Let $A\in\mathbb{R}^{n_1\times n_2\times n_3 \times n_4}$ be a tensor. Suppose that $k$ is the minimum integer there exist matrices $X_1,\ldots,X_j\in\mathbb{R}^{i_1\times i_2}$ and $Y_1,\ldots,Y_j\in\mathbb{R}^{i_3\times i_4}$ such that
$$A(i_1,i_2,i_3,i_4) = \sum_{j=1}^k X_j(i_1,i_2)Y_j(i_3,i_4).$$
Does the integer $k$ have a name in the literature? I'm tempted to say that $k$ is the rank of the associated matrix $A(\{i_1,i_2\},\{i_3,i_4\})\in\mathbb{R}^{n_1n_2\times n_3n_4}$, but there's probably a term that already exists. Is it a tensor-train rank?