11 problems

Let $n$ be the number of particles in a multipartite qubit state, and let the overlap of the state with the set of product states be its maximal overlap with any product state. Aub…

For a tensor dimension $n_1\times n_2\times\dots\times n_d$, let $\phi(\mathbb{R}^{n_1\times n_2\times\dots\times n_d}_+)$ be the infimum of the spectral-to-Frobenius norm ratio ov…

Let $2\le n=\prod_{k=1}^{s_n}p_k$ be the prime factorization, with $2\le p_1\le p_2\le\dots\le p_{s_n}$. Consider a tensor…

Let $\boldsymbol{\beta}$ denote the lower bound in equation (main) for the ratio of the spectral norm to the Frobenius norm. An unfolded permutation tensor is a tensor obtained by…

Shang et al.'s conjecture. If the entanglement of $\rho$ is detected by the matrix unit tester, then it is also detected by the SIC POVM tester:

Let $T\in\bigotimes_{i=1}^t\mathbb R^n$ be a tensor of CP rank $r$, and let $|T|_\infty=\max_{i_1,\ldots,i_t}|T_{i_1,\ldots,i_t}|$. Write $\|T\|_\mathrm{max}$ for its tensor max-no…

Projective/injective ratio conjectures. The lower bound $r(n,m)\geqslant19/18$ can be improved to $r(n,m)\geqslant\sqrt{2}$, and the exponent $1/8$ in the known asymptotic lower bo…

Tensor norm-gap entanglement conjecture. Any tensor $z$ satisfying

Chebyshev extremality conjecture. The polynomial $T_n(2x-1)$ is extremal in the argument above, so equality holds in both displayed inequalities. This is explicitly left as an open…

For positive integers $n$ and $m$, let $r(n,m)$ denote the projective/injective ratio defined in the paper for pairs of normed spaces of dimensions $n$ and $m$. Absolute minimum co…

Let the weak Banach–Mazur compactum be the space of $n$-dimensional Banach spaces equipped with the weak Banach–Mazur distance, and let its radius denote the corresponding radius.…