linear response theory for thermally driven quantum open systems this note is a continuation of our recent paper v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press where we have proven the green kubo formula and the onsager reciprocity relations for heat fluxes in thermally driven quantum open systems in this note we extend the derivation of the green kubo formula to heat and charge fluxes and discuss some other generalizations of the model and results of v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press

introduction to special issue journal of mathematical physics turns 50 jmp turns 50 today in this rst issue of year 51 of jmp the rst issue of the next 50 years so to speak you will nd articles by leading researchers inmathematical physics which collectively give a good impression of where the subject standstoday as a discipline mathematical physics is characterized by a combination of rigorous math ematics and a clear connection to physics all areas of theoretical physics are included and

quantum macrostates equivalence of ensembles and an h theorem before the thermodynamic limit macroscopic averages need not commute for a quantum system as a consequence aspects of macroscopic fluctuations or of constrained equilibrium require a careful analysis when dealing with several observables we propose an implementation of ideas that go back to john von neumann s writing about the macroscopic measurement we apply our scheme to the relation between macroscopic autonomy and an h theorem and to the problem of equivalence of ensembles in particular we show how the latter is related to the asymptotic equipartition theorem the main point of departure is an expression of a law of large numbers for a sequence of states that start to concentrate as the size of the system gets larger on the macroscopic values for the different macroscopic observables deviations from that law are governed by the entropy

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

large deviations for quantum spin systems we consider high temperature kms states for quantum spin systems on a lattice we prove a large deviation principle for the distribution of empirical averages bar x lambda frac 1 lambda sigma i in lambda x i where the x i s are copies of a self adjoint element x level one large deviations from the analyticity of the generating function we obtain the central limit theorem we generalize to a level two large deviation principle for the distribution of frac 1 lambda sigma i in lambda delta x i

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

unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti we present a new simpler proof of the unconditional uniqueness of solutions to the cubic gross pitaevskii hierarchy in one of the main tools in our analysis is the quantum de finetti theorem our uniqueness result is equivalent to the one established in the celebrated works of erd s schlein and yau 2015 wiley periodicals inc

lie algebra of anomalously scaled fluctuations for quantum lattice systems it is proven that anomalously scaled fluctuations have a natural lie algebra structure the harmonic lattice in the ground state is given as an illustration of the general theorem

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

randomization and the gross pitaevskii hierarchy we study the gross pitaevskii hierarchy on the spatial domain mathbb t 3 by using an appropriate randomization of the fourier coefficients in the collision operator we prove an averaged form of the main estimate which is used in order to contract the duhamel terms that occur in the study of the hierarchy in the averaged estimate we do not need to integrate in the time variable an averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary the range of regularity exponents that we obtain is alpha frac 3 4 it was shown in our previous joint work with gressman j funct anal 266 7 4705 4764 2014 that the range alpha 1 is sharp in the corresponding deterministic spacetime estimate this is in contrast to the non periodic setting which was studied by klainerman and machedon commun math phys 279 1 169 185 2008 where the spacetime estimate is known to hold whenever alpha geq 1 the goal of our paper is to extend the range of in this class of estimates in a probabilistic sense we use the new estimate and the ideas from its proof in order to study randomized forms of the gross pitaevskii hierarchy more precisely we consider hierarchies similar to the gross pitaevskii hierarchy but in which the collision operator has been randomized for these hierarchies we show convergence to zero in low regularity sobolev spaces of duhamel expansions of fixed deterministic density matrices we believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization

ruijsenaars commuting difference operators and invariant subspace spanned by theta functions we study a family of mutually commutative difference operators introduced by ruijsenaars the conjugations of these operators with an appropriate function give the hamiltonians of some relativistic quantum systems these operators can be regarded as elliptic analogs of the macdonald operators and their coefficients consist of the jacobi theta functions we show that these operators act on the space of meromorphic functions on the cartan subalgebra of affine lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators

on the connection between the one dimensional s 1 2 heisenberg chain and haldane shastry model extra integrals of motion and the lax representation are found for interacting spin systems with the hamiltonianh j 2 j k 1 j k n p j k j k where one of the periods of the weierstrassp function is equal ton the heisenberg and haldane shastry chains appear as limiting cases of these systems at some values of the second period the simplest eigenvectors and eigenvalues ofh corresponding to the scattering of two spin waves are presented explicitly for these finite dimensional systems and for their infinite dimensional version

quantum de finetti theorems under local measurements with applications quantum de finetti theorems are a useful tool in the study of correlations in quantum multipartite states in this paper we prove two new quantum de finetti theorems both showing that under tests formed by local measurements in each of the subsystems one can get an exponential improvement in the error dependence on the dimension of the subsystems we also obtain similar results for non signaling probability distributions we give several applications of the results to quantum complexity theory polynomial optimization and quantum information theory the proofs of the new quantum de finetti theorems are based on information theory in particular on the chain rule of mutual information the results constitute improvements and generalizations of a recent de finetti theorem due to brandao christandl and yard

mathematical structures for long range dynamics and symmetry breaking the algebraic dynamics of systems with long range instantaneous interactions requires an enlargement of the quasi local algebra which in most relevant cases includes variables at infinity that enter in an essential way in the time evolution of local variables the mathematical structures emerging for the treatment of long range dynamics are investigated also in connection with spontaneous symmetry breaking

new non isospectral integrable couplings of the akns system abstract by means of semi direct sums of lie algebras and the compatibility condition of lax pair an efficient way to construct non isospectral integrable couplings is proposed as its applications new non isospectral integrable couplings of the akns system are obtained the method can be devoted to constructing non isospectral integrable couplings of other nonlinear evolution equations

non commutative central limits non commutative central limit theorems are derived the ccr c algebra of fluctuations is analyzed in detail the stability of the central limit is studied by means of the notion of relative entropy

effective coupling for open billiards we derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard that is opened by attaching a waveguide the wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account the influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random matrix theory

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

connection between yangian symmetry and the quantum inverse scattering method the quantum nonlinear schrodinger model with two component fermions exhibits a yangian symmetry when considered on an infinite interval we construct the generators of the yangian using dunkl operators we show an equivalence between the monodromy matrix constructed from the dunkl operators and that of the quantum inverse scattering method under the yangian algebra the space of states with a fixed particle number forms a tensor product representation of fundamental representations

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

algebraic structure of quantum fluctuations on the basis of the existence of second and third moments of fluctuations we prove a theorem about the lie algebraic structure of fluctuation operators this result gives insight into the quantum character of fluctuations we illustrate the presence of a lie algebra of fluctuation operators in a model of the anharmonic crystal and show the dependence of the lie algebra structure on the fine structure of the fluctuation operator algebra the result is also applied to construct the normal goldstone mode in the ideal bose gas for bose einstein condensation

the pauli principle revisited by the pauli exclusion principle no quantum state can be occupied by more than one electron one can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1 shortly after its discovery the pauli principle was replaced by anti symmetry of the multi electron wave function in this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix that goes far beyond the original pauli principle our approach uses berenstein and sjamaar s theorem on the restriction of an adjoint orbit onto a subgroup and allows us to treat any type of permutational symmetry

statistics of resonances and delay times in random media beyond random matrix theory we review recent developments in quantum scattering from mesoscopic systems various spatial geometries whose closed analogues show diffusive localized or critical behaviour are considered these are the features that cannot be described by the universal random matrix theory results instead one has to go beyond this approximation and incorporate them in a non perturbative way here we pay particular attention to the traces of these non universal characteristics in the distribution of the wigner delay times and resonance widths the former quantity captures time dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix

skew young diagram method in spectral decomposition of integrable lattice models the spectral decomposition of the path space of the vertex model associated to the vector representation of the quantized affine algebra is studied we give a one to one correspondence between the spin configurations and the semistandard tableaux of skew young diagrams as a result we obtain a formula of the characters for the degeneracy of the spectrum in terms of skew schur functions we conjecture that our result describes the module contents of the yangian module structures of the level 1 integrable modules of the affine lie algebra an analogous result is obtained also for a vertex model associated to the quantized twisted affine algebra where characters appear for the degeneracy of the spectrum the relations to the spectrum of the haldane shastry and the polychronakos models are also discussed

the bilinear biquadratic model on the complete graph we study the spin 1 bilinear biquadratic model on the complete graph of n sites i e when each spin is interacting with every other spin with the same strength because of its complete permutation invariance this hamiltonian can be rewritten as the linear combination of the quadratic casimir operators of 3 and 2 using group representation theory we explicitly diagonalize the hamiltonian and map out the ground state phase diagram of the model furthermore the complete energy spectrum with degeneracies is obtained analytically for any number of sites

quasi order of clocks and their synchronism and quantum bounds for copying timing information the statistical state of any classical or quantum system with nontrivial time evolution can be interpreted as the pointer of a clock the quality of such a clock is given by the statistical distinguishability of its states at different times if a clock is used as a resource for producing another one the latter can at most have the quality of the resource we show that this principle formalized by a quasi order implies constraints on many physical processes similarly the degree to which two quantum or classical clocks are synchronized can be formalized by a quasi order of synchronism copying timing information is restricted by quantum no cloning and no broadcasting theorems since classical clocks can only exist in the limit of infinite energy we show this quantitatively by comparing the fisher timing information of two output systems to the input s timing information for classical signal processing in the quantum regime our results imply that a signal looses its localization in time if it is amplified and distributed to many devices

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

integrability of the one dimensional bariev model we investigate the exact integrability of the one dimensional 1d bariev model in the framework of the quantum inverse scattering method qism using the jordan wigner transformation the 1d bariev model can be regarded as a coupled spin model we construct the higher conserved currents which commute with the hamiltonian the explicit form of the conserved currents helps us to infer the l operator of the 1d bariev model from the l operator we construct a transfer matrix which is a generating function of the conserved currents we also find the corresponding r matrix which satisfies the yang baxter relation thus the exact integrability of the 1d bariev model is established the r matrix does not have the difference property for the spectral parameter as in the case of the 1d hubbard model we also provide the lax representation and the fermionic formulation of the yang baxter relation

about the exactness of the linear response theory for quantum lattice systems with finite range potentials and integrable space clustering we prove the linearity of the response theory when dealing with fluctuation observables

a non commutative central limit theorem we prove a non commutative version of the central limit theorem for spin 1 2 variables the non commutative spins satisfy bose einstein statistics the proof involves the feynman kac formula for a hopping process and an explicit calculation of the fourier transform of the corresponding measure the limit process is a modification of the ornstein uhlenbeck process the fourier transform of which was essentially already calculated by feynman however his derivation is not rigorous and we therefore present an alternative derivation here

translation invariant pure state on b j zmd j c and its split property in this paper we prove that a real lattice symmetric reflection positive translation invariant pure state of b j zmd j c admits split property if and only if its two point spatial correlation functions decay exponentially we use amalgamated representation of cuntz algebras to represent two point spatial correlation functions on an augmented hilbert space the underling symmetries and reflection positive property of the pure state make it possible to investigate its split and decaying two point correlation functions properties as spectral properties of a contractive self adjoint operator on the augmented hilbert space haag duality property of the pure state is crucially used in the analysis

unitary equivalence of temperature dynamics for ideal and locally perturbed fermi gas we consider the local perturbation v varepsilon sum limits x y in mathbb z v v x y chi omega x chi omega y a x a y a y a x of the ideal fermi gas on the lattice v where is a finite subset of v and is its indicator the invertibility of moller morphisms for small is proven it follows that in the cyclic gns representation with respect to kms states the dynamics of ideal and locally perturbed fermi gas are unitary equivalent

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

scaling of fluctuations and critical exponents we present a new technique to describe the abnormal behavior of certain fluctuation observables in the critical regime of quantum statistical systems which undergo a phase transition the idea is to rescale the local fluctuation operators by a relevant external parameter of the system in addition to the usual scaling with the inverse square root of the volume the scaling indices used in this scaling procedure are directly related to the critical exponents furthermore it is explained that this new method of scaling preserves the ccr structure of the algebra of macroscopic fluctuations finally scaling indices are computed for the relevant microscopic observables at all temperatures in a mean field approximation for a quantum anharmonic crystal these indices yield the same critical exponents as predicted by mean field theory

phase transitions and algebra of fluctuation operators in an exactly soluble model of a quantum anharmonic crystal a complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions in the one phase region the fluctuations are normal and its algebra is non abelian in the two phase region and on the critical line t c 0 the momentum fluctuation is normal the displacement is critical and the algebra is abelian att c 0 quantum phase transition this algebra is non abelian with abnormal displacement and supernormal squeezed momentum fluctuation operators both being dimension dependent

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

a central limit theorem in many body quantum dynamics we study the many body quantum evolution of bosonic systems in the mean field limit the dynamics is known to be well approximated by the hartree equation so far the available results have the form of a law of large numbers in this paper we go one step further and we show that the fluctuations around the hartree evolution satisfy a central limit theorem interestingly the variance of the limiting gaussian distribution is determined by a time dependent bogoliubov transformation describing the dynamics of initial coherent states in a fock space representation of the system

integrability of difference calogero moser systems a general class of n particle difference calogero moser systems with elliptic potentials is introduced besides the step size and two periods the hamiltonian depends on nine coupling constants we prove the quantum integrability of the model for n 2 and present partial results for n 3 in degenerate cases rational hyperbolic or trigonometric limit the integrability follows for arbitrary particle number from previous work connected with the multivariable q polynomials of koornwinder and macdonald liouville integrability of the corresponding classical systems follows as a corollary limit transitions lead to various well known models such as the nonrelativistic calogero moser systems associated with classical root systems and the relativistic calogero moser system

central limit theorems for the large spin asymptotics of quantum spins we use a generalized form of dyson s spin wave formalism to prove several central limit theorems for the large spin asymptotics of quantum spins in a coherent state

gaussian non gaussian critical fluctuations in the curie weiss model it is known that at the critical temperature the curie weiss mean field model has non gaussian fluctuations and that internal fluctuations can be gaussian here we compute the distribution of theq mode magnetization fluctuations as a function of the temperature the wave vectorq and a fading out external field we obtain new classes of probability distributions generated by this external field as well as new critical behavior in terms of its rate of fading out we discuss also the susceptibility as the limitq tending to zero

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

finite de finetti theorem for conditional probability distributions describing physical theories we work in a general framework where the state of a physical system is defined by its behavior under measurement and the global state is constrained by no signaling conditions we show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions generalizing the classical finite de finetti theorem of diaconis and freedman our results apply to correlations obtained from quantum states even when there is no bound on the local dimension so that known quantum de finetti theorems cannot be used

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

topics in quantum integrable systems recent developments in the theory of quantum integrable particle systems in one dimension with inverse square interactions are reviewed first the yangian symmetry is introduced and the energy spectra of the related spin models are discussed the character of the su n 1 wznw theory is shown to be closely related with the rogers szego polynomial second the infinite dimensional representation for solutions of the yang baxter equation and the reflection equation is given based on the representation the dunkl operators associated with the classical root systems are constructed the macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case finally some results on short range interacting systems are mentioned

a noncommutative central limit theorem for ccr algebras abstract a noncommutative generalization of the central limit theorem for even completely positive mappings between two ccr algebras is proved quasi free completely positive mappings are found to be the generalizations of the gaussian distributed random variables

su p q coherent states and a gaussian de finetti theorem we prove a generalization of the quantum de finetti theorem when the local space is an infinite dimensional fock space in particular instead of considering the action of the permutation group on n copies of that space we consider the action of the unitary group u n on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in u n our first result is a complete characterization of this subspace which turns out to be spanned by a family of generalized coherent states related to the special unitary group su p q of signature p q more precisely this construction yields a unitary representation of the noncompact simple real lie group su p q we therefore find a dual unitary representation of the pair of groups u n and su p q on an n p q mode fock space the gaussian su p q coherent states resolve the identity on the symmetric subspace which implies a gaussian de finetti theorem stating that tracing over a few modes of a unitary invariant state yields a state close to a mixture of gaussian states as an application of this de finetti theorem we show that the n n upper left submatrix of an n n haar invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3 o m we prove a generalization of the quantum de finetti theorem when the local space is an infinite dimensional fock space in particular instead of considering the action of the permutation group on n copies of that space we consider the action of the unitary group u n on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in u n our first result is a complete characterization of this subspace which turns out to be spanned by a family of generalized coherent states related to the special unitary group su p q of signature p q more precisely this construction yields a unitary representation of the noncompact simple real lie group su p q we therefore find a dual unitary representation of the pair of groups u n and su p q on an n p q mode fock space the gaussian su p q coherent states resolve the identity on the symmetric subspace which implies a gaussian de finetti theorem stating that tr

symmetry of the calogero model confined in the harmonic potential yangian and w algebra the su nu calogero model confined in the harmonic potential is considered we show that the model has yangian symmetry y su nu unifying with the creation annihilation operators which constitute the su nu loop algebra we reveal the algebraic structure of this system the relation with the energy spectrum is also discussed

central limit theorem for locally interacting fermi gas we consider a locally interacting fermi gas in its natural non equilibrium steady state and prove the quantum central limit theorem qclt for a large class of observables a special case of our results concerns finitely many free fermi gas reservoirs coupled by local interactions the qclt for flux observables together with the green kubo formulas and the onsager reciprocity relations previously established jop4 complete the proof of the fluctuation dissipation theorem and the development of linear response theory for this class of models

de finetti representation theorem for infinite dimensional quantum systems and applications to quantum cryptography we show that the quantum de finetti theorem holds for states on infinite dimensional systems provided they satisfy certain experimentally verifiable conditions this result can be applied to prove the security of quantum key distribution based on weak coherent states or other continuous variable states against general attacks

affine hecke algebra macdonald polynomials and quantum many body systems the macdonald operators associated with the classical root systems are constructed based on the infinite dimensional representation for solutions of the yang baxter equation and the reflection equation

representation of the yangian invariant motif and the macdonald polynomial the representation of the yangian invariant motif is considered the relationship with the rogers szego polynomial is studied whose one parameter deformation is the macdonald polynomial we propose the deformation of the motifs which provides a new realization of the macdonald polynomials for the one row young diagrams

quantum integrability of the generalized elliptic ruijsenaars models the quantum integrability of the generalized elliptic ruijsenaars models is shown these models are mathematically related to the macdonald operator and the macdonald koornwinder operator which appeared in the q orthogonal polynomial theories we construct these integrable families by using the yang baxter equation and the reflection equation

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

fluctuation operators and spontaneous symmetry breaking we develop an alternative approach to this field which was to a large extent developed by verbeure et al it is meant to complement their approach which is largely based on a noncommutative central limit theorem and coordinate space estimates in contrast to that we deal directly with the limits of l point truncated correlation functions and show that they typically vanish for l 3 provided that the respective scaling exponents of the fluctuation observables are appropriately chosen this direct approach is greatly simplified by the introduction of a smooth version of spatial averaging which has a much nicer scaling behavior and the systematic development of fourier space and energy momentum spectral methods we both analyze the regime of normal fluctuations the various regimes of poor clustering and the case of spontaneous symmetry breaking or goldstone phenomenon

dynamics of fluctuations for quantum lattice systems for short range interactions and forl1 space clustering states it is proved that there exists a bonafide time evolution on the set of normal fluctuations this dynamics is applied to derive the notion of equilibrium state of the algebra of fluctuations

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

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

quantum de finetti theorem under fully one way adaptive measurements we prove a version of the quantum de finetti theorem permutation invariant quantum states are well approximated as a probabilistic mixture of multifold product states the approximation is measured by distinguishability under measurements that are implementable by fully one way local operations and classical communication locc our result strengthens brandao and harrow s de finetti theorem where a kind of partially one way locc measurements was used for measuring the approximation with essentially the same error bound as main applications we show i a quasipolynomial time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant measured with a variant of the relative entropy of entanglement and ii a proof that in quantum merlin arthur proof systems polynomially many provers are not more powerful than a single prover when the verifier is restricted to one way locc operations

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

a fermion levy theorem the notion of quantum process with continuous trajectories is defined in terms of mutual quadratic variations and it is proved that for classical stochastic processes this notion of continuity of trajectories coincides with the usual one our main result is that any continuous trajectory difference martingale m which is a grassmann measure with scalar non atomic brackets is isomorphic to a fermion white noise mean zero fermi gaussian family whose covariance coincides with the brackets of m this is a fermion version of the levy representation theorem for classical brownian motion

the brachistochrone problem in open quantum systems recently the quantum brachistochrone problem has been discussed in the literature by using non hermitian hamilton operators of different types here it is demonstrated that the passage time is tunable in realistic open quantum systems due to the biorthogonality of the eigenfunctions of the non hermitian hamilton operator as an example the numerical results obtained by bulgakov et al for the transmission through microwave cavities of different shapes are analyzed from the point of view of the brachistochrone problem the passage time is shortened in the crossover from the weak coupling to the strong coupling regime where the resonance states overlap and many branch points exceptional points in the complex plane exist the effect can not be described in the framework of the standard quantum mechanics with the hermitian hamilton operator and consideration of s matrix poles

large deviations for ideal quantum systems we consider a general d dimensional quantum system of non interacting particles in a very large formally infinite container we prove that in equilibrium the fluctuations in the density of particles in a subdomain of the container are described by a large deviation function related to the pressure of the system that is untypical densities occur with a probability exponentially small in the volume of with the coefficient in the exponent given by the appropriate thermodynamic potential furthermore small fluctuations satisfy the central limit theorem

a tracial quantum central limit theorem we prove a central limit theorem for non commutative random variables in a von neumann algebra with a tracial state any non commutative polynomial of averages of i i d samples converges to a classical limit the proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the baker campbell hausdorff expansion the result can be considered a generalization of johansson s theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity math co 9906120 math pr 9909104

product state approximations to quantum states we show that for any many body quantum state there exists an unentangled quantum state such that most of the two body reduced density matrices are close to those of the original state this is a statement about the monogamy of entanglement which cannot be shared without limit in the same way as classical correlation our main application is to hamiltonians that are sums of two body terms for such hamiltonians we show that there exist product states with energy that is close to the ground state energy whenever the interaction graph of the hamiltonian has high degree this proves the validity of mean field theory and gives an explicitly bounded approximation error if we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the hamiltonian satisfies one or more of the following properties 1 high degree 2 small expansion or 3 a ground state where the blocks in the partition have sublinear entanglement previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero our approximations allow an extensive error in energy which is the scale considered by the quantum pcp probabilistically checkable proof and nlts no low energy trivial state conjectures thus our results put restrictions on the possible hamiltonians that could be used for a possible proof of the qpcp or nlts conjectures by contrast the classical pcp constructions are often based on constraint graphs with high degree likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum hamiltonians unless qpcp is false the main technical tool behind our results is a collection of new classical and quantum de finetti theorems which do not make any symmetry assumptions on the underlying states

the green kubo formula and the onsager reciprocity relations in quantum statistical mechanics we study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the green kubo formula and the onsager reciprocity relations for heat fluxes generated by temperature differentials our derivation is axiomatic and the key assumptions concern ergodic properties of non equilibrium steady states

central limit theorem for mixing quantum systems and the ccr algebra of fluctuations we analyse macroscopic fluctuations of an infinite quantum system and introduce the ccr c algebra of normal fluctuations a non commutative central limit theorem for mixing quantum systems is proved

hierarchies integrable decompositions and conservation laws of geng equation abstract in this paper two hierarchies of the geng equations are presented including positive non isospectral hierarchy and negative non isospectral hierarchy moreover integrable couplings of the corresponding geng hierarchies are also constructed by enlarging the associated matrix spectral problem three new integrable decompositions and conservation laws of the isospectral geng equation are also obtained the gauge transformations are used to obtain the associated binary symmetry constraints of the geng equation at the first time

boundary k matrix elliptic dunkl operator and quantum many body systems the infinite dimensional representation of the boundary k operator the solution of the reflection equation is considered the trigonometric k matrix is studied as a degenerate case of the elliptic operator a method to construct the elliptic dunkl operator is proposed and the relationship with the quantum many body problem is also discussed

s matrix theory for transmission through billiards in tight binding approach in the tight binding approximation we consider multi channel transmission through a billiard coupled to leads following dittos we derive the coupling matrix the scattering matrix and the effectiv

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

a quantum central limit theorem for sums of independent identically distributed random variables we formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables

infinitely divisible completely positive mappings we generalise the theory of infinitely divisible positive definite functions f g on a group g to a theory of infinite divisibility for completely positive mappings g taking values in the algebra of bounded operators on some hilbert space

on the central limit theorem for stationary mixing random fields a simple proof of a central limit theorem for stationary random fields under mixing conditions is given generalizing some results obtained by more complicated methods e g bernstein s method

markov shift in non commutative probability abstract we consider a class of quantum dissipative semigroup on a von neumann algebra which admits a normal invariant state we investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical markov shift in case the normal invariant state is also faithful we also extend the notion of quantum detailed balance introduced by frigerio gorini and prove that forward weak markov process and backward weak markov process are equivalent by an anti unitary operator

s matrix formalism of transmission through two quantum billiards coupled by a waveguide we consider a system that consists of two single quantum billiards qbs coupled by a waveguide and study the transmission through this system as a function of length and width of the waveguide to interpret the numerical results for the transmission we explore a simple model with a small number of states which allows us to consider the problem analytically the transmission is described in the s matrix formalism by using the non hermitian effective hamilton operator for the open system the coupling of the single qbs to the internal waveguide characterizes the internal coupling strength u of the states of the system while that of the system as a whole to the attached leads determines the external coupling strength v of the resonance states via the continuum waves in the leads the transmission is resonant for all values of v u in relation to the effective hamiltonian it depends strongly on the ratio v u via the eigenvalues and eigenfunctions of the effective hamiltonian the results obtained are compared qualitatively with those from simulation calculations for larger systems most interesting is the existence of resonance states with vanishing widths that may appear at all values of v u they cause zeros in the transmission through the double qb due to trapping of the particle in the waveguide

entropy and the central limit theorem in quantum mechanics gives a short proof that any infinitely divisible even representation of a clifford algebra or a car algebra is quasi free and of hudson s central limit theorem 1973 for fermions the author shows that the entropy of any even state of the car clifford or ccr algebra is less than or equal to the entropy of the quasi free state with the same two point function