Sharp nonzero lower bounds for the Schur product theorem

By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product $M \circ N$ of two positive semidefinite matrices $M,N$ is again positive. Vybiral [Adv. Math. 2020] improved on this by showing the uniform lower bound $M \circ \overline{M} \geq E_n / n$ for all $n \times n$ real or complex correlation matrices $M$, where $E_n$ is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions on groups. Vybiral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of $M$, or for $M \circ N$ when $N \neq M, \overline{M}$. A natural third question is to obtain a tighter lower bound that need not vanish as $n \to \infty$, i.e. over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybiral's result to lower-bound $M \circ N$, for arbitrary complex positive semidefinite matrices $M, N$. Specifically: we provide tight lower bounds, improving on Vybiral's bounds. Second, our proof is 'conceptual' (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy-Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert-Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs-Krieg-Novak-Vybiral [J. Complexity, in press], which yields improvements in the error bounds in certain tensor product (integration) problems.