Quantum Marginal Inequalities and the Conjectured Entropic Inequalities

A conjecture – the modified super-additivity inequality of relative entropy – was proposed in Zhang et al. (Phys. Lett. A 377:1794–1796, 2013): There exist three unitary operators \(U_{A}\in \mathrm {U}(\mathcal {H}_{A}), U_{B}\in \mathrm {U}(\mathcal {H}_{B})\) , and \(U_{AB}\in \mathrm {U}(\mathcal {H}_{A}\otimes \mathcal {H}_{B})\) such that $$\mathrm{S}\left(U_{AB}\rho_{AB}U^{\dagger}_{AB}||\sigma_{AB}\right)\geqslant \mathrm{S}\left(U_{A}\rho_{A}U^{\dagger}_{A}||\sigma_{A}\right) + \mathrm{S}\left(U_{B}\rho_{B}U^{\dagger}_{B}||\sigma_{B}\right), $$ where the reference state σ is required to be full-ranked. A numerical study on the conjectured inequality is conducted in this note. The results obtained indicate that the modified super-additivity inequality of relative entropy seems to hold for all qubit pairs.     

Some Special Functions of Noncommuting Variables

In this paper we deal with q-commuting variables x and y satisfying the relation xy = qyx + (q – 1)y ² with q complex, 0 < |q| < 1. We study various functional equations for q-exponentials and we deduce some identities for q-special functions involving q-commuting variab les.     

Entropic Inequalities for Two Coupled Superconducting Circuits

We discuss the known construction of two interacting superconducting circuits based on Josephson junctions, which can be precisely engineered and easily controlled. In particular, we use the parametric excitation of two circuits realized by an instant change of the qubit coupling to study entropic and information properties of the density matrix of a composite system. We obtain the density matrix from the initial thermal state and perform its analysis in the approximation of small perturbation parameter and sufficiently low temperature. We also check the subadditivity condition for this system both for the von Neumann entropy and deformed entropies and check the dependence of mutual information on the system temperature. Finally, we discuss the applicability of this approach to describe the two coupled superconducting qubits as harmonic oscillators with limited Hilbert space.     

Exhibition of Monogamy Relations between Entropic Non-contextuality Inequalities

We exhibit the monogamy relation between two entropic non-contextuality inequalities in the scenario where compatible projectors are orthogonal. We show the monogamy relation can be exhibited by decomposing the orthogonality graph into perfect induced subgraphs. Then we find two entropic non-contextuality inequalities are monogamous while the KCBS-type non-contextuality inequalities are not if the orthogonality graphs of the observable sets are two odd cycles with two shared vertices.     

Weighted Information and Weighted Entropic Inequalities for Qutrit States

We review the notion of weighted quantum entropy and consider the weighted quantum entropy for bipartite and noncomposite quantum systems. We extend the subadditivity condition, the inequality known for the weighted entropy information, to the case of indivisible qudit system, such as a qutrit. We discuss the new inequality for the qutrit density matrix for different weights and states, as well as the role of weighted entropy with respect to nonlinear quantum channels.     

Entropic and Information Inequalities for Indivisible Qudit Systems<Superscript>*</Superscript>

We present the idea that in both classical and quantum systems all correlations available for composite multipartite systems, e.g., bipartite systems, exist as “hidden correlations” in indivisible (noncomposite) systems. The presence of correlations is expressed by entropic-information inequalities known for composite systems like the subadditivity condition. We show that the mathematically identical subadditivity condition and the mutual information nonnegativity are available as well for noncomposite systems like a single-qudit state. We demonstrate an explicit form of the subadditivity condition for a qudit with j = 2 or the five-level atom. We consider the possibility to check the subadditivity condition (entropic inequality) in experiments where such a system is realized by the superconducting circuit based on Josephson-junction devices.     

熵测不准关系与光场的熵压缩

用熵作为光场量子涨落的量度,根据熵测不准关系,建立了熵压缩的概念,具体研究了光场与原子相互作用时的熵压缩,结果显示,熵压缩实现了对光场压缩效应的高灵敏量度。     

Factorization Method and Special Orthogonal Functions

We present a general construction for ladder operators for the special orthogonal functions based on Nikiforov-Uvarov mathematical formalism. A list of creation and annihilation operators are provided for the well known special functions. Furthermore, we establish the dynamic group associated with these operators.     

On the Grassmannian Special Functions I

A generalization of the notion of a special function to the case of anticommuting variables is presented. In particular, Grassmann-Hermite multinomials are obtained and their elementary properties are displayed.Work partially supported by Research Program of the Polish Ministry of Higher Education, RPI 10.     

Extended Rearrangement Inequalities and Applications to Some Quantitative Stability Results

In this paper, we prove a new functional inequality of Hardy–Littlewood type for generalized rearrangements of functions. We then show how this inequality provides quantitative stability results of steady states to evolution systems that essentially preserve the rearrangements and some suitable energy functional, under minimal regularity assumptions on the perturbations. In particular, this inequality yields a quantitative stability result of a large class of steady state solutions to the Vlasov–Poisson systems, and more precisely we derive a quantitative control of the L¹ norm of the perturbation by the relative Hamiltonian (the energy functional) and rearrangements. A general non linear stability result has been obtained by Lemou et al. (Invent Math 187:145–194, 2012) in the gravitational context, however the proof relied in a crucial way on compactness arguments which by construction provides no quantitative control of the perturbation. Our functional inequality is also applied to the context of 2D-Euler systems and also provides quantitative stability results of a large class of steady-states to this system in a natural energy space.     

Special Temporal Functions on Globally Hyperbolic Manifolds

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings in Minkowski spaces.     

The Brjuno Functions and Their Regularity Properties

We show that various possible versions of the Brjuno function, based on different kinds of continued fraction developments, are all equivalent and we study their regularity (L ^p, BMO and H?lder) properties, through a systematic analysis of the functional equation which they fulfill. Received: 21 March 1995 / Accepted: 8 August 1996     

Properties of Generalized Univariate Hypergeometric Functions

Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E ₇ (elliptic, hyperbolic) and of type E ₆ (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.     

Concentration Inequalities for Functions of Gibbs Fields with Application to Diffraction and Random Gibbs Measures

We derive useful general concentration inequalities for functions of Gibbs fields in the uniqueness regime. We also consider expectations of random Gibbs measures that depend on an additional disorder field, and prove concentration w.r.t. the disorder field. Both fields are assumed to be in the uniqueness regime, allowing in particular for non-independent disorder fields. The modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the Dobrushin contraction coefficient, and are explicitly computable. On the basis of these inequalities, we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in Euclidean space, restricted to a finite volume. Here we also allow for thermal dislocations of the scatterers around their equilibrium positions. Extending recent results for independent scatterers, we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean. The bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set.Work supported by the DFG     

Quantum Surfaces,Special Functions,and the Tunneling Effect

The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2-form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. This is interpreted as a tunneling effect in the quantum geometry.     

Some Remarks on Special Conformal and Special Projective Symmetries in General Relativity

Those space-times admitting special conformal vector fields and those admitting special projective vector fields have recently been studied. In this paper these two classes of space times are shown to be very closely related to each other. Certain uniqueness features of (and necessary extra symmetries contained in) the associated Lie algebras are discussed and the dimensionality of each of the algebras is computed.     

On Some Harris-FKG Type Correlation Inequalities for a Non-Attractive Model

We consider the following non-attractive one-dimensional model whose evolution is given by ξ_n+1(x)=ξ_n(x+1)+ξ_n(x?1) (mod 2) with probability p, =0 with probability 1?p. In the present letter, we prove that some Harris-FKG type correlation inequalities hold for this model. Moreover it is shown that a correlation inequality is also correct for the more general non-attractive class.     

Entropic characteristics of photon tomograms

We study the electromagnetic-field tomograms for classical and quantum states. We use the violation of the positivity of entropy for the photon-probability distributions for distinguishing the classical and quantum domains. We show that the photon-probability distribution expressed in terms of optical or symplectic tomograms of the photon quantum state must be a nonnegative function, which yields the nonnegative Shannon entropy. We also show that the optical tomogram of the photon classical state provides the expression for the Shannon entropy, which can be nonpositive.