the amount of classical information specifying the subspace for a signal from the source

the scott model of linear logic is the extensional collapse of its relational model we show that the extensional collapse of the relational model of linear logic is the model of prime algebraic complete lattices a natural extension to linear logic of the well known scott semantics of the lambda calculus

entanglement information rate distortion of a quantum gaussian source quantum distortion operator is introduced based on canonical operators as the lower bound of quantum information rate distortion the entanglement information rate distortion is achieved by gaussian map for gaussian source the entanglement information rate distortion function is calculated for one mode gaussian source it is achieved by a nontrivial unitary transformation the rate distortion is achievable at zero distortion point in contrast to the distortion defined via fidelity our definition of the distortion makes it possible to calculate the entanglement information rate distortion function for gaussian source

asymptotic number of scattering resonances for generic schrodinger operators let v be the schrodinger operator acting on l 2 mathbb r d mathbb c with d geq 3 odd here v is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a let n v r denote the number of resonances of v with modulus r we show that if the potential v is generic in a sense of pluripotential theory then n v r c d a dr d o r d 3 over 16 epsilon quad mbox as r to infty for any e 0 where c d is a dimensional constant

on the resonances of the laplacian on waveguides abstract the resonances for the dirichlet and neumann laplacian are studied on compactly perturbed waveguides in the absence of resonances an upper bound is proven for the localised resolvent this is then used to prove that the existence of a quasimode whose asymptotics is bounded away from the thresholds implies the existence of resonances converging to the real axis the following upper bound to the number of resonances is also proven k j res dist k j physical plane k j 2 k j 3

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

a local trace formula for resonances of perturbed periodic schrodinger operators abstract we give a local trace formula for the pair p 1 h p 0 w hy p 0 where p 0 is a periodic schrodinger operator w is a decreasing perturbation and h is a small positive parameter we apply this result to establish the existence of h n resonances near some energy of p 0

resonances in scattering by two magnetic fields at large separation and a complex scaling method abstract we study the quantum resonances in magnetic scattering in two dimensions the scattering system consists of two obstacles by which the magnetic fields are completely shielded the trajectories trapped between the two obstacles are shown to generate the resonances near the positive real axis when the distance between the obstacles goes to infinity the location is described in terms of the backward amplitudes for scattering by each obstacle a difficulty arises from the fact that even if the supports of the magnetic fields are largely separated from each other the corresponding vector potentials are not expected to be well separated to overcome this we make use of a gauge transformation and develop a new type of complex scaling method we can cover the scattering by two solenoids at large separation as a special case the obtained result heavily depends on the magnetic fluxes of the solenoids this indicates that the aharonov bohm effect influences the location of resonances