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I've seen in some papers and notes that we can write the projector onto the symmetric subspace as $$ \Pi^{d, 2}_{sym} = \frac{1}{2}(I+F) $$ but I can't really figure out how specifically this follows from the characteristics of a projector onto the symmetric subspace. I've seen some say this can be proven using representation theory/Schur's Lemma or with Wick's Theorem, and maybe I don't understand those two fully, but I'm not sure how to apply either here?

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I'm assuming $F$ is the Swap operator here, acting on some finite-dimensional space $H\otimes H$. In bra-ket notation, this reads $F\equiv \sum_{ij} |ij\rangle\!\langle ji|$.

  1. Observe that if $\dim H=2$, then $$F(|01\rangle\pm |10\rangle)=\pm (|01\rangle\pm|10\rangle), \\ F|00\rangle=|00\rangle, \qquad F|11\rangle=|11\rangle, $$ meaning $F$ has eigenvalues $\pm1$, with eigenspaces the symmetric and skew-symmetric subspaces, respectively. In more general dimensions, observe that the eigenspace of $F$ corresponding to the eigenvalue $+1$ is the symmetric subspace, spanned by the vectors $\{|ij\rangle+|ji\rangle\}_{i<j}$, and $\{|ii\rangle\}_i$.

  2. Let $P$ be a generic orthogonal projection (on a finite-dimensional space). This means that $P^\dagger=P$ and $P^2=P$. Any such operator can only have as eigenvalues $0$ and/or $1$.

  3. Let $F$ be a generic unitary operator with eigenvalues $\pm1$. Then $\frac12(I\pm F)$ are orthogonal projections, projecting on the eigenspaces of $F$ with eigenvalues $+1$ and $-1$, respectively.

So to summarise, the Swap $F$ always has eigenvalues $\pm1$, the eigenspace corresponding to the $+1$ eigenvalue is the symmetric space, and $(I+F)/2$ is the projector onto such space.

See for more details Watrous' book. Specifically, chapter 6 (example 6.10) and chapter 7 are particularly relevant here.

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