Suppose I had 3 entangled qubits and I wanted to apply a SWAP gate on the first and third qubit.

Because it's entangled I can't decompose it into individual states, and because the qubits are not adjacent I can't simply take a Kronecker product with an identity matrix.

How would I go about creating the matrix for applying this transformation? Also is there a generalization for swapping 2 arbitrary qubits in an n qubit system?


Gates don't depend on states they act on. So it doesn't matter if the state is entangled or not. The gate you've described (let's call it $U$) acts on a standard basis by this rules $$ U |0x0\rangle = |0x0\rangle $$ $$ U |0x1\rangle = |1x0\rangle $$ $$ U |1x0\rangle = |0x1\rangle $$ $$ U |1x1\rangle = |1x1\rangle $$ for each $x=0$ and $x=1$.

You can write this matrix as $$ U=\sum_{x=0}^1 \big(|0x0\rangle\langle 0x0 |+|1x0\rangle\langle 0x1 |+|0x1\rangle\langle 1x0 |+|1x1\rangle\langle 1x1 | \big) = $$ $$ = |0\rangle\langle 0| \otimes I \otimes |0\rangle\langle 0| + |1\rangle\langle 0| \otimes I \otimes |0\rangle\langle 1| + |0\rangle\langle 1| \otimes I \otimes |1\rangle\langle 0| + |1\rangle\langle 1| \otimes I \otimes |1\rangle\langle 1| $$


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