Universal state-inversion positivity conjecture for multipartite quadratic forms

Let $\mathscr{H}$ be a finite-dimensional Hilbert space with a tensor decomposition

$$\mathscr{H}=\mathscr{H}_1\otimes\cdots\otimes\mathscr{H}_n,$$

where $\dim(\mathscr{H}_i)=d_i\geq2$. For $v\in\\{0,1\\}^n$, let $q_v(\alpha,C)$ be the quadratic form defined from the partial traces of $C\in L(\mathscr{H})$, and write $r=\operatorname{rank}(C)$.

Universal state-inversion positivity conjecture. For every $C\in L(\mathscr{H})$ and every $v\in\\{0,1\\}^n$,

$$q_v(\alpha,C)\geq0$$

whenever

$$|\alpha|\leq\alpha_{\mathrm{opt}}=\frac{1}{\min\\{r,\max\\{d_1,\ldots,d_n\\}\\}}.$$

The conjecture unifies positivity claims for the quadratic forms arising from different choices of symmetrization and antisymmetrization. The paper presents it as the main conjecture, with the preceding rank-specific statement as a special case; its general validity remains open.