Quantum states can be represented (or 'written down') in several ways. Each one of them has its advantages. One way is to represent them as matrices. When these matrices have many zero entries, we might express them as "kets". We can even represent them geometrically, on the so-called "Bloch sphere". Here we give an overview of these various representations. Do practice with them; this will significantly increase your understanding of quantum states!
- Fundamentals of linear algebra
Sometimes, one representation is more useful than the other. When might ket notation be easier than matrices?
Let's practice with some states and gates!
Here is an overview of the gates used in the questions:
- What happens when you apply a Hadamard gate on the plus state in matrix representation? How does that look on the Bloch sphere?
- What if you do an X gate on the 0 state? And on the -? What does that look like on the Bloch Sphere?
- Try multiplying the following matrices: HZH. To what gate is the resulting matrix equal? Applying this matrix to a state is the same as Hadamard, then a Z gate, and then a Hadamard on a state. Try this on the 0 state, on the Bloch sphere. Can you figure out why HZH is equal to some other gate?
- What does the truth table of the CNOT gate look like?