1. Biased coin as a qubit

Basic Probability

Suppose we have a biased coin which has a 0.64 probability of heads and 0.36 probability of tails. What is a qubit state that reproduces the same behavior as that of the biased coin? What is the amplitude of the state representing heads?

Hint

Write down the qubit state as a superposition of $|0\rangle$ and $|1\rangle$ and make sure it's normalized.

Solution

Here is a state that generates the same probability distribution:

$$ |\psi\rangle = 0.8|0\rangle + 0.6|1\rangle. $$

We always want to make sure that the state of the qubit is normalized that is

$$ \langle \psi | \psi \rangle = 0.8^2 + 0.6^2 = 0.64 + 0.36 = 1 $$