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Povzetek: A $\ast$-linear map $\Phi$ between matrix spaces is cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V \rangle:={\rm tr}(UV)=0$ implies $\langle\Phi (U),V \rangle \ge 0$, and is completely cross-positive if all its ampliations $I_n \otimes \Phi$ are cross-positive. (Completely) cross-positive maps arise in the theory of operator semigroups, where they are sometimes called exponentially-positive maps, and are also important in the theory of affine processes on symmetric cones in mathematical finance. To each $\Phi$ as above a bihomogeneous form is associated by $p_\Phi (x,y)=y^T\Phi (xx^T)y$. Then $\Phi$ is cross-positive if and only if $p_\Phi$ is nonnegative on the variety of pairs of orthogonal vectors $\{(x,y) | x^Ty = 0\}$. Moreover, $\Phi$ is shown to be completely cross-positive if and only if $p_\Phi$ is a sum of squares modulo the principal ideal $(x^Ty)$. These observations bring the study of cross-positive maps into the powerful setting of real algebraic geometry. Here this interplay is exploited to prove quantitative bounds on the fraction of cross-positive maps that are completely cross-positive. Detailed results about cross-positive maps $\Phi$ mapping between $3\times3$ matrices are given. Finally, an algorithm to produce cross-positive maps that are not completely cross-positive is presented.
Ključne besede: positive polynomials, sum of squares, positive maps, completely positive maps, one-parameter semigroups, convex cones
Objavljeno v DiRROS: 17.10.2025; Ogledov: 229; Prenosov: 110
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Povzetek: The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance $10^{-7}$) on all QMC instances with uniform edge weights on graphs with at most 8 vertices. The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.
Ključne besede: Quantum Max Cut, swap operators, noncommutative polynomials, symmetric group, Gröbner bases
Objavljeno v DiRROS: 04.06.2024; Ogledov: 938; Prenosov: 493
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