# Proof that the projector onto the symmetric subspace of the Swap $F$, with $n=2$, equals $\frac{1}{2}(I+F)$

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?

## 1 Answer

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 $$\{|ij\rangle+|ji\rangle\}_{i, and $$\{|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.