Lieb Inequality Research Articles

Quadrature uncertainty and information entropy of quantum elliptical vortex states

We study the quadrature uncertainty of the quantum elliptical vortex state using the associated Wigner function. Deviations from the minimum uncertainty states were observed due to the absence of Gaussianity. We further observed that there exists an optimum value of ellipticity which gives rise to the maximum entanglement of the two modes of the quantum elliptical vortex states. In our study of entropy, we noticed that with increasing vorticity, entropy increases for both the modes. A further increase in ellipticity reduces the entropy thereby resulting in a loss of information carrying capacity. We check the validity of the entropic inequality relations, namely the subaddivity and the Araki–Lieb inequality. The latter was satisfied only for a very small range of the ellipticity of the vortex, while the former seemed to be valid at all values.

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa–Leindler and Brascamp–Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy–Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.

Quantum Information Theoretical Analysis of Quantum Secret Sharing

Quantum secret sharing (QSS) schemes are analyzed from an information theoretical perspective centered on the Araki—Lieb inequality. Based on this inequality, mathematical characterizations of QSS schemes and quantum error-correcting codes (QECCs) are given. Furthermore, we present a proof of the relation between QSS schemes and QECCs. This information theoretic description of QSS schemes is used to derive the quantum Singleton bound.

Global Nonlinear Brascamp–Lieb Inequalities

We prove global versions of certain known nonlinear Brascamp–Lieb inequalities under a natural homogeneity assumption. We also establish a conditional theorem allowing one to generally pass from local to global nonlinear Brascamp–Lieb estimates under such a homogeneity assumption.

Geometric Brascamp–Lieb Has the Optimal Best Constant

We study the optimal best constant for the Brascamp–Lieb inequality and show that it is furnished exactly by the geometric Brascamp–Lieb inequality.

Some nonlinear Brascamp–Lieb inequalities and applications to harmonic analysis

We use the method of induction-on-scales to prove certain diffeomorphism-invariant nonlinear Brascamp–Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru–Herr–Tataru and Bennett–Carbery–Wright.

The Brascamp–Lieb Polyhedron

AbstractA set of necessary and sufficient conditions for the Brascamp–Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank 1. This complements the result of Barthe concerning the case where the linear maps all have rank 1. Pushing our analysis further, we describe the case where the maps have either rank 1 or rank 2.A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp–Lieb inequality to hold. We present an algorithm which generates such a list.

VOLUME ESTIMATES FOR L p ‐ZONOTOPES AND BEST BEST CONSTANTS IN BRASCAMP–LIEB INEQUALITIES

Given some unit vectors a1,…,am∈ℝn that span all ℝn and some positive numbers θ1,…,θm, we consider for every p≥1 the convex body We will give some upper bounds for the volume of Kp and some lower bounds for the volume of its polar, depending on some parameters, which improve the ones obtained using the Brascamp–Lieb inequality. We will also see how the best choice of these parameters is related to the transformation which takes Kp to a special position which, for instance, when p=∞, is John's position.

Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $${\mathbb {R}^n}$$ , and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schrödinger operators. Though we focus mainly on the case of random variables in $${\mathbb {R}^n}$$ in this paper, we discuss extensions to other settings as well.

Finsler interpolation inequalities

We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.

A new lower bound inequality for the norm of the symmetric product

If V is a complex finite dimensional inner product space, and q is a positive integer, then T q (V) denotes the set of all symmetric q-linear ℂ-valued functions with domain V × V × ··· × V- (q copies), and S q (V) denotes the subspace of T q (V) whose elements are the fully symmetric q-linear ℂ-valued functions. If A ∈ S n (V) and B ∈ S p (V), then we define the tensor product, A ⊗ B, and the symmetric product, A · B, in the usual way, and we obtain an inner product, and norm, on T q (V) by extending the inner product on V. If x ∈ V and C ∈ S q (V), then C(x) denotes the member of S q − 1(V) such that C(x) (v 1, v 2, … , v q − 1) = C(x, v 1, v 2, … , v q − 1) for all v 1, v 2, … , v q − 1 ∈ V. We prove that if is an orthonormal basis for V, then for all A ∈ S n (V) and all B ∈ S p (V). This inequality improves upon Neuberger's inequality, which states that if A ∈ S n (V) and B ∈ S p (V), then . It also implies a known refinement to Lieb's inequality, and generates several new permanental inequalities as well.

Pfaffians, hafnians and products of real linear functionals

We prove pfaffian and hafnian versions of Lieb's inequalities on determinants and permanents of positive semi-definite matrices. We use the hafnian inequality to improve the lower bound of R\'ev\'esz and Sarantopoulos on the norm of a product of linear functionals on a real Euclidean space (this subject is sometimes called the `real linear polarization constant' problem).

The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals

We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. Our main tool is a monotonicity formula for positive solutions to heat equations in linear and multilinear settings, which was first used in this type of setting by Carlen, Lieb, and Loss [CLL]. In that paper, the heat flow method was used to obtain the rank-one case of Lieb’s fundamental theorem concerning exhaustion by gaussians; we extend the technique to the higher-rank case, giving two new proofs of the general-rank case of Lieb’s theorem.

Integration by Parts Formulae for Wiener Measures on a Path Space between two Curves

This paper is concerned with the integration by parts formulae for the pinned or the standard Wiener measures restricted on a space of paths staying between two curves. The boundary measures, concentrated on the set of paths touching one of the curves once, are specified. Our approach is based on the polygonal approximations. In particular, to establish the convergence of boundary terms, a uniform estimate is derived by means of comparison argument for a sequence of random walks conditioned to stay between two polygons. Applying the Brascamp–Lieb inequality, the stochastic integrals of Wiener type are constructed relative to the three-dimensional Bessel bridge or the Brownian meander.

On stochastic domination in the BrascampLieb framework

We exploit a recent approach to Brascamp–Lieb inequalities, due to Caffarelli [5], and reconsider earlier approaches to establish stochastic domination inequalities between Gaussian variables and random variables with density of the form a log-concave or log-convex function. These extend to inequalities on random vectors via a classical result by Prékopa and Leindler and they complement the Brascamp–Lieb moment inequalities. Some applications to a class of Gibbs measures, the anharmonic crystals, are developed.

Orbital magnetism of a two-dimensional noncommutative confined system

We study a system of spinless electrons moving in a two-dimensional noncommutative space subject to a perpendicular magnetic field and confined by a harmonic potential type 1/2mw0 r 2. We look for the orbital magnetism of the electrons in different regimes of temperature T, magnetic field and noncommutative parameter θ. We prove that the degeneracy of Landau levels can be lifted by the θ-term appearing in the electron energy spectrum at weak magnetic field. Using the Berezin–Lieb inequalities for thermodynamical potential, it is shown that in the high-temperature limit, the system exibits a magnetic θ-dependent behaviour, which is missing in the commutative case. Moreover, a correction to susceptibility at low T is observed. Using the Fermi–Dirac trace formulas, a generalization of the thermodynamical potential, the average number of electrons and the magnetization is obtained. There is a critical point where the thermodynamical potential becomes infinite in both of the two methods above. So at this point we deal with the partition function by adopting another approach. The standard results in the commutative case for this model can be recovered by switching off the θ-parameter.

From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities

We develop several applications of the Brunn—Minkowski inequality in the Prekopa—Leindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Prekopa—Leindler theorem the Brascamp—Lieb inequality for strictly convex potentials. We deduce similarly the logarithmic Sobolev inequality for uniformly convex potentials for which we deal more generally with arbitrary norms and obtain some new results in this context. Applications to transportation cost and to concentration on uniformly convex bodies complete the exposition.

On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality

A study is made of an integral identity of Helffer and Sj{\"o}strand, which for some class of probability measures yields a formula for the covariance of two functions (of a stochastic variable). In comparison with the Brascamp--Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten-Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp--Lieb's inequality is obtained. For a certain class of measures occurring in statistical mechanics, explicit criteria for the means are found from the Persson--Agmon formula, from compactness of embeddings and from the Weyl calculus of pseudo-differential operators, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.

Entropic repulsion for massless fields

We consider the anharmonic crystal, or lattice massless field, with 0-boundary conditions outside D N=ND∩ Z d, D⊆ R d and N a large natural number, that is the finite volume Gibbs measure P N on {ϕ∈ R Z d :ϕ x=0 for every x∉ D N } with Hamiltonian ∑ x∼yV(ϕ x−ϕ y), V a strictly convex even function. We establish various bounds on P N(Ω +(D N)) , where Ω +(D N)={ϕ:ϕ x⩾0 for all x∈ D N }. Then we extract from these bounds the asymptotics ( N→∞) of P N(·|Ω +(D N)) : roughly speaking we show that the field is repelled by a hard-wall to a height of O( log N ) in d⩾3 and of O( log N) in d=2. If we interpret ϕ x as the height at x of an interface in a ( d+1)-dimensional space, our results on the conditioned measure P N(·|Ω +(D N)) clarify some aspects of the effect of a hard-wall on an interface. Besides classical techniques, like the FKG inequalities and the Brascamp–Lieb inequalities for log-concave measures, we exploit a representation of the random field in term of a random walk in dynamical random environment (Helffer–Sjöstrand representation).

Remarks on Decay of Correlations and Witten Laplacians Brascamp–Lieb Inequalities and Semiclassical Limit

As it appears in recent articles by Helffer or Sjöstrand and Naddaf–Spencer, the analysis, in the context of the statistical mechanics, of measures of the type exp−Φ(x)dxis connected with the analysis of suitable Witten Laplacians on 1-forms. For illustrating this point of view, we present here remarks about the Brascamp–Lieb inequalities and its extensions and prove the decay of the correlation in some cases whenΦis weakly non convex.