7
$\begingroup$

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: Loop Equation

The relevant equations are as follows:Yanking 4.6 Yanking 4.8 Yanking 4.9

As an aside, I would really appreciate it if anyone knows where to find the solutions of this book.

Thank you!

Improve this question
$\endgroup$

2 Answers 2

Reset to default
5
$\begingroup$

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).

enter image description here

Improve this answer
$\endgroup$
0
$\begingroup$
  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.

Improve this answer
$\endgroup$
5
  • $\begingroup$ So it's okay to bend a diagram sideways? What would be the proof of that? $\endgroup$
    – Mahathi Vempati
    Dec 30, 2019 at 17:54
  • $\begingroup$ 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)? $\endgroup$
    – Nick Decroos
    Dec 30, 2019 at 18:11
  • $\begingroup$ In step(2), the resulting diagram is like diagram 4.9 once you bend it sideways, right? $\endgroup$
    – Mahathi Vempati
    Dec 31, 2019 at 14:22
  • 1
    $\begingroup$ 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. $\endgroup$
    – Nick Decroos
    Dec 31, 2019 at 15:16
  • $\begingroup$ 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. $\endgroup$
    – Mahathi Vempati
    Jan 4, 2020 at 14:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.