Optimal Bound Research Articles

Multipartite entanglement detection based on the generalized state-dependent entropic uncertainty relation for multiple measurements

We present the generalized state-dependent entropic uncertainty relations in multiple measurements setting, and the optimal lower bound is obtained by considering different measurement sequences. We then apply this uncertainty relation to witness entanglement, and give the experimentally accessible lower bounds on both bipartite and tripartite entanglements. This method of detecting entanglement is applied to a physical system of two particles on a one-dimensional lattice, and GHZ-Werner state. It is shown that, for measurements that are not in mutually unbiased bases, this new entropic uncertainty relation is superior to the previous state-independent one in entanglement detection. Furthermore, we conduct an online experiment to detect multipartite entanglement of GHZ states up to 10 qubits on Quafu cloud quantum computation platform. Our results might play important roles in detecting multipartite entanglement experimentally.

Gradient estimates of the solution to the Cauchy problem for degenerate parabolic equations with nonpower nonlinearities

ABSTRACT We prove the uniform gradient estimates of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The large-time estimates in terms of initial data mass and nonlinearity structure were obtained. In the particular case, when the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound.

Optimal Bounds for Numerical Approximations of Infinite Horizon Problems Based on Dynamic Programming Approach

In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that, considering a time discretization with a positive step size , an error bound of size can be proved for the difference between the value function (viscosity solution of the Hamilton–Jacobi–Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size , an error bound of size can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under assumptions similar to those of the time discrete case, that the error of the fully discrete case is in fact , which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behavior from the bound has not been observed.

On the Kolmogorov Distance for the Least Squares Estimator in the Fractional Ornstein-Uhlenbeck Process

The paper shows that the distribution of the normalized least squares estimator of the drift parameter in the fractional Ornstein-Uhlenbeck process observed over [0, T] converges to the standard normal distribution with an uniform optimal error bound of the order O(T −1/2) for 0.5 ≤ H ≤ 0.63 and of the order O(T4H-3) for 0.63 < H < 0.75 where H is the Hurst exponent of the fractional Brownian motion driving the Ornstein-Uhlenbeck process. For the normalized quasi-least squares estimator, the error bound is of the order O(T−1/4) for 0.5 ≤ H ≤ 0.69 and of the order O(T4H−3) for 0.69 < H < 0.75.

Actions of tame abelian product groups

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When [Formula: see text] for countable abelian [Formula: see text], Solecki [Equivalence relations induced by actions of Polish groups, Trans. Amer. Math. Soc. 347 (1995) 4765–4777] gave a characterization for when G is tame. In [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343], Ding and Gao showed that for such G, the orbit equivalence relation must in fact be potentially [Formula: see text], while conjecturing that the optimal bound could be [Formula: see text]. We show that the optimal bound is [Formula: see text] by constructing an action of such a group G which is not potentially [Formula: see text], and show how to modify the analysis of [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343] to get this slightly better upper bound. It follows, using the results of Hjorth et al. [Borel equivalence relations induced by actions of the symmetric group, Ann. Pure Appl. Logic 92 (1998) 63–112], that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets.

An Optimized LTE-Based Technique for Drone Base Station Dynamic 3D Placement and Resource Allocation in Delay-Sensitive M2M Networks

Drones are expected to facilitate extending wireless networks’ access for both human users and smart machine-type-communication devices (MTCDs) with strict and diverse quality of service (QoS) requirements. In this paper, we propose an optimal solution for the dynamic placement of an LTE drone-mounted base station to maximize the coverage of the MTCDs deployed over a large geographical area. The resulting technique jointly optimizes the drone's 3D positioning to maximize coverage and allocates the network resources in such a way that gives high priority to the delay-sensitive machine-to-machine (M2M) traffic. This optimization algorithm determines the optimal bound of the solution. Since it cannot be used in real-time operations due to its computational complexity, we also introduce a heuristic technique that offers a near-optimal solution with much reduced complexity. We conduct several simulation evaluations to assess the proposed techniques. These results are compared to those of other drone placement approaches. The comparisons show that the proposed techniques offer significantly better results in the communication coverage while fulfilling the diverse QoS requirements of the deployed M2M network. Moreover, the heuristic-based technique is shown to succeed in finding solutions close to the optimal bound with a considerable reduction of complexity over the exact algorithm.

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

Optimal estimates for the conductivity problem close to the boundary in dimension greater than two

<p style='text-indent:20px;'>In high-contrast composite materials, the electric field (or stress field) may blow up in the narrow region when a convex inclusion is closely to the boundary of the matrix domain. The gradient of solutions depends on the parameters–the distance between the inclusions and boundaries. An optimal upper bound was obtained for convex inclusions of arbitrary shape in dimension greater than two. Our results extend that in [<xref ref-type="bibr" rid="b3">3</xref>], where the disk inclusion case in 2D is considered.</p>

Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables where the disturbance does not enter through the boundary.For such systems, we define a notion of H2-norm using an initial state-to-output framework and show that this notion reduces to more traditional concepts under the assumption of existence of a strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE representation and for such systems show that computing a minimal upper bound on the H2-norm of delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation results are available in the literature. The numerical results demonstrate the accuracy of the computed upper bound on the H2-norm.

Probing chaos by magic monotones

There is a property of a quantum state called ``magic.'' As shown by the Gottesman-Knill theorem, so-called stabilizer states, which are composed of only Clifford gates, can be efficiently computed on a classical computer, and thus quantum computation gives no advantage. Nonstabilizer states are called magic states, which are necessary to achieve the universal quantum computation. Magic (monotone) is the measure of the amount of nonstabilizer resource, and it measures how difficult it is for a classical computer to simulate the state. We study magic of states in the integrable and chaotic regimes of the higher-spin generalization of the Ising model through two quantities: ``mana'' and ``robustness of magic'' (RoM). We find that in the chaotic regime, mana increases monotonically in time in the early-time region, and at late times these quantities oscillate around some nonzero value that increases linearly with respect to the system size. Our result also suggests that under chaotic dynamics, any state evolves to a state whose mana almost saturates the optimal upper bound; i.e., the state becomes ``maximally magical.'' We find that RoM also shows similar behaviors. On the other hand, in the integrable regime, mana and RoM behave periodically in time in contrast to the chaotic case. In addition to mana and RoM, for the early-time behavior of magic, we study the stabilizer R\'enyi entropy, which can be numerically computed for larger systems than mana and RoM. In the anti-de Sitter/conformal field theory correspondence, classical spacetime emerges from the chaotic nature of the dual quantum system. Our results suggest that magic of quantum states is strongly involved in the emergence of spacetime geometry.

Fractional Helly Theorem for Cartesian Products of Convex Sets

Helly’s theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff’s result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $$\alpha \in (1-1/t^d,1]$$ and a finite family $${\mathcal {F}}$$ of Cartesian products of convex sets $$\prod _{i\in [t]}A_i$$ in $${\mathbb {R}}^{td}$$ with $$A_i\subset {\mathbb {R}}^d$$ , if at least $$\alpha $$ -fraction of the $$(d+1)$$ -tuples in $${\mathcal {F}}$$ are intersecting, then at least $$(1-(t^d(1-\alpha ))^{1/(d+1)})$$ -fraction of sets in $${\mathcal {F}}$$ are intersecting. This is a special case of a more general result on intersections of d-Leray complexes. We also provide a construction showing that our result on d-Leray complexes is optimal. Interestingly, the extremal example is representable as a family of Cartesian products of convex sets, implying that the bound $$\alpha >1-1/t^d$$ and the fraction $$(1-(t^d(1-\alpha ))^{1/(d+1)})$$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $${\mathbb {R}}^d$$ does not have $$(p,d+1)$$ -condition for sublinear p, that is, it contains a linear-size subfamily with no intersecting $$(d+1)$$ -tuple. Inspired by this, we give constructions showing that, somewhat surprisingly, imposing an additional $$(p,d+1)$$ -condition has a negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.

Optimal non-asymptotic analysis of the Ruppert–Polyak averaging stochastic algorithm

This paper is devoted to the non-asymptotic analysis of the Ruppert–Polyak averaging method introduced in Polyak and Juditsky (1992) and Ruppert (1988)[26] for the minimization of a smooth function f with a stochastic algorithm. We first establish a general non-asymptotic optimal bound: if θˆn is the position of the algorithm at step n, we prove that E|θˆn−argmin(f)|2⩽Tr(Σ⋆)n+Cd,fn−rβ,where Σ⋆ is the limiting covariance matrix of the CLT demonstrated in Polyak and Juditsky (1992) and Cd,fn−rβ is a new state-of-the-art second order term that translates the effect of the dimension. We also identify the optimal gain of the baseline SGD γn=γn−3/4, leading to a second-order term with r3/4=5/4. Second, we show that this result holds under some Kurdyka-Łojiasewicz-type condition (Kurdyka, 1988; Lojasiewicz, 1963) for function f, which is far more general than the standard uniformly strongly convex case. In particular, it makes it possible to handle some pathological examples such as on-line learning for logistic regression and recursive quantile estimation.

Optimal quantum control robust against pulse inhomogeneities: Analytic solutions

We study optimal quantum control robust against pulse inhomogeneities for various partial population transfers and single-qubit gates by inverse optimization. We show that the pulse is constant for time or energy minimization and we provide the analytic form of the detuning as Jacobi elliptic cosines. The performance of composite pulse techniques, which we optimize for the case of complete population transfer, is compared to this optimal bound.

A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation

In this paper, an adaptive grid method for a singularly perturbed Volterra integro-differential equation is studied. Firstly, this problem is discretized by a new second-order finite difference scheme, for which a truncation error analysis is conducted. Then, based on this truncation error bound and the mesh equidistribution principle, we show that there is a mesh that provides an optimal error bound of O(N−2), which is robust with respect to the perturbation parameter. Finally, based on an approximation monitor function, an adaptive grid generation algorithm is constructed and some numerical results are given to support our theoretical results.

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

This article considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body K of unit diameter in Euclidean d -dimensional space (where d is a constant) and an error parameter ε > 0, the objective is to determine a convex polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity , we mean the total number of faces of all dimensions. Classical constructions by Dudley and Bronshteyn/Ivanov show that O (1/ε ( d -1)/2 ) facets or vertices are possible, respectively, but neither achieves both bounds simultaneously. In this article, we show that it is possible to construct a polytope with O (1/ε ( d -1)/2 ) combinatorial complexity, which is optimal in the worst case. Our result is based on a new relationship between ε-width caps of a convex body and its polar body. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are “essentially different.” We achieve our main result by combining this with a variant of the witness-collector method and a novel variable-thickness layered construction of the economical cap covering.

A non-uniform event-triggered distributed filtering scheme for discrete-time nonlinear systems over wireless sensor networks

This paper investigates the distributed set-membership filtering problem for discrete-time nonlinear systems subject to one-sided Lipschitz (OSL) and quadratic inner-boundedness (QIB) constraints over wireless sensor networks (WSNs). The investigation takes into account both system and sensor nonlinearities under unknown but bounded (UBB) perturbations. Moreover, a novel non-uniform event-triggering condition is introduced to achieve resource conservation while maintaining filter accuracy. The proposed triggering condition is more general and can be customized as per design requirements. Furthermore, a recursive condition for set-membership filter design is derived that guarantees the existence of optimal bounding ellipsoids for true system states. In contrast to the existing methods, the proposed approach considers set-membership filtering for discrete-time systems subject to OSL system and sensor nonlinearities over a WSN and deals with non-uniform amplitude-based triggering. Finally, a numerical example is presented to validate filter’s efficacy.

Dual Certificates and Efficient Rational Sum-of-Squares Decompositions for Polynomial Optimization over Compact Sets

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. For a theoretical application, we give a short new proof of Powers's theorems on the existence of rational WSOS certificates of positive polynomials. For a computational application, we show that exact WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. We also present an algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.

Performance analysis of control allocation using data‐driven integral quadratic constraints

AbstractA new method is presented for evaluating the performance of a nonlinear control allocation system within a linear control loop. To that end, a worst‐case gain analysis problem is formulated that can be readily solved by means of well‐established methods from robustness analysis using integral quadratic constraints (IQCs). It exploits the fact that control allocation systems are in general memoryless mappings that can be bounded by IQCs. A data‐driven approach is used to find an optimal bound of the input/output mapping of the control allocation. Additionally, an iterative procedure based on local IQCs is introduced to determine meaningful sampling limits for less conservative yet accurate results. The effectiveness of the proposed data‐driven performance analysis is shown at the example of an actively controlled flexible wing in a wind tunnel.

One-parameter generalization of the Böttcher-Wenzel inequality and its application to open quantum dynamics

In this paper, we introduce a one-parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a q-deformed commutator. For n×n matrices A and B, we consider the inequalityRe〈[B,A],[B,A]q〉≤c(q)‖A‖2‖B‖2, where 〈A,B〉=tr(A⁎B) is the Hilbert-Schmidt inner product, ‖A‖ is the Frobenius norm, [A,B]=AB−BA is the commutator, and [A,B]q=AB−qBA is the q-deformed commutator. We prove that when n=2, or when A is normal with any size n, the optimal bound is given byc(q)=(1+q)+2(1+q2)2. We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for n up to 15 by numerical optimization. When q=1, this inequality is exactly the BW inequality. When q=0, this inequality leads the sharp bound for the r-function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics.

Upper bounds on the Boolean rank of Kronecker products

The Boolean rank of a 0,1-matrix A , denoted R B ( A ) , is the smallest number of monochromatic combinatorial rectangles needed to cover the 1-entries of A . In 1988, de Caen, Gregory, and Pullman asked if the Boolean rank of the Kronecker product C n ⊗ C n is strictly smaller than the square of R B ( C n ) , where C n is the n × n matrix with zeros on the main diagonal and ones everywhere else (de Caen et al., 1988). A positive answer was given by Watts for n = 4 (Watts, 2001). A result of Karchmer, Kushilevitz, and Nisan, motivated by direct-sum questions in non-deterministic communication complexity, implies that the Boolean rank of C n ⊗ C n grows linearly in that of C n (Karchmer et al., 1995), and thus R B ( C n ⊗ C n ) < R B ( C n ) 2 for every sufficiently large n . Their proof relies on a probabilistic argument. In this work, we present a general method for proving upper bounds on the Boolean rank of Kronecker products of 0,1-matrices. We use it to affirmatively settle the question of de Caen et al. for all integers n ≥ 7 . We further provide an explicit construction of a cover of C n ⊗ C n , whose number of rectangles nearly matches the optimal asymptotic bound. Our method for proving upper bounds on the Boolean rank of Kronecker products might find applications in different settings as well. We express its potential applicability by extending it to the wider framework of spanoids , recently introduced by Dvir et al. (2020).