The Exercise

Let $A$ be an observable of one quantum system, a Hermitian operator on its state space $V$, and $B$ an observable of a second system, a Hermitian operator on its state space $W$. Their tensor product $A \otimes B$ acts on $V \otimes W$, the state space of the two systems together. The exercise asks to show that $A \otimes B$ is Hermitian, an observable of the joint system.

Hermitian Matrices and the Kronecker Product

The library states the claim for matrices. With $m$ and $n$ the dimensions of $V$ and $W$, a basis of each makes $A$ an $m \times m$ matrix and $B$ an $n \times n$ matrix, and in the product basis of $V \otimes W$, $A \otimes B$ becomes the matrix built from $A$ and $B$ this way, their Kronecker product.

A Hermitian matrix $A$ of size $m$ is an $m \times m$ matrix equal to its adjoint $A^\dagger$, the transpose of its complex conjugate: $$A^\dagger = A, \qquad (A^\dagger)_{ij} = A_{ji}^{*}.$$(1)

The Kronecker product of an $m \times m$ matrix $A$ and an $n \times n$ matrix $B$ is the $mn \times mn$ matrix $A \otimes B$ whose rows and columns are labelled by pairs, an index of $A$ followed by an index of $B$, with entries $$(A \otimes B)_{(i,k),(j,l)} = A_{ij}\, B_{kl}.$$(2) With the pairs listed in dictionary order it is the block matrix whose block in place $(i,j)$ is $A_{ij} B$.

The Claim

What Is Checked

The claim is stated over Hermitian matrices as in (1) and their Kronecker product (2), with entries that are complex numbers or, more generally, elements of any commutative ring with a conjugation. The library proves (3) for every pair of sizes $m$ and $n$, equal or not.