Tight Inequalities Research Articles

All tight correlation Bell inequalities have quantum violations

It is by now well-established that there exist non-local games for which the best entanglement-assisted performance is not better than the best classical performance. Here we show in contrast that any two-player XOR game, for which the corresponding Bell inequality is tight, has a quantum advantage. In geometric terms, this means that any correlation Bell inequality for which the classical and quantum maximum values coincide, does not define a facet, i.e. a face of maximum dimension, of the local Bell polytope. Indeed, using semidefinite programming duality, we prove upper bounds on the dimension of these faces, bounding it far away from the maximum. In the special case of non-local computation games, it had been shown before that they are not facet-defining; our result generalises and improves this. As a by-product of our analysis, we find a similar upper bound on the dimension of the faces of the convex body of quantum correlation matrices, showing that (except for the trivial ones expressing the non-negativity of probability) it does not have facets.

Sharp Beckner-type inequalities for Cauchy and spherical distributions

Using some harmonic extensions on the upper-half plane, and probabilistic representations, and curvature-dimension inequalities with some negative dimensions, we obtain some new opimal functional inequalities of the Beckner type for the Cauchy type distributions on the Euclidean space. These optimal inequalities appear to be equivalent to some non tight optimal Beckner inequalities on the sphere, and the family appear to be a new form of the Sobolev inequality.

Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind t_{mu ,nu }(x). In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function t_{mu ,nu }(x), for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.

A minimax approach to one-shot entropy inequalities

One-shot information theory entertains a plethora of entropic quantities, such as the smooth max-divergence, hypothesis testing divergence, and information spectrum divergence, that characterize various operational tasks in quantum information theory and are used to analyze their asymptotic behavior. Tight inequalities between these quantities are thus of immediate interest. In this note, we use a minimax approach (appearing previously, for example, in the proofs of the quantum substate theorem), to simplify the quantum problem to a commutative one, which allows us to derive such inequalities. Our derivations are conceptually different from previous arguments and in some cases lead to tighter relations. We hope that the approach discussed here can lead to progress in open problems in quantum Shannon theory and exemplify this by applying it to a simple case of the joint smoothing problem.

Relating broadcast independence and independence

An independent broadcast on a connected graph G is a function f:V(G)→N0 such that, for every vertex x of G, the value f(x) is at most the eccentricity of x in G, and f(x)>0 implies that f(y)=0 for every vertex y of G within distance at most f(x) from x. The broadcast independence number αb(G) of G is the largest weight ∑x∈V(G)f(x) of an independent broadcast f on G. Clearly, αb(G) is at least the independence number α(G) for every connected graph G. Our main result implies αb(G)≤4α(G). We prove a tight inequality and characterize all extremal graphs.

Polygamy relations of multipartite entanglement beyond qubits

We investigate the polygamy relations related to the concurrence of assistance for any multipartite pure states. General polygamy inequalities given by the th power of concurrence of assistance is first presented for multipartite pure states in arbitrary-dimensional quantum systems. We further show that the general polygamy inequalities can even be improved to be tighter inequalities under certain conditions on the assisted entanglement of bipartite subsystems. Based on the improved polygamy relations, lower bound for distribution of bipartite entanglement is provided in a multipartite system. Moreover, the th () power of polygamy inequalities are obtained for the entanglement of assistance as a by-product, which are shown to be tighter than the existing ones. A detailed example is presented.

Uncertainties of genuinely incompatible triple measurements based on statistical distance

We investigate the measurement uncertainties of a triple of positive-operator-valued measures based on statistical distance and formulate state-independent tight uncertainty inequalities satisfied by the three measurements in terms of triplewise joint measurability. In particular, uncertainty inequalities for three unbiased qubit measurements are presented with analytical lower bounds which relates to the necessary and sufficient condition of the triplewise joint measurability of the given triple. We show that the measurement uncertainties for a triple measurement are essentially different from the ones obtained by pairwise measurement uncertainties by comparing the lower bounds of different measurement uncertainties.

Inequalities for some integrals involving modified Bessel functions

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double inequality, involving the modified Bessel function of the first kind, for a generalized hypergeometric function. We also present some open problems that arise from this research.

The cone of supermodular games on finite distributive lattices

In this article we study supermodular functions on finite distributive lattices. Relaxing the assumption that the domain is a powerset of a finite set, we focus on geometrical properties of the polyhedral cone of such functions. Specifically, we generalize the criterion for extremality and study the face lattice of the supermodular cone. An explicit description of facets by the corresponding tight linear inequalities is provided.

Inequalities for Integrals of the Modified Struve Function of the First Kind II

Simple inequalities are established for integrals of the type int _0^x mathrm {e}^{-gamma t} t^{-nu } mathbf {L}_nu (t),mathrm {d}t, where x>0, 0le gamma <1, nu >-frac{3}{2} and mathbf {L}_{nu }(x) is the modified Struve function of the first kind. In most cases, these inequalities are tight in certain limits. As a consequence we deduce a tight double inequality, involving the modified Struve function mathbf {L}_{nu }(x), for a generalized hypergeometric function.

Tight steering inequalities from generalized entropic uncertainty relations

We establish a general connection between entropic uncertainty relations, Einstein-Podolsky-Rosen steering, and joint measurability. Specifically, we construct steering inequalities from any entropic uncertainty relation, given that the latter satisfies two natural properties. We obtain steering inequalities based on R\'enyi entropies. These turn out to be tight in many scenarios, using max- and min-entropy. Considering steering tests with two noisy measurements, our inequalities exactly recover the noise threshold for steerability. This is the case for any pair of qubit 2-outcome measurements, as well as for pairs of mutually unbiased bases in any dimension. This shows that easy-to-evaluate quantities such as entropy can optimally witness steering, despite the fact that they are coarse-grained representations of the underlying statistics.

Finer distribution of quantum correlations among multiqubit systems

We study the distribution of quantum correlations characterized by monogamy relations in multipartite systems. By using the Hamming weight of the binary vectors associated with the subsystems, we establish a class of monogamy inequalities for multiqubit entanglement based on the $\alpha$th ($\alpha\geq 2$) power of concurrence, and a class of polygamy inequalities for multiqubit entanglement in terms of the $\beta$th ($0\leq \beta\leq2$) power of concurrence and concurrence of assistance. Moveover, we give the monogamy and polygamy inequalities for general quantum correlations. Application of these results to quantum correlations like squared convex-roof extended negativity (SCREN), entanglement of formation and Tsallis-$q$ entanglement gives rise to either tighter inequalities than the existing ones for some classes of quantum states or less restrictions on the quantum states. Detailed examples are presented.

Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling

Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets via quantifier elimination and polyhedral projection algorithms is of primal importance. In this work, before considering specific problems, we review polyhedral projection algorithms with focus on applications in information theory, and, alongside known algorithms, we also present a newly developed geometric algorithm which walks along the face lattice of the polyhedron in the projection space. One important application of this is in the field of quantum non-locality, where marginal problems arise in the computation of Bell inequalities. We apply the discussed algorithms to discover many tight entropic Bell inequalities of the tripartite Bell scenario as well as more complex networks arising in the field of causal inference. Finally, we analyze the usefulness of these inequalities as nonlocality witnesses by searching for violating quantum states.

Universal bound on the cardinality of local hidden variables in networks

We present an algebraic description of the sets of local correlations in arbitrary networks, when the parties have finite inputs and outputs. We consider networks generalizing the usual Bell scenarios by the presence of multiple uncorrelated sources. We prove a finite upper bound on the cardinality of the value sets of the local hidden variables. Consequently, we find that the sets of local correlations are connected, closed and semialgebraic, and bounded by tight polynomial Bell-like inequalities.

Hamming weight and tight constraints of multi-qubit entanglement in terms of unified entropy

We establish a characterization of multi-qubit entanglement constraints in terms of non-negative power of entanglement measures based on unified-(q, s) entropy. Using the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of tight monogamy inequalities of multi-qubit entanglement based on the αth-power of unified-(q, s) entanglement for α ≥ 1. For 0 ≤ β ≤ 1, we establish a class of tight polygamy inequalities of multi-qubit entanglement in terms of the βth-power of unified-(q, s) entanglement of assistance. Thus our results characterize the monogamy and polygamy of multi-qubit entanglement for the full range of non-negative power of unified entanglement.

Subspace concentration of dual curvature measures of symmetric convex bodies

We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body. © 2018 International Press of Boston, Inc. All Rights Reserved.

Entanglement polygon inequality in qubit systems

We prove a set of tight entanglement inequalities for arbitrary N-qubit pure states. By focusing on all bi-partite marginal entanglements between each single qubit and its remaining partners, we show that the inequalities provide an upper bound for each marginal entanglement, while the known monogamy relation establishes the lower bound. The restrictions and sharing properties associated with the inequalities are further analyzed with a geometric polytope approach, and examples of three-qubit GHZ-class and W-class entangled states are presented to illustrate the results.

Inequalities for Integrals of the Modified Struve Function of the First Kind

Simple inequalities for some integrals involving the modified Struve function of the first kind mathbf {L}_{nu }(x) are established. In most cases, these inequalities have best possible constant. We also deduce a tight double inequality, involving the modified Struve function mathbf {L}_{nu }(x), for a generalized hypergeometric function.

Weighted polygamy inequalities of multiparty entanglement in arbitrary-dimensional quantum systems

We provide a generalization for the polygamy constraint of multiparty entanglement in arbitrary dimensional quantum systems. By using the $\beta$th-power of entanglement of assistance for $0\leq \beta \leq1$ and the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of weighted polygamy inequalities of multiparty entanglement in arbitrary dimensional quantum systems. We further show that our class of weighted polygamy inequalities can even be improved to be tighter inequalities with some conditions on the assisted entanglement of bipartite subsystems.

Unified picture for spatial, temporal, and channel steering

Quantum steering describes how local actions on a quantum system can affect another, space-like separated, quantum state. Lately, quantum steering has been formulated also for time-like scenarios and for quantum channels. We approach all the three scenarios as one using tools from Stinespring dilations of quantum channels. By applying our technique we link all three steering problems one-to-one with the incompatibility of quantum measurements, a result formerly known only for spatial steering. We exploit this connection by showing how measurement uncertainty relations can be used as tight steering inequalities for all three scenarios. Moreover, we show that certain notions of temporal and spatial steering are fully equivalent and prove a hierarchy between temporal steering and macrorealistic hidden variable models.