Optimal Bounds on the Positivity of a Matrix from a Few Moments

Abstract

In many contexts one encounters Hermitian operators M on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such M is positive semidefinite? Here we approach this problem by deriving asymptotically optimal bounds to the distance to the positive semidefinite cone in Schatten p-norm for all integer $p\in [1,\infty )$, assuming that we know the moments $\mathbf {tr}(M^k)$ up to a certain order $k=1,\ldots , m$. We then provide three methods to compute these bounds and relaxations thereof: the sos polynomial method (a semidefinite program), the Handelman method (a linear program relaxation), and the Chebyshev method (a relaxation not involving any optimization). We investigate the analytical and numerical performance of these methods and present a number of example computations, partly motivated by applications to tensor networks and to the theory of free spectrahedra.

Extension to Von Neumann Algebras

Let ${\mathcal {N}}$ be a von Neumann algebra equipped with a faithful normal trace $\mathbf {tr}$ satisfying $\mathbf {tr}(1) = 1$. For example, ${\mathcal {N}}$ may be the group von Neumann algebra of a discrete group $\varGamma$, defined as the weak operator closure of the group algebra ${\mathbb {C}}[\varGamma ]$ as acting on $\ell ^2(\varGamma )$; the trace is given by $\mathbf {tr}(x) := \langle e,xe\rangle$ with $e\in \varGamma$ being the unit. In the case where ${\mathcal {N}}= M_s({\mathbb {C}})$ is just the matrix algebra, the discussion presented here specializes to that of the main text.

For a Hermitian element $M\in {\mathcal {N}}$, the Schatten p-norm is again defined as

$$\begin{aligned} \Vert M \Vert _p := \left( \mathbf {tr}(|M|^p)\right) ^{1/p}, \end{aligned}$$

where the absolute value and power functions are defined in terms of functional calculus. We can now ask the same question as in the main text: suppose we have M of which we only know the values of the first m moments

$$\begin{aligned} \mathbf {tr}(M^k) \text{ for } k = 0,\ldots ,m. \end{aligned}$$

Then what can we say about the p-distance from M to the cone of positive elements?

It is straightforward to see that essentially all of the methods of the main text still apply without any change. The only differences are the following:

• Remark 1 no longer applies, since the estimate that we used there involves the matrix size explicitly. Hence there is no bound on $\Vert M\Vert _\infty$ that could be computed from the moments. So in order to scale M such that $\Vert M\Vert _\infty \le 1$ is guaranteed, an a priori bound on $\Vert M\Vert _\infty$ needs to be known in addition to the moments. Otherwise our methods will not apply in their current form.

• The proof of Proposition 1 is still essentially the same, but $\sigma$ needs to be generalized to the spectral scale, and the corresponding inequality is [12, Corollary 3.3(1)].

Everything else is completely unchanged, including the fact that we are secretly addressing a version of the Hausdorff moment problem.