# Diagrammatic Quantum Reasoning: Proving the loop equation using yanking equations

I'm trying to study the book: Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning, and would like some help with Exercise 4.12: As an aside, I would really appreciate it if anyone knows where to find the solutions of this book.

Thank you!

Here is the solution. The trick is to use "the only connectivity matters" rule. The swap rule of 4.9 helps us reorder the inputs, which then makes it topologically equivalent to the next diagram (match the first and second wires of the states). 1. Replace in the diagram in exercise 4.12 the triangles with the corresponding bending wires.
2. It seems that the resulting diagram is like the second identity in 4.9, but with two added straight horizontal wires. (The diagram is rotated, but that doesn't matter.) Apply this identity.
3. The resulting diagram is a cap with added straight wires. It is permitted to straighten the wire.

Another way using yanking out crossings:

Start from the given diagram. Rotate the bottom bending with the cup in 3D around the vertical axis , until it the diagrams looks like the last diagram in 4.6. That is equal to a straight wire.

I hope the rewrite process I described is clear.

• So it's okay to bend a diagram sideways? What would be the proof of that? Dec 30, 2019 at 17:54
• Sorry, it isn't clear to me what you mean with bending sideways. Can you say at which step this is (if it isn't a general remark)? Dec 30, 2019 at 18:11
• In step(2), the resulting diagram is like diagram 4.9 once you bend it sideways, right? Dec 31, 2019 at 14:22
• If you mean with bending it sideways, rotating the whole diagram 90 degrees. Then yes. But I'm still not sure if I understand you. Dec 31, 2019 at 15:16
• I don't think you can rotate the whole diagram 90 degrees. For example, in the first diagram of Equation 4.9, both the wires mean refer to the output. Rotating it would make one the input and another the output. Jan 4, 2020 at 14:31