Transformation matrix for a two-qubit operation where there are more than two qubits

Question

I have an OPENQASM program that performs entanglement swapping. It has five qubits: the data qubit and four link qubits. It works, but I want to see the details of the Bell measurement transformation. So what I am looking for is a reference or technique that will help me construct the 32 by 32 matrix from the four-by-for matrix for the Bell measurement basis. I understand that this is written as a direct product or tensor product, but I don't know what that means in terms of matrix operations.

Something like $B \otimes I$? Again, what does this mean?

Of course, the general problem is, given an $n$-qubit unitary transformation, how to construct the $m$-qubit matrix where $m>n$?

Answer 1

Parallel composition of operator $A$ acting on subsystem $a$ and operator $B$ acting on subsystem $b$ is the tensor product $A\otimes B$ on the joint system $ab$. In particular, parallel composition of $A$ on $a$ and identity on $b$ is indeed $A\otimes I$ as anticipated in the question. This applies to unitary gates, measurement operators, pairs of Kraus operators etc.

We can compute the matrix of operator $A\otimes B$ using Kronecker product

$$ A\otimes B=\begin{bmatrix}a_{11}B&\dots&a_{1n}B\\&\dots&\\a_{n1}B&\dots&a_{nn}B\end{bmatrix} $$

of matrix $A$ with matrix $B$. Note that if $A$ is $n\times n$ and $B$ is $m\times m$ then $A\otimes B$ is $nm\times nm$.