8 Quantum graphs
A quantum graph is a pair \((B,A)\), where \(B\) is a quantum set and \(A\) is a linear map \(B\to B\) such that \(A\bullet A=A\).
A quantum graph \((B,A)\) is real if \(A\) is a real linear map.
Given a quantum set \(B\), we have
\[ m^*m=\sum _i\operatorname {rmul}(u_i)\otimes \operatorname {lmul}(u_i^*), \]
where \((u_i)_i\) is an orthonormal basis of \(B\).
Proof
Using Lemma 5.0.16, we get
\[ m^*m=\Phi (\operatorname {id})=\operatorname {rmul}(u_i)^*\otimes \operatorname {lmul}(u_i). \]
Taking adjoints of the above then gives us our desired result.
Given a quantum graph \((B,A)\), we have \((B,A)\) is real if and only if \(A\) is a positive map.
Proof
If \(A\) is a positive map, then it is real by Theorem 7.0.13. So suppose \(A\) is real. Now let \(x\in {B}\). Then using Lemma 8.0.3, we compute,
\begin{align*} A(x^*x) & = (A \bullet A)m(x^*\otimes x)\\ & = \sum _im(A\otimes A)(x^*u_i\otimes u_i^*x)\\ & = \sum _iA(x^*u_i)A(u_i^*x)=\sum _i{A(u_i^*x)}^*A(u_i^*x) \geq 0. \end{align*}