Reminder: Class on Friday, October 23rd will again be on Zoom, due to my illness. The link is in Canvas. If you need a quiet place to join, you are welcome to use our usual classroom. The class will be recorded, but please attend synchronously if you can.

Welcome to Module 4: Quantum! Today we met complex numbers, qubits, measurement, the $L$ and $V$ bases, and the three filter experiment.

Lecture notes and video: my notes from today’s Zoom lecture are posted (also on the Notes/Videos page and the Archive). The link to the Zoom recording is on discord.

Relevant Textbook Sections:

The opening of Chapter 6 (Quantum): complex numbers, single qubit states, and measurement.

To Do:

1. Complex numbers worksheet. Work through whatever parts are most useful or interesting to you of the online Complex Numbers Worksheet. My suggestion: if you’re comfortable with complex numbers as done in class, then start at “Domain Colouring,” which is just lovely. If complex numbers are new to you, the earlier parts of the worksheet will give you the practice you need for the rest of the module.
2. Compute the length of the complex number $\frac{3}{5} + \frac{4}{5}i$. (The answer should be $1$; please make sure you got this!)
3. Consider the qubit superposition $i|0\rangle + \sqrt{3}|1\rangle$.
   1. Normalize it so that $|a|^2 + |b|^2 = 1$.
   2. Measure it in the basis $|0\rangle$, $|1\rangle$. What are the possible outcomes and their probabilities?
4. Recall the $L$ basis $|\uparrow\rangle$, $|\rightarrow\rangle$ and the $V$ basis $|\nearrow\rangle$, $|\nwarrow\rangle$, where $|\nearrow\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow\rangle + |\rightarrow\rangle\right)$ and $|\nwarrow\rangle = \frac{1}{\sqrt{2}}\left(|\uparrow\rangle - |\rightarrow\rangle\right)$.
   1. Measure $|\nearrow\rangle$ in the $L$ basis (i.e. give the outcomes and their probabilities).
   2. Measure $|\nearrow\rangle + i|\nwarrow\rangle$ in the $L$ basis. (Normalize first!)
5. Consider the state $|\psi\rangle = \frac{1}{\sqrt{5}}|\uparrow\rangle + \frac{2}{\sqrt{5}}|\rightarrow\rangle$.
   1. Measure it in the $L$ basis. What are the possible outcomes and their probabilities?
   2. Measure it in the $V$ basis. What are the possible outcomes and their probabilities? (To do this, you will first need to write $|\psi\rangle$ as a normalized linear combination of $|\nearrow\rangle$ and $|\nwarrow\rangle$.)