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Optimality of the pretty good measurement for port-based teleportation

LETTERS IN MATHEMATICAL PHYSICS 2022年第5期112卷 98-98页

作者： Leditzky, Felix Univ Illinois Dept Math Urbana IL 61801 USA Univ Illinois IQUIST Urbana IL 61801 USA Univ Waterloo Dept Combinator & Optimizat Waterloo ON Canada Univ Waterloo Inst Quantum Comp Waterloo ON Canada Perimeter Inst Theoret Phys Waterloo ON Canada

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.

关键词： Quantum information theory Quantum teleportation Quantum state discrimination Representation theory semidefinite programming

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An experimental comparison of methods for computing the numerical radius

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RESULTS IN APPLIED MATHEMATICS 2024年 21卷

作者： Mitchell, Tim Overton, Michael L. CUNY Queens Coll Dept Comp Sci 65-30 Kissena Blvd Flushing NY 11367 USA NYU Courant Inst Math Sci 251 Mercer St New York NY 10012 USA

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.

关键词： semidefinite programming Convex optimization Nonconvex optimization Field of values Numerical range

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Estimating the Fundamental Matrix using L_∞ Minimization Algorithm

Estimating the Fundamental Matrix using <i>L</i><sub>∞</sub...

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7th World Congress on Intelligent Control and Automation

作者： Yang, Min Nanjing Univ Posts & Telecommun Coll Automat Nanjing 210003 Peoples R China

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.

关键词： computer vision fundamental matrix semidefinite programming L-infinity minimization

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semidefinite Optimization Providing Guaranteed Bounds on Linear Functionals of Solutions of Linear Integral Equations with Smooth Kernels

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JOURNAL OF APPLIED MATHEMATICS 2014年第unknown期000卷 1-8页

作者： Zhou, Guangming Deng, Chao Wu, Kun Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Hunan Peoples R China

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.

关键词： semidefinite programming BOUNDS (Mathematics) LINEAR systems INTEGRAL equations SMOOTHNESS of functions KERNEL functions

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An SOS counterexample to an inequality of symmetric functions

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JOURNAL OF PURE AND APPLIED ALGEBRA 2021年第8期225卷 106656-106656页

作者： Heaton, Alexander Shankar, Isabelle Max Planck Inst Math Sci Leipzig Germany Univ Calif Berkeley Evans Hall Berkeley CA 94720 USA

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.

关键词： Sums of squares Symmetric polynomials semidefinite programming

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Compositionally Verifiable Vector Neural Lyapunov Functions for Stability Analysis of Interconnected Nonlinear Systems

Compositionally Verifiable Vector Neural Lyapunov Functions ...

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American Control Conference (ACC)

作者： Jun Liu Yiming Meng Maxwell Fitzsimmons Ruikun Zhou Department of Applied Mathematics Faculty of Mathematics University of Waterloo Waterloo Ontario Canada Coordinated Science Laboratory University of Illinois Urbana-Champaign Urbana IL USA

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.

关键词： semidefinite programming Control design Partial differential equations Neural networks Stability analysis Vectors Polynomials

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QUANTUM GENERALIZATIONS OF THE POLYNOMIAL HIERARCHY WITH APPLICATIONS TO QMA(2)

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COMPUTATIONAL COMPLEXITY 2022年第2期31卷 13-13页

作者： Gharibian, Sevag Santha, Miklos Sikora, Jamie Sundaram, Aarthi Yirka, Justin Paderborn Univ Paderborn North Rhine Wes Germany Virginia Commonwealth Univ Richmond VA USA Univ Paris CNRS IRIF Paris France Natl Univ Singapore Ctr Quantum Technol Singapore Singapore Virginia Polytech Inst & State Univ Blacksburg VA 24061 USA Perimeter Inst Theoret Phys Waterloo ON Canada Microsoft Quantum Redmond WA USA Univ Texas Austin Austin TX 78712 USA

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'').

关键词： Complexity theory Quantum computing Polynomial hierarchy QMA(2) semidefinite programming Toda's theorem

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Existence and Uniqueness of the Positive Definite Solution for the Matrix Equation X = Q plus A*((X)over-cap-C)^-1 A

ABSTRACT AND APPLIED ANALYSIS

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ABSTRACT AND APPLIED ANALYSIS 2013年第Spe4期000卷 1-4页

作者： Gao, Dongjie Heze Univ Dept Math Heze 274015 Shandong Peoples R China

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.

关键词： EXISTENCE theorems UNIQUENESS (Mathematics) NUMERICAL solutions to equations semidefinite programming FIXED point theory ITERATIVE methods (Mathematics)

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EXPERIMENTAL EVALUATION OF SIMULTANEOUS 3D LOCALIZATION OF SENSOR NODES AND TRACKING MOVING TARGETS

EXPERIMENTAL EVALUATION OF SIMULTANEOUS 3D LOCALIZATION OF S...

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European Signal Processing Conference

作者： Pinar Oguz Ekim Joao Gomes Joao Xavier Paulo Oliveira Institute for Systems and Robotics - Instituto Superior Tecnico (ISR/IST) Technical University of Lisbon (UTL)

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.

关键词： Localization Tracking Sensor networks semidefinite programming Real indoor experiments

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Sensor Localization in NLOS Environments with Anchor Uncertainty and Unknown Clock Parameters

Sensor Localization in NLOS Environments with Anchor Uncerta...

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International Conference on Communication Workshop

作者： Siamak Yousefi Reza Monir Vaghefi Xiao-Wen Chang Benoit Champagne R. Michael Buehrer Department of Electrical and Computer Engineering McGill University Montreal Quebec Canada Department of Electrical and Computer Engineering Virginia Tech Blacksburg VA USA School of Computer Science McGill University Montreal Quebec Canada

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.

关键词： Convex relaxation joint localization and synchronization non-line of sight semidefinite programming

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