an element $f\in V^{\otimes k}$, we write the (canonical) symmetrization of $f$ as $\operatorname{Sym}(f)$. Namely, the map $V^{\otimes k}\ni f\mapsto \operatorname{Sym}(f)\in V^{\otimes k}$ is characterized as the unique continuous linear operator on $V^{\otimes k}$ such that $$\operatorname{Sym}(f_1\otimes\cdots\otimes f_k)=\frac{1}{k!}\sum_{\tau\in\mathcal{S}_k}f_{\tau(1)}\otimes\cdots\otimes f_{\tau(k)}$$ for all $f_1,\dots,f_k\in V$, where $\mathcal{S}_k$ denotes the set of all permutations of $\{1,\dots,k\}$, i.e. the symmetric group of degree $k$. An element $f\in V^{\otimes k}$ is said to be *symmetric* if $\operatorname{Sym}(f)=f$. We refer to Appendix E of [@Janson1997] for details on Hilbert space tensor products. 0

Tensors
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In this subsection we introduce some notation related to tensors (or multi-way arrays) which are necessary to state our main results.

We denote by $\mathbb{K}$ the real field $\mathbb{R}$ or the complex field $\mathbb{C}$ and consider a vector space $V$ over $\mathbb{K}$. Let $q$ be a positive integer $q$. We denote by $\mathcal{T}^q_d(V)$ the set of all $V$-valued functions on $\{1,\dots,d\}^q$ ($V$-valued $d$-dimensional $q$-way tensors). For a tensor $T\in\mathcal{T}^q_d(V)$ and indices $i_1,\dots,i_q\in\{1,\dots,d\}$ we write $T(i_1,\dots,i_q)$ as $T_{i_1,\dots,i_q}$ and $T$ itself as $T=(T_{i_1,\dots,i_q})_{1\leq i_1,\dots,i_q\leq d}$. We identify $\mathcal{T}^1_d(V)$ with $V^d$ and $\mathcal{T}^2_d(V)$ with the set of all $V$-valued $d\times d$ matrices, respectively. When $V=\mathbb{K}$, $\mathcal{T}^q_d(\mathbb{K})$ is naturally identified with the Hilbert space 