Here $S^n $ denotes the unit sphere in an $n$-dimensional (complex) Hilbert space. We have $m$ copies of this sphere. By $S_m $ we mean the permutation group of order $m$.
By the quotient, we identify the m-tuple $(v_1, v_2, \ldots, v_m )$ with $(v_{\sigma_1}, v_{\sigma_2}, \ldots, v_{\sigma_m })$ for an arbitrary element $\sigma \in S_m $.
Without the quotient, the product space $S^n \otimes S^n \otimes \ldots \otimes S^n $ is a compact space with a metric. The question is, how to define a metric on the quotient space to make it a compact metric space?
Let $\pi$ denote the quotient map and $d$ the metric on the product. One obvious scheme is, for $[p']$ and $[q']$ elements of the quotient, to define $d'([p'], [q'])$ to be the distance between the finite preimages $\pi^{-1}[p']$ and $\pi^{-1}[q']$, i.e., the minimum of the distances $d(p, q)$ such that $\pi(p) = p'$ and $\pi(q) = q'$.
Unfortunately, that's probably cumbersome to use in practice because generic preimages contain $m!$ elements.