The tracial moment problem and trace-optimization of polynomials

Burgdorf, Sabine; Cafuta, Kristijan; Klep, Igor; Povh, Janez

doi:10.1007/s10107-011-0505-8

Cited by 34 publications

(37 citation statements)

References 43 publications

(50 reference statements)

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“…This fact does not generalize to higher powers or more variables (see examples in [KS08a,KS08b,Qua15]). In [BK12,BCKP13] the authors obtain the tracial analogs of the results on the classical moment problem of Curto and Fialkow, Stochel, Bayer and Teichmann, Fialkow and Nie [FN10], providing powerful means to tackle the special cases of the TTMP in a way analogous way to the classical one.…”

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confidence: 99%

The Singular Bivariate Quartic Tracial Moment Problem

Bhardwaj

Zalar

2017

Complex Anal. Oper. Theory

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The (classical) truncated moment problem, extensively studied by Curto and Fialkow, asks to characterize when a finite sequence of real numbers indexes by words in commuting variables can be represented with moments of a positive Borel measure µ on R n . In [BK12] Burgdorf and Klep introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of µ with tracial moments of matrices. In the bivariate quartic case, where indices run over words in two variables of degree at most four, every sequence with a positive definite 7 × 7 moment matrix M 2 can be represented with tracial moments [BK10,BK12]. In this article the case of singular M 2 is studied. For M 2 of rank at most 5 the problem is solved completely; namely, concrete measures are obtained whenever they exist and the uniqueness question of the minimal measures is answered. For M 2 of rank 6 the problem splits into four cases, in two of which it is equivalent to the feasibility problem of certain linear matrix inequalities. Finally, the question of a flat extension of the moment matrix M 2 is addressed. While this is the most powerful tool for solving the classical case, it is shown here by examples that, while sufficient, flat extensions are mostly not a necessary condition for the existence of a measure in the tracial case. INTRODUCTION1.1. Context. The Moment problem (MP) is a classical question in analysis and concerns the existence of a positive Borel measure µ supported on a subset K of R n , representing a given sequence of real or complex numbers indexed by monomials as the integration of the corresponding monomials w.r.t. µ; nice expositions on the MP are [Akh65, KN77]. The solution to the MP on R n is given by Haviland's theorem [Hav35], which establishes the duality with positive polynomials and relates the MP to real algebraic geometry (RAG). One of the cornerstones of RAG is the celebrated Schmüdgen theorem [Sch91], which solves the problem on compact basic closed semialgebraic sets and is the beginning of extensive research of the MP in RAG; we refer the reader to [Put93, PV99, DP01, PS01, KM02, Lau05, Lau09, Mar08, Las09] and the references therein for further details. Another important aspect of the MP is uniqueness of the representing measures. For compact sets the measure is unique (see e.g., [Mar08]), while for noncompact sets, the question of uniqueness is highly nontrivial (see [PS06,PS08]). There are various generalizations of the MP. Functional analysis studies various versions of matrix and operator MPs; see [Kre49, Kov83, AV03, Vas03, BW11, CZ12, KW13] and references therein. The quantum MP from quantum physics is considered in [DLTW08]. The rational MP, which extends Schmüdgen theorem from the polynomial algebra to its localizations, is solved in [CMN11], while [GKM16] investigates the MP for the polynomial algebra in infinitely many variables. The MP on semialgebraic sets of generalized function is considered in [IKR14]. The beginning of free RAG is the sol...

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The Singular Bivariate Quartic Tracial Moment Problem

Bhardwaj

Zalar

2017

Complex Anal. Oper. Theory

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“…. , X n ))/d (so that the identity matrix has trace one) [7,6,8,21]. The moment approach for these problems relies on minimizing L(f ), where L is a linear functional on the space of noncommutative polynomials satisfying some necessary conditions, and L(f ) models ψ * f (X 1 , .…”

Section: Techniques From Noncommutative Polynomial Optimizationmentioning

confidence: 99%

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

2018

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In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.

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“…Similarly we can use semidefinite programming to test whether a given nc polynomial f ∈ R X is an element of Θ 2 as first observed in [19], see also [10,7,6]. The method behind it is a variant of the Gram matrix method:…”

Section: Sums Of Hermitian Squares (With Commutators) and Semidefinitmentioning

confidence: 99%

Rational sums of hermitian squares of free noncommutative polynomials

Cafuta

Klep

Povh³

2015

Ars Math. Contemp.

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In this paper we consider polynomials in noncommuting variables that admit sum of hermitian squares and commutators decompositions. We recall algorithms for finding decompositions of this type that are based on semidefinite programming. The main part of the article investigates how to find such decomposition with rational coefficients if the original polynomial has rational coefficients. We show that the numerical evidence, obtained by the Gram matrix method and semidefinite programming, which is usually an almost feasible point, can be frequently tweaked to obtain an exact certificate using rational numbers. In the presence of Slater points, the Peyrl-Parrilo rounding and projecting method applies. On the other hand, in the absence of strict feasibility, a variant of the facial reduction is proposed to reduce the size of the semidefinite program and to enforce the existence of Slater points. All these methods are implemented in our open source computer algebra package NCSOStools. Throughout the paper many worked out examples are presented to illustrate our results.

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The tracial moment problem and trace-optimization of polynomials

Cited by 34 publications

References 43 publications

The Singular Bivariate Quartic Tracial Moment Problem

The Singular Bivariate Quartic Tracial Moment Problem

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

Rational sums of hermitian squares of free noncommutative polynomials

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