Showing papers in "Communications in Mathematical Physics in 2020"

Recovering the QNEC from the ANEC

Abstract: We study the relative entropy in QFT comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy (ANE) of possible purifications. This variational expression can be used to easily prove the quantum null energy condition (QNEC). We formulate Wall’s conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this context. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.

Bit Threads and Holographic Monogamy

Abstract: Bit threads provide an alternative description of holographic entanglement, replacing the Ryu–Takayanagi minimal surface with bulk curves connecting pairs of boundary points. We use bit threads to prove the monogamy of mutual information property of holographic entanglement entropies. This is accomplished using the concept of a so-called multicommodity flow, adapted from the network setting, and tools from the theory of convex optimization. Based on the bit thread picture, we conjecture a general ansatz for a holographic state, involving only bipartite and perfect-tensor type entanglement, for any decomposition of the boundary into four regions. We also give new proofs of analogous theorems on networks.

Vertex Algebras for S-duality

Abstract: We define new deformable families of vertex operator algebras $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras are equipped with two $$\mathfrak {g}$$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of vertex operator algebras equipped with a $$\mathfrak {g}$$ affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of vertex operator algebras. The space of conformal blocks (in the derived sense, i.e. chiral homology) for $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras.

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

Abstract: We prove that for each $$\gamma \in (0,2)$$, there is an exponent $$d_\gamma > 2$$, the “fractal dimension of $$\gamma $$-Liouville quantum gravity (LQG)”, which describes the ball volume growth exponent for certain random planar maps in the$$\gamma $$-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $$\gamma $$-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that $$d_\gamma $$ is a continuous, strictly increasing function of $$\gamma $$ and prove upper and lower bounds for $$d_\gamma $$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $$\gamma =\sqrt{2}$$ (which corresponds to spanning-tree weighted planar maps) our bounds give $$3.4641 \le d_{\sqrt{2}} \le 3.63299$$ and in the limiting case we get $$4.77485 \le \lim _{\gamma \rightarrow 2^-} d_\gamma \le 4.89898$$.

Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

Abstract: Materials science and the study of the electronic properties of solids are a major field of interest in both physics and engineering. The starting point for all such calculations is single-electron, or non-interacting, band structure calculations, and in the limit of strong on-site confinement this can be reduced to graph-like tight-binding models. In this context, both mathematicians and physicists have developed largely independent methods for solving these models. In this paper we will combine and present results from both fields. In particular, we will discuss a class of lattices which can be realized as line graphs of other lattices, both in Euclidean and hyperbolic space. These lattices display highly unusual features including flat bands and localized eigenstates of compact support. We will use the methods of both fields to show how these properties arise and systems for classifying the phenomenology of these lattices, as well as criteria for maximizing the gaps. Furthermore, we will present a particular hardware implementation using superconducting coplanar waveguide resonators that can realize a wide variety of these lattices in both non-interacting and interacting form.

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

Abstract: In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell $$F_4$$ type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.

VOAs and Rank-Two Instanton SCFTs

Abstract: We analyze the $$\mathcal {N}=2$$ superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.

Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms

Abstract: We formulate a family of spin Topological Quantum Field Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf–Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group G symmetry. We use the fact that the torsion part of the classification is given by Pontryagin duals to spin-bordism groups of the classifying space BG. We also consider unoriented analogues, that is G-equivariant invertible $$\hbox {pin}^\pm $$ -TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian G using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in 3, 4 and other dimensions. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of ’t Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary spin-TQFTs (surface fermionic topological orders). We explore SPT and symmetry enriched topologically (SET) ordered states, and crystalline SPTs protected by space-group (e.g. translation $${\mathbb {Z}}$$ ) or point-group (e.g. reflection, inversion or rotation $$C_m$$ ) symmetries, via the layer-stacking construction.

Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity

Abstract: This second part of the series treats spin $$\pm 2$$ components (or extreme components), that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowly rotating Kerr black hole. For each of these two components, after performing a first-order differential operator once and twice, the resulting equations together with the Teukolsky master equation itself constitute a linear spin-weighted wave system. An energy and Morawetz estimate for spin $$\pm 2$$ components is proved by treating this system. This is a first step in a joint work (Andersson et al. in Stability for linearized gravity on the Kerr spacetime, arXiv:1903.03859 , 2019) in addressing the linear stability of slowly rotating Kerr metrics.

Quantum Spectral Methods for Differential Equations

Abstract: Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a d-dimensional system of linear equations or linear differential equations with complexity $${{\,\mathrm{poly}\,}}(\log d)$$. While several of these algorithms approximate the solution to within $$\epsilon $$ with complexity $${{\,\mathrm{poly}\,}}(\log (1/\epsilon ))$$, no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity $${{\,\mathrm{poly}\,}}(\log d, \log (1/\epsilon ))$$.

Entanglement Entropy and Berezin–Toeplitz Operators

Abstract: We consider Berezin–Toeplitz operators on compact Kahler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being proportional to the Riemannian volume of the boundary. As a consequence, we deduce the area law for the entanglement entropy of integer quantum Hall states. Another application is for the determinantal processes with correlation kernel the Bergman kernels of a positive line bundle: we prove that the number of points in a smooth domain is asymptotically normal.

Ranks of Operators in Simple C ∗ -algebras with Stable Rank One

Abstract: Let A be a simple $$C^*$$-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of A is realized as the rank of an operator in the stabilization of A. Assuming moreover that A has locally finite nuclear dimension, we deduce that A is $$\mathcal {Z}$$-stable if and only if it has strict comparison of positive elements. In particular, the Toms–Winter conjecture holds for simple, approximately subhomogeneous $$C^*$$-algebras with stable rank one.

A Bethe Ansatz type formula for the superconformal index

Abstract: Inspired by recent work by Closset, Kim, and Willett, we derive a new formula for the superconformal (or supersymmetric) index of 4D $${\mathcal {N}}=1$$ theories. Such a formula is a finite sum, over the solution set of certain transcendental equations that we dub Bethe Ansatz Equations, of a function evaluated at those solutions.

Products of Many Large Random Matrices and Gradients in Deep Neural Networks

Abstract: We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $$\ell _2$$ norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity. Depending on the scaling limit considered, the mean and variance of the limiting Gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn. We also obtain explicit error bounds on the moments of the norm and the Kolmogorov-Smirnov distance to a Gaussian. Finally, we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with ReLU activations. This provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with ReLU activations and a given architecture.

Optimal Rate for Bose–Einstein Condensation in the Gross–Pitaevskii Regime

Abstract: We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential.

Equivariant K-Theory and Refined Vafa-Witten Invariants

Abstract: In Maulik and Thomas (in preparation) the Vafa–Witten theory of complex projective surfaces is lifted to oriented $${\mathbb {C}}^*$$ -equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in $$t^{1/2}$$ invariant under $$t^{1/2}\leftrightarrow t^{-1/2}$$ which specialise to numerical Vafa–Witten invariants at $$t=1$$ . On the “instanton branch” the invariants give the virtual $$\chi ^{}_{-t}$$ -genus refinement of Gottsche–Kool, extended to allow for strictly semistable sheaves. Applying modularity to their calculations gives predictions for the contribution of the “monopole branch”. We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of Gottsche–Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon–Northcott complexes, and show they calculate refined Vafa–Witten invariants. Using this Laarakker (Monopole contributions to refined Vafa–Witten invariants. arXiv:1810.00385 ) proves universality results for the invariants.

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

Abstract: We study stability times for a family of parameter dependent nonlinear Schrodinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.

Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies

Abstract: We study symmetry-enriched topological (SET) phases in 2+1 space-time dimensions with spatial reflection and/or time-reversal symmetries. We provide a systematic construction of a wide class of reflection and time-reversal SET phases in terms of a topological path integral defined on general space-time manifolds. An important distinguishing feature of different topological phases with reflection and/or time-reversal symmetry is the value of the path integral on non-orientable space-time manifolds. We derive a simple general formula for the path integral on the manifold $$\Sigma ^2 \times S^1$$, where $$\Sigma ^2$$ is a two-dimensional non-orientable surface and $$S^1$$ is a circle. This also gives an expression for the ground state degeneracy of the SET on the surface $$\Sigma ^2$$ that depends on the reflection symmetry fractionalization class, generalizing the Verlinde formula for ground state degeneracy on orientable surfaces. Consistency of the action of the mapping class group on non-orientable manifolds leads us to a constraint that can detect when a time-reversal or reflection SET phase is anomalous in (2+1)D and, thus, can only exist at the surface of a (3+1)D symmetry protected topological (SPT) state. Given a (2+1)D reflection and/or time-reversal SET phase, we further derive a general formula that determines which (3+1)D reflection and/or time-reversal SPT phase hosts the (2+1)D SET phase as its surface termination. A number of explicit examples are studied in detail.

Magnetic Skyrmions at Critical Coupling

Abstract: We introduce a family of models for magnetic skyrmions in the plane for which infinitely many solutions can be given explicitly. The energy defining the models is bounded below by a linear combination of degree and total vortex strength, and the configurations attaining the bound satisfy a first order Bogomol’nyi equation. We give explicit solutions which depend on an arbitrary holomorphic function. The simplest solutions are the basic Bloch and Neel skyrmions, but we also exhibit distorted skyrmions and anti-skyrmions as well as line defects and configurations consisting of skyrmions and anti-skyrmions.

Quantum fields and local measurements

Abstract: The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the “system”, using another, the “probe”. The measurement process involves a dynamical coupling of “system” and “probe” within a bounded spacetime region. The resulting “coupled theory” determines a scattering map on the uncoupled combination of the “system” and “probe” by reference to natural “in” and “out” spacetime regions. No specific interaction is assumed and all constructions are local and covariant. Given any initial state of the probe in the “in” region, the scattering map determines a completely positive map from “probe” observables in the “out” region to “induced system observables”, thus providing a measurement scheme for the latter. It is shown that the induced system observables may be localized in the causal hull of the interaction coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. Post-selected states conditioned on measurement outcomes are obtained using Davies–Lewis instruments that depend on the initial probe state. Composite measurements involving causally ordered coupling regions are also considered. Provided that the scattering map obeys a causal factorization property, the causally ordered composition of the individual instruments coincides with the composite instrument; in particular, the instruments may be combined in either order if the coupling regions are causally disjoint. This is the central consistency property of the proposed framework. The general concepts and results are illustrated by an example in which both “system” and “probe” are quantized linear scalar fields, coupled by a quadratic interaction term with compact spacetime support. System observables induced by simple probe observables are calculated exactly, for sufficiently weak coupling, and compared with first order perturbation theory.

Anomalies and Bosonization

Abstract: Recently, general methods of bosonization beyond 1+1 dimensions have been developed. In this article, we review these bosonizations and extend them to systems with boundary. Of particular interest is the case when the bulk theory is a G-symmetry protected topological phase and the boundary theory has a G ’t Hooft anomaly. We discuss how, when the anomaly is not realizable in a bosonic system, the G symmetry algebra becomes modified in the bosonization of the anomalous theory. This gives us a useful tool for understanding anomalies of fermionic systems, since there is no way to compute a boundary gauge variation of the anomaly polynomial, as one does for anomalies of bosonic systems. We take the chiral anomalies in 1+1D as case studies and comment on our expectations for parity/time reversal anomalies in 2+1D. We also provide a derivation of new constraints in SPT phases with domain defects decorated by $$p+ip$$ superconductors and Kitaev strings, which is necessary to understand the bosonized symmetry algebras which appear.

S-duality for the Large N = 4 Superconformal Algebra

Abstract: We prove some conjectures about vertex algebras which emerge in gauge theory constructions associated to the geometric Langlands program. In particular, we present the conjectural kernel vertex algebra for the $$S T^2 S$$ duality transformation in SU(2) gauge theory. We find a surprising coincidence, which gives a powerful hint about the nature of the corresponding duality wall. Concretely, we determine the branching rules for the small $$N=4$$ superconformal algebra at central charge $$-9$$ as well as for the generic large $$N=4$$ superconformal algebra at central charge $$-6$$. Moreover we obtain the affine vertex superalgebra of $$\mathfrak {osp}(1|2)$$ and the $$N=1$$ superconformal algebra times a free fermion as quantum Hamiltonian reductions of the large $$N=4$$ superconformal algebras at $$c=-6$$.

The Structure of Fluctuations in Stochastic Homogenization

Abstract: Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. In case of random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify another crucial intrinsic quantity, motivated by H-convergence, which we refer to as the homogenization commutator and is related to variational quantities first considered by Armstrong and Smart. In the simplified setting of the random conductance model, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability, which is expressed in form of a suitable two-scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization. In addition, we show that the (rescaled) homogenization commutator converges in law to a Gaussian white noise, and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method. This collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates. Extensions to the (non-symmetric) continuum setting are also discussed, the details of which are postponed to forthcoming works.

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

Abstract: In this paper we are concerned with resolvent estimates for the Laplacian $$\Delta $$ in Euclidean spaces. Uniform resolvent estimates for $$\Delta $$ were shown by Kenig et al. (Duke Math J 55(2):329–347, 1987) who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp $$L^p$$–$$L^q$$ bounds depending on z has not been considered in a general framework which admits all possible p, q. In this paper, we present a complete picture of sharp $$L^p$$–$$L^q$$ resolvent estimates, which may depend on z. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner–Riesz operators of negative index.

Dissipative Solutions and Semiflow Selection for the Complete Euler System

Abstract: To circumvent the ill-posedness issues present in various models of continuum fluid mechanics, we present a dynamical systems approach aiming at the selection of physically relevant solutions. Even under the presence of infinitely many solutions to the full Euler system describing the motion of a compressible inviscid fluid, our approach permits to select a system of solutions (one trajectory for every initial condition) satisfying the classical semiflow property. Moreover, the selection respects the well accepted admissibility criteria for physical solutions, namely, maximization of the entropy production rate and the weak–strong uniqueness principle. Consequently, strong solutions are always selected whenever they exist and stationary states are stable and included in the selection as well. To this end, we introduce a notion of dissipative solution, which is given by a triple of density, momentum and total entropy defined as expectations of a suitable measure-valued solution.

Weighted Hurwitz Numbers and Topological Recursion

Abstract: The KP and 2D Toda $$\tau $$-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel–Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations in the case of polynomial weight functions.

Stochastic PDE Limit of the Six Vertex Model

Abstract: We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $$\Delta \rightarrow 1^+$$ so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the Kardar–Parisi–Zhang (KPZ) equation. We also prove that the one-dimensional family of stochastic Gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic Burgers equation. Our proofs rely upon the Markov (self) duality of our model. The starting point is an exact microscopic Hopf–Cole transform for the stochastic six vertex model which follows from the model’s known one-particle Markov self-duality. Given this transform, the crucial step is to establish self-averaging for specific quadratic function of the transformed height function. We use the model’s two-particle self-duality to produce explicit expressions (as Bethe ansatz contour integrals) for conditional expectations from which we extract time-decorrelation and hence self-averaging in time. The crux of our Markov duality method is that the entire convergence result reduces to precise estimates on the one-particle and two-particle transition probabilities. Previous to our work, Markov dualities had only been used to prove convergence of particle systems to linear Gaussian SPDEs (e.g. the stochastic heat equation with additive noise).

A Many-Body Index for Quantum Charge Transport

Abstract: We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb–Schultz–Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.

Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses

Abstract: In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock–Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2-log-Sobolev and 1-log-Sobolev inequalities respectively. We then prove some tensorization-type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers. Finally as an application of these results, we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting. We prove strong converse bounds for the problems of quantum hypothesis testing and classical-quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive.

Regularization Estimates and Cauchy Theory for Inhomogeneous Boltzmann Equation for Hard Potentials Without Cut-Off

Abstract: In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant framework for this equation. Moreover, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.