Port-based teleportation (PBT) is a protocol in which Alice teleports an unknown quantum state to Bob using measurements on a shared entangled multipartite state called the port state and forward classical communicati...
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Port-based teleportation (PBT) is a protocol in which Alice teleports an unknown quantum state to Bob using measurements on a shared entangled multipartite state called the port state and forward classical communication. In this paper, we give an explicit proof that the so-called pretty good measurement, or square-root measurement, is optimal for the PBT protocol with independent copies of maximally entangled states as the port state. We then show that the very same measurement remains optimal even when the port state is optimized to yield the best possible PBT protocol. Hence, there is one particular pretty good measurement achieving the optimal performance in both cases. The following well-known facts are key ingredients in the proofs of these results: (i) the natural symmetries of PBT, leading to a description in terms of representation-theoretic data;(ii) the operational equivalence of PBT with certain state discrimination problems, which allows us to employ duality of the associated semidefinite programs. Along the way, we rederive the representation-theoretic formulas for the performance of PBT protocols proved in Studzinski et al. (Sci Rep 7(1):1-11, 2017) and Mozrzymas et al. (N J Phys 20(5):053006, 2018) using only standard techniques from the representation theory of the unitary and symmetric groups. Providing a simplified derivation of these beautiful formulas is one of the main goals of this paper.
We make an experimental comparison of methods for computing the numerical radius of an n x n complex matrix, based on two well-known characterizations, the first a nonconvex optimization problem in one real variable a...
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We make an experimental comparison of methods for computing the numerical radius of an n x n complex matrix, based on two well-known characterizations, the first a nonconvex optimization problem in one real variable and the second a convex optimization problem in n(2) + 1 real variables. We make comparisons with respect to both accuracy and computation time using publicly available software.
Fundamental matrix estimation is a central problem in computer vision and forms the basis of tasks such as stereo imaging and structure from motion. A new method for the estimation of the fundamental matrix from point...
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ISBN:
(纸本)9781424421138
Fundamental matrix estimation is a central problem in computer vision and forms the basis of tasks such as stereo imaging and structure from motion. A new method for the estimation of the fundamental matrix from point correspondences is presented. The minimization of an objective function closer to the geometric distance is performed based L. minimization framework. The fundamental matrix is optimally computed with taking into account the rank-two constraint, and the method is no need for normalization of the image coordinates. It is shown how this nonlinearly estimating the fundamental matrix can be solved avoiding local minima by using semidefinite programming. Experiments on real images show that this method provides a more accurate estimate of the fundamental matrix and superior to previous approaches.
Based on recent progress on moment problems, semidefinite optimization approach is proposed for estimating upper and lower bounds on linear functionals defined on solutions of linear integral equations with smooth ker...
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Based on recent progress on moment problems, semidefinite optimization approach is proposed for estimating upper and lower bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. The approach is also suitable for linear integrodifferential equations with smooth kernels. Firstly, the primal problem with smooth kernel is converted to a series of approximative problems with Taylor polynomials obtained by expanding the smooth kernel. Secondly, two semidefinite programs (SDPs) are constructed for every approximative problem. Thirdly, upper and lower bounds on related functionals are gotten by applying SeDuMi 1.1R3 to solve the two SDPs. Finally, upper and lower bounds series obtained by solving two SDPs, respectively infinitely approach the exact value of discussed functional as approximative order of the smooth kernel increases. Numerical results show that the proposed approach is effective for the discussed problems.
It is known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. The only excep...
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It is known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. The only exception is the case of homogeneous symmetric functions where it is only known that dominance of the partitions implies nonnegativity of the corresponding difference of symmetric functions. It was conjectured by Cuttler, Greene, and Skandera in 2011 that the converse also holds, as in the cases of the monomial, elementary, power-sum, and Schur bases. In this paper we provide a counterexample, showing that homogeneous symmetric functions break the pattern. We use semidefinite programming to find an explicit sums of squares decomposition of the polynomial H-44 - H-521 as a sum of 41 squares. This rational certificate of nonnegativity disproves the conjecture, since a polynomial which is a sum of squares cannot be negative, and since the partitions 44 and 521 are incomparable in dominance order. (C) 2020 Elsevier B.V. All rights reserved.
While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due t...
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ISBN:
(数字)9798350382655
ISBN:
(纸本)9798350382662
While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due to the curse of dimensionality. In this paper, we demonstrate that by leveraging the compositional structure of interconnected nonlinear systems, it is possible to verify neural Lyapunov functions for high-dimensional systems beyond the capabilities of current satisfiability modulo theories (SMT) solvers using a monolithic approach. Our numerical examples employ neural Lyapunov functions trained by solving Zubov's partial differential equation (PDE), which characterizes the domain of attraction for individual subsystems. These examples show a performance advantage over sums-of-squares (SOS) polynomial Lyapunov functions derived from semidefinite programming.
The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two qua...
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The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, Q Sigma(3), into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs "equivalent''), then QMA(2) is in the counting hierarchy (specifically, in P-pppp). Second, because QMA(2)subset of Q Sigma(3), QMA(2) is strictly contained in NEXP unless QMA(2)=Q Sigma(3) (i.e., alternating quantifiers do not help in the presence of "unentanglement'').
We consider the nonlinear matrix equation X - Q + A* ((X) over cap -C)(-1) A, where Q is positive definite, C is positive semidefinite, and (X) over cap is the block diagonal matrix defined by (X) over cap = diag(X, X...
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We consider the nonlinear matrix equation X - Q + A* ((X) over cap -C)(-1) A, where Q is positive definite, C is positive semidefinite, and (X) over cap is the block diagonal matrix defined by (X) over cap = diag(X, X, ..., X). We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.
In this work we carry out an experimental performance characterization of a simultaneous localization and tracking (SLAT) algorithm for sensor networks, whose aim is to determine the positions of sensor nodes and a mo...
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ISBN:
(纸本)9781467310680
In this work we carry out an experimental performance characterization of a simultaneous localization and tracking (SLAT) algorithm for sensor networks, whose aim is to determine the positions of sensor nodes and a moving target in a network, given incomplete and inaccurate range measurements between the target and each of the sensors. To achieve this, we propose to iteratively maximize a likelihood function (ML) of positions given the observed ranges, which requires initialization with an approximate solution to avoid convergence towards local extrema. A modified Euclidean Distance Matrix (EDM) completion problem is solved for a block of target range measurements to approximately set up initial sensor/target positions, and the likelihood function is then iteratively refined through Majorization-Minimization (MM). To reduce the computational load, an incremental scheme is used whereby each new target or sensor position is estimated from range measurements, providing additional initialization for ML without the need for solving an expanded EDM completion problem. The proposed algorithms are experimentally evaluated with a series of 3D indoor tests for a range of operation of up to ten meters using a Crossbow Cricket location system and a robotic or human target. Centimetric accuracy is obtained under realistic conditions.
In this paper, joint sensor localization and synchronization in non-cooperative wireless sensor networks (WSNs) using time-of-arrival (TOA) measurements is studied. In addition to zero-mean errors in TOA measurements ...
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In this paper, joint sensor localization and synchronization in non-cooperative wireless sensor networks (WSNs) using time-of-arrival (TOA) measurements is studied. In addition to zero-mean errors in TOA measurements we consider other sources of error such as non-line-of-sight (NLOS) propagation and anchor uncertainty to make our technique more useful in practice, where the presence of these errors is inevitable. The proposed technique is based on semi-definite programming (SDP) relaxation which can be solved in polynomial time and guarantees convergence to the global minimum. It is shown that the optimal accuracy is obtained by discarding the NLOS measurements and applying the maximum likelihood (ML) technique to jointly estimate the positions of anchor nodes, and the position and clock parameters of the sensor node. The results show that the proposed SDP technique, which does not require prior identification of NLOS links, is robust against NLOS errors and its performance is close to that of optimal accuracy.
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