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73
BiQuadratic Optimization over Unit Spheres and Semidefinite Programming Relaxations
, 2008
"... Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl ..."
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Cited by 32 (15 self)
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Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl
Approximation algorithms for homogeneous polynomial optimization with quadratic constraints
, 2009
"... In this paper, we consider approximation algorithms for optimizing a generic multivariate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, appr ..."
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Cited by 25 (11 self)
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In this paper, we consider approximation algorithms for optimizing a generic multivariate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are nonconvex in general, the problems under consideration are all NPhard. In this paper we shall focus on polynomialtime approximation algorithms. In particular, we first study optimization of a multilinear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worstcase performance ratios, which are shown to depend only on the dimensions of the models. The methods are then extended to optimize a generic multivariate homogeneous polynomial function with spherical constraints. Likewise, approximation algorithms are proposed with provable relative approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of cocentered ellipsoids. In particular, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomialtime approximation algorithms with provable worstcase performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.
COMPUTATIONAL COMPLEXITY OF THE QUANTUM SEPARABILITY PROBLEM
, 2007
"... Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or separable, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) qu ..."
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Cited by 19 (0 self)
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Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or separable, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the onesided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper’s main focus. I then give a thorough treatment of the problem’s relationship with NP, NPcompleteness, and coNP. I also discuss extensions of Gurvits ’ NPhardness result to strong NPhardness of certain related problems. A major open question is whether the NPcontained formulation (QSEP) of the quantum separability problem is KarpNPcomplete; QSEP may be the first natural example of a problem that is TuringNPcomplete but not KarpNPcomplete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem [1, 2, 3]; and the entanglementwitness search (via interiorpoint algorithms and global optimization) [4, 5]. These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal pointcoverings of the sphere.
Strong NPHardness of the Quantum Separability Problem
, 2009
"... Given the density matrix ρ of a bipartite quantum state, the quantum separability problem asks whether ρ is entangled or separable. In 2003, Gurvits showed that this problem is NPhard if ρ is located within an inverse exponential (with respect to dimension) distance from the border of the set of se ..."
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Cited by 18 (1 self)
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Given the density matrix ρ of a bipartite quantum state, the quantum separability problem asks whether ρ is entangled or separable. In 2003, Gurvits showed that this problem is NPhard if ρ is located within an inverse exponential (with respect to dimension) distance from the border of the set of separable quantum states. In this paper, we extend this NPhardness to an inverse polynomial distance from the separable set. The result follows from a simple combination of works by Gurvits, Ioannou, and Liu. We apply our result to show (1) an immediate lower bound on the maximum distance between a bound entangled state and the separable set (assuming P ̸ = NP), and (2) NPhardness for the problem of determining whether a completely positive tracepreserving linear map is entanglementbreaking. 1
Maximum block improvement and polynomial optimization
 SIAM Journal on Optimization
"... Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with t ..."
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Cited by 15 (5 self)
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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.
Entanglement detection
 Physics Reports
"... How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalitie ..."
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How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalities, entanglement witnesses, the determination of nonlinear properties of a quantum state via measurements on several copies, and spin squeezing inequalities. An emphasis is given to the theory and application of entanglement witnesses. We also discuss several experiments, where some of the presented methods have been implemented.
Combinatorial and algorithmic aspects of hyperbolic polynomials
"... Let p(x1,..., xn) = ∑ (r1,...,rn)∈In,n a ∏ (r1,...,rn) 1≤i≤n xri i be homogeneous polynomial of degree n in n real variables with integer nonnegative coefficients. The support of such polynomial p(x1,..., xn) is defined as supp(p) = {(r1,..., rn) ∈ In,n: a (r1,...,rn) ̸ = 0}. The convex hull CO(s ..."
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Cited by 12 (6 self)
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Let p(x1,..., xn) = ∑ (r1,...,rn)∈In,n a ∏ (r1,...,rn) 1≤i≤n xri i be homogeneous polynomial of degree n in n real variables with integer nonnegative coefficients. The support of such polynomial p(x1,..., xn) is defined as supp(p) = {(r1,..., rn) ∈ In,n: a (r1,...,rn) ̸ = 0}. The convex hull CO(supp(p)) of supp(p) is called the Newton polytope of p. We study the following decision problems, which are farreaching generalizations of the classical perfect matching problem: • Problem 1. Consider a homogeneous polynomial p(x1,..., xn) of degree n in n real variables with nonnegative integer coefficients given as a black box (oracle). Is it true that (1, 1,.., 1) ∈ supp(p)? • Problem 2. Consider a homogeneous polynomial p(x1,..., xn) of degree n in n real variables with nonnegative integer coefficients given as a black box (oracle). Is it true that (1, 1,.., 1) ∈ CO(supp(p))?
Algebraic Set Kernels with Application to Inference Over Local Image Representations
 In
, 2005
"... This paper presents a general family of algebraic positive definite similarity functions over spaces of matrices with varying column rank. The columns can represent local regions in an image (whereby images have varying number of local parts), images of an image sequence, motion trajectories in a mu ..."
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This paper presents a general family of algebraic positive definite similarity functions over spaces of matrices with varying column rank. The columns can represent local regions in an image (whereby images have varying number of local parts), images of an image sequence, motion trajectories in a multibody motion, and so forth. The family of set kernels we derive is based on a group invariant tensor product lifting with parameters that can be naturally tuned to provide a cookbook of sorts covering the possible ”wish lists ” from similarity measures over sets of varying cardinality. We highlight the strengths of our approach by demonstrating the set kernels for visual recognition of pedestrians using local parts representations. 1
Entanglement Theory and the Quantum Simulation Of Manybody Physics
, 2008
"... Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum theory has changed in an equally dramatic manner our understanding of information processing and computation. On one hand, the fundamental prop ..."
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Cited by 12 (2 self)
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Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum theory has changed in an equally dramatic manner our understanding of information processing and computation. On one hand, the fundamental properties of quantum systems can be harnessed to transmit, store, and manipulate information in a more efficient and secure way than possible in the realm of classical physics. On the other hand, the development of systematic procedures to manipulate systems of a large number of particles in the quantum regime, crucial to the implementation of quantumbased information processing, has triggered new possibilities in the exploration of quantum manybody physics and related areas. In this thesis, we present new results relevant to two important problems in quantum information science: the development of a theory of entanglement, intrinsically quantum correlations, and the exploration of the use of controlled quantum systems to the computation and simulation of quantum manybody phenomena. In the first part we introduce a new approach to the study of entanglement
NPhardness of deciding convexity of quartic polynomials and related problems
, 2010
"... We show that unless P=NP, there exists no polynomial time (or even pseudopolynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexi ..."
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Cited by 11 (2 self)
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We show that unless P=NP, there exists no polynomial time (or even pseudopolynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NPhard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.