scholarly journals A note on completely positive relaxations of quadratic problems in a multiobjective framework

2021 ◽  
Author(s):  
Gabriele Eichfelder ◽  
Patrick Groetzner
Keyword(s):  
Positive Semidefinite ◽  
Weighted Sum ◽  
Quadratic Problem ◽  
Completely Positive ◽  
Nonconvex Problems ◽  
Completely Positive Matrices ◽  
Convex Problems ◽  
Positive Matrices ◽  
Nondominated Points ◽  
Single Objective

AbstractIn a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.

scholarly journals Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

2019 ◽  
Vol 35 ◽  
pp. 156-180 ◽  
Author(s):  
Nathaniel Johnston ◽  
Olivia MacLean
Keyword(s):  
Sufficient Conditions ◽  
Positive Semidefinite ◽  
Quantum States ◽  
Completely Positive ◽  
Positive Partial Transpose ◽  
Completely Positive Matrices ◽  
Partial Transposition ◽  
Partial Transpose ◽  
Positive Matrices ◽  
Necessary And Sufficient

A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.

scholarly journals Zero-one completely positive matrices and the A(R, S) classes

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
G. Dahl ◽  
T. A. Haufmann
Keyword(s):  
Related Concept ◽  
Completely Positive ◽  
Completely Positive Matrices ◽  
Positive Matrices

AbstractA matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.

Factorizations of completely positive matrices

1971 ◽  
Vol 69 (1) ◽  
pp. 53-58 ◽  
Author(s):  
Thomas L. Markham
Keyword(s):  
Quadratic Form ◽  
Positive Matrix ◽  
Completely Positive ◽  
Positive Form ◽  
Completely Positive Matrices ◽  
Completely Positive Matrix ◽  
Positive Matrices ◽  
Real Quadratic

1. Introduction. DEFINITION. If with aij = aji, is a real quadratic form inx1 …,xn, andwhere Lk = cklx1 + … + cknxn (ckj ≥ 0 for k = 1, …, t), then Q is called a completely positive form, and A = (aij) is called a completely positive matrix.

Completely Positive Matrices: Real, Rational, and Integral

2018 ◽  
Vol 43 (4) ◽  
pp. 629-639 ◽  
Author(s):  
Abraham Berman ◽  
Naomi Shaked-Monderer
Keyword(s):  
Completely Positive ◽  
Completely Positive Matrices ◽  
Positive Matrices

Completely positive matrices associated withM-matrices

1994 ◽  
Vol 37 (4) ◽  
pp. 303-310 ◽  
Author(s):  
John H Drew ◽  
Charles R. Johnson ◽  
Raphael Loewy
Keyword(s):  
Completely Positive ◽  
Completely Positive Matrices ◽  
Positive Matrices

scholarly journals Diagonal unitary and orthogonal symmetries in quantum theory

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 519
Author(s):  
Satvik Singh ◽  
Ion Nechita
Keyword(s):  
Sufficient Conditions ◽  
Quantum Channels ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Completely Positive Matrices ◽  
Linear Maps ◽  
Positive Matrices ◽  
Invariant States ◽  
Class Of Matrices ◽  
Necessary And Sufficient

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

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