10 problems

Let $(a,b)$ and $(a',b')$ lie in $\mathbf{U}_3\cup\mathbf{V}_3\cup\mathbf{W}_3$, and consider the bridge graph with bond dimension $3$ and boundary dimensions $a,b,b',a'$. Three-bo…

Let $w$ be a positive integer and let $(a,b)$ and $(a',b')$ lie in $\mathbf{U}_w\cup\mathbf{V}_w\cup\mathbf{W}_w$, with $ab'\leq a'b$. Consider the corresponding bridge graph with…

Let $w$ be a positive integer and let $(a,b)$ and $(a',b')$ lie in the union of the regions $\mathbf{U}_w$, $\mathbf{V}_w$, and $\mathbf{W}_w$. Consider the bridge graph with bond…

Positivity and definiteness conjecture. The dominant eigenvalue $\eta$ is real and positive, and

Dimension conjecture. For all such $d$, $\mathbf{m}$, and $\mathbf{n}$, the displayed dimension formula holds outside precisely the listed exceptional cases.

Let $G_K$ be a hypergraph defining a $K$-party state construction, and let $\Ket{G_K}$ denote the resulting $K$-party quantum state. The construction uses AME tensors at bulk verti…

Let $\Sigma$ be a network with $n\geq2$ boundary vertices. An extremal-ray realization is a subnetwork whose entropy vector realizes an extremal ray of the $n$-boundary-region holo…

Let $\Sigma$ be a network with four boundary vertices $A_1,\ldots,A_4$, and let $I(A_i:A_j)$ denote the mutual information between $A_i$ and $A_j$ on $\Sigma$. Let…

Large bond-dimension conjecture. For $d=2$ and $k$ sufficiently large, there exists a tensor $T \in \operatorname{TNS}(G_{2 \times N},d,k)$ such that all flattenings of $T$ have ma…

Let $N$ be a $v$-valent graph with $k$ input cut edges and $l$ output cut edges, with the incoming edges at each vertex locally ordered. For a $v$-index tensor $W$ on a vector spac…