Positive univariate trace polynomials

Abstract

A univariate trace polynomial is a polynomial in a variable x and formal trace symbols $\mathrm{Tr}(x^j)$. Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

Introduction

Univariate trace polynomials are real polynomials in x and $\mathrm{Tr}(x^j)$ for $j\in \mathbb{N}$. We view x as a matrix variable of unspecified size, and evaluate a (univariate) trace polynomial $f(x,\mathrm{Tr}(x),\mathrm{Tr}(x^2),\dots )$ at any $n\times n$ matrix X as $f(X,\frac{1}{n}\mathrm{tr}(X),\frac{1}{n}\mathrm{tr}(X^2),\dots )$. That is, the trace symbol Tr evaluates as the normalized trace of a matrix. Trace polynomials as matricial functions originated in invariant theory [23], and more recently emerged in free probability [11] and quantum information theory [22], [10]. In this paper we characterize trace polynomials that have positive semidefinite values at symmetric matrices of all sizes.

Trace polynomials form a commutative polynomial ring (in countably many variables), and several sum-of-squares positivity certificates (Positivstellensätze) for multivariate polynomials on semialgebraic sets are provided by real algebraic geometry [3], [18], [21], [19], [2]. However, this theory does not appear to directly apply to our setup. First, matrix evaluations of trace polynomials are just a special class of homomorphisms from trace polynomials. Second, the dimension-free context addresses positivity on symmetric matrices of all sizes, hence on a countable disjoint union of real affine spaces; there is no bound (with respect to the degree of a trace polynomial) on the size of matrices for which positivity needs to be verified (Remark 3.6).

Therefore a different approach is required. To demonstrate it, consider the inequality

$$\mathrm{Tr}(X^4)(\mathrm{Tr}(X^2)-\mathrm{Tr}{(X)}^2)+2\mathrm{Tr}(X^3)\mathrm{Tr}(X^2)\mathrm{Tr}(X)-\mathrm{Tr}{(X^3)}^2-\mathrm{Tr}{(X^2)}^3\ge 0$$

which holds for all symmetric matrices X. One way to certify (1.1) is by noticing that $f=\mathrm{Tr}(x^4)(\mathrm{Tr}(x^2)-\mathrm{Tr}{(x)}^2)+2\mathrm{Tr}(x^3)\mathrm{Tr}(x^2)\mathrm{Tr}(x)-\mathrm{Tr}{(x^3)}^2-\mathrm{Tr}{(x^2)}^3$ is the determinant of the Hankel matrix

$$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill \mathrm{Tr}(x)\hfill & \hfill \mathrm{Tr}(x^2)\hfill \\ \hfill \mathrm{Tr}(x)\hfill & \hfill \mathrm{Tr}(x^2)\hfill & \hfill \mathrm{Tr}(x^3)\hfill \\ \hfill \mathrm{Tr}(x^2)\hfill & \hfill \mathrm{Tr}(x^3)\hfill & \hfill \mathrm{Tr}(x^4)\hfill \end{array}\right)$$

which is positive semidefinite for every matrix evaluation (since it is obtained by applying the normalized partial trace to a positive semidefinite matrix). Another certificate of (1.1), in the spirit of sum-of-squares representations in real algebraic geometry, is

$$\mathrm{Tr}\left({(x-\mathrm{Tr}(x))}^2\right)\cdot f=\mathrm{Tr}\left({((\mathrm{Tr}{(x)}^2-\mathrm{Tr}(x^2))x^2+(\mathrm{Tr}(x^3)-\mathrm{Tr}(x^2)\mathrm{Tr}(x))x+\mathrm{Tr}{(x^2)}^2-\mathrm{Tr}(x^3)\mathrm{Tr}(x))}^2\right),$$

where we view Tr as an idempotent linear endomorphism of trace polynomials in a natural way. Thus f is a quotient of traces of squares. The main result of this paper shows that these characterizations apply to all positive trace polynomials.

Corollary 3.8

Let f be a univariate trace polynomial. Then $f(X)$ is positive semidefinite for all symmetric matrices X if and only if f is a quotient of sums of products of squares and traces of squares of trace polynomials.

Corollary 3.8 is a special case of Theorem 3.2, a tracial Positivstellensatz that characterizes positivity of trace polynomials subject to tracial constraints under certain mild regularity assumptions. The proof of Theorem 3.2 splits into two parts: every tracial inequality is a consequence of a certain tracial Hankel matrix being positive semidefinite (Proposition 3.1), and this positive semidefiniteness is in turn certified by traces of squares (Proposition 2.4).

Since we are addressing trace polynomials in only one matrix variable and the trace is invariant under conjugation, we could of course restrict evaluations to diagonal matrices and reach the same positivity conclusions. From this viewpoint, Corollary 3.8 pertains to positive symmetric polynomials and mean inequalities in combinatorics and statistics [4], [28], [9], [20], [27]: a positive trace polynomial corresponds to a sample-size independent power mean (or moment) inequality.

Nevertheless there are benefits to working with general matrix evaluations of trace polynomials. Besides the algebraic structure, there is an intimate connection with the emerging area of free analysis [13] (cf. Proposition 2.1). Evaluations on arbitrary symmetric matrices put the positivity of univariate trace polynomials under the umbrella of multivariate trace polynomials and noncommutative tracial inequalities induced by them. If one considers only tuples of matrices of fixed size, Positivstellensätze on arbitrary tracial semialgebraic sets are known [24], [25], [6], [17]. On the other hand, multivariate trace positivity on matrices of all finite sizes is not understood well. Namely, the failure of Connes' embedding conjecture [12] implies that there is a noncommutative polynomial whose trace is positive on all matrix contractions, but negative on a tuple of operator contractions from a tracial von Neumann algebra [15], which obstructs the existence of a clean trace-of-squares certificate for matrix positivity in general. There are however Positivstellensätze for positivity of multivariate trace polynomials on von Neumann algebras subject to archimedean constraints [14]. In a different direction, tracial inequalities of analytic functions are heavily studied in relation to monotonicity, convexity and entropy in quantum statistical mechanics [5]. The results of this paper fill the gap in understanding univariate trace polynomials by demonstrating that in one operator variable, global positivity on matrices of all sizes implies global operator positivity (in tracial von Neumann algebras).