Let $n,d > 1$ be integers. We consider a vector space $V \cong \mathbb C^d$, and the ways in which we may decompose vectors $$\Psi \in V^{\otimes n} := \bigotimes_{j=1}^n V$$ into a sum of product vectors, $$ \Psi = \sum_{t=1}^M s_t \Bigl(\psi^{(1)}_t \otimes \cdots \otimes \psi^{(n)}_t\Bigr)\;,$$ for $s_t \in \mathbb C$, where $\psi^{(j)}_t \in V$ are unit vectors for each $1 \leqslant j \leqslant n$ and $1 \leqslant t \leqslant M$. I'm specifically interested in the conditions in which we can bound $M$ from above for all $\Psi$.<br><br>

1. Let $\alpha \geqslant 0$ be a function of $n$ such that $1/(1 - \alpha)$ increases at most polynomially with $n$. Suppose that for each $1 \leqslant j \leqslant n$ and for all $1 \leqslant t,u \leqslant M$, we require that $$\Bigl\langle \psi^{(j)}_t , \psi^{(j)}_u \Bigr\rangle\Bigl\langle \psi^{(j)}_u , \psi^{(j)}_t \Bigr\rangle \ \in\  [0,\alpha] \cup \{1\} \ .$$
What is the smallest value of $M$ for which all vectors in $V^{\otimes n}$ have such a decomposition?<br><br>

2. What is the smallest such value of $M$ if we set $\alpha := 0$ for all $n$? Equivalently: if for each $1 \leqslant j \leqslant n$ and for all $1 \leqslant t,u \leqslant M$, we require that $\psi^{(j)}_t$ and $\psi^{(j)}_u$ are either equal or orthogonal?<br><br>

For the second question above, in the case $n = 2$ we may set $M = d$ (and as a corollary restrict each scalar $s_t$ to be a non-negative real); this is just the Schmidt decomposition. And we may always bound $M$ from above by $M \leqslant d^n$ by simply chosing an orthonormal basis $\mathbf B$ for $V$, and requiring $\psi^{(j)}_t \in \mathbf B$ for each $(j,t)$. Are there any good upper bounds on $M$ for $n > 2$, for either for $\alpha = 0$, $\alpha > 0$ some constant, or for some function $\alpha \in 1 - \Omega(1/\mathrm{poly}(n))$?