The "$S_{n,n}$ conjecture" asserts that there does not exist a simple graph on $n$ vertices with Laplacian spectrum $\{0,1,2,\ldots,n-1\}$ for any integer $n \geq 2$. This conjecture has already been ...

A multiset of pure quantum states is said to be k-learnable if there is a measurement strategy that always narrows an unknown sample drawn from the list down to one of at most k candidates. The parame...

We construct PPT states on $\mathbb{C}^m \otimes\mathbb{C}^n$ that have Schmidt number asymptotically approaching the smaller local dimension. More specifically, we construct a PPT state with Schmidt ...

Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bi...

A graph with Laplacian matrix $L$ is called Laplacian integral if the eigenvalues of $L$ are all integers, and it is called $\{-1,0,1\}$-diagonalizable if $L$ has a full set of eigenvectors with entri...

We introduce a convergent hierarchy of lower bounds on the minimum value of a real form over the unit sphere. The main practical advantage of our hierarchy over the real sum-of-squares (RSOS) hierarch...

A matrix is said to have factor width at most k if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single k×k…

YouTube video by Nathaniel Johnston

YouTube video by Nathaniel Johnston

An endless crafting game

YouTube video by Nathaniel Johnston

A graph is called Laplacian integral if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Lapl…