Lower bound of the expectation of the product of inner products of random vectors

Question

I encountered the following value in my research:

Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$. Denote $$ L = \mathop{\mathrm{E}}_x[ \prod_{1\leq j\leq m} \langle \alpha_j,x \rangle^2], $$ where $x \in \mathbb{R}^n$ is a random vector whose $\ell$th component is i.i.d. uniformly over $\{-1,1\}$.

My observation is that if $m=n$, $L$ may be $0$. For instance, let $m=n=2$, $\alpha_1=\frac{1}{\sqrt{2}}(1\ 1)$ and $\alpha_2=\frac{1}{\sqrt{2}}(1\ {-1})$. However, if, say, $m=o(n)$ or even $n/2$, then $L$ is seemingly lower bounded by $\Omega(1)$. How can I lower bound $L$ for those small $m$ (relative to $n$)?

For small and concrete $m$, I can manually lower bound $L$ by first expand the RHS and then apply the random subsum principle (namely, $\mathrm{E}_x[x_{\ell_1}x_{\ell_2} \dots x_{\ell_c}]=0$, where $x_{\ell_j}$ denotes the $\ell$th component of $x$). But I am not sure how to generalize this approach to arbitrary $m$.

Any hint or reference will be greatly appreciated (let alone an answer).

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For those who are curious about the background: I am working on quantum query complexity, and I am trying to use the polynomial method to solve some certain problems. If you are interested but not familiar with these terms, refer to [BdW02].

Answer 1

The exact lower bound on $L$ is $0$ for any $n\ge2$ and $m\ge2$. Indeed, for $j=1,\dots, n$, let $a_j:=\alpha_j$ be any unit vectors in $\mathbb R^n$ such that $a_1=\frac1{\sqrt{2}}(1,1,0,\dots,0)$ and $a_2=\frac1{\sqrt{2}}(1,-1,0,\dots,0)$. Then $\prod_{1\leq j\leq m} \langle \alpha_j,x \rangle^2=0$ and hence $L=0$.