Exercise 2.30

Suppose \(A\) is a \(m\times m\) Hermitian operator and \(B\) is a \(n\times n\) Hermitian operator, we could obtain tensor product between \(A\) and \(B\) as \(A\otimes B\), and the adjoint conjugate of \(A\otimes B\) as

(1)\[ (A\otimes B)^{\dagger} = A^{\dagger}\otimes B^{\dagger} = A\otimes B \]

where we use \(A^{\dagger} = A \) and \(B^{\dagger}= B\). Thus, we conclude that the tensor product of two Hermitian operators is Hermitian.