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Recall that $a=(q+ip)/\sqrt{2}$ in some dimensionless units (Weedbrook might change the units because I think they like $\hbar=2$; I'm using $[q,p]=i$ and $[a,a^\dagger]=1$). We can thus find the equality $$a^\dagger a=(q-ip)(q+ip)/2=[q^2+p^2+i(qp-pq)]/2=\frac{q^2+p^2-1}{2}.$$ This relates the total number of photons to the trace of your matrix $\sigma$. ...