# Simplify bra-ket notation with kronecker product and kronecker sum

I am taking a quantum informatics and communication course, this is the first time I have faced with Dirac's Bra-ket notation.

I have the following equation(Swap gate with 3 cnot):

First equation

$$|\phi_2\rangle = |i\bigoplus(i\bigoplus k)\rangle\bigotimes|i\bigoplus k\rangle$$

the textbook derived this simplier formula :

$$|\phi_2\rangle=|k\rangle\bigotimes|i\bigoplus k\rangle$$

I came to the same answer , so my question is the rules I applied are they logically correct?

Here is my method:

1. "Get rid of" the Kronecker sums by using the following property:

$$i\bigoplus k = i\bigotimes I+I\bigotimes k$$

but I also now $$|i\rangle$$ and $$|k\rangle \in \{0,1\}$$ so identity matrix(I) become the skalar 1(since $$|k\rangle$$ and $$|i\rangle$$ is also scalar)

$$|i\bigotimes I + I \bigotimes (i\bigotimes k)\rangle\bigotimes|i \bigoplus k \rangle$$=

=$$|i\bigotimes (i\bigotimes k)\rangle\bigotimes |i\bigoplus k\rangle$$=

since Kronecker product is associative

=$$|(i\bigotimes i)\bigotimes k\rangle\bigotimes |i\bigoplus k\rangle$$

I am not sure with the following, $$i\bigotimes i = 1$$if i is scalar

and then $$1 \bigotimes k$$ = k?

=$$|k\rangle\bigotimes|i\bigoplus k\rangle$$

• Something with this notation seems off here. If the $\oplus$ really operates on the numbers $i,k\in\lbrace 0,1\rbrace$, then $i\oplus k=i+k$ by definition (as $I=1\in\mathbb R$ as you pointed out). On the other hand, $|i\rangle\oplus|k\rangle$ classically is only defined (via $A\oplus B=A\otimes I+I\otimes B$) if $|i\rangle,|k\rangle$ are square matrices of some size and not vectors in $\mathbb R^n$. That'd be the first thing we would have to clear up... then again I've never been into quantum logic gates too much so maybe I'm not the right person to help here in the first place? – Frederik vom Ende Sep 23 '18 at 9:24