Two-Qubit Quantum States

Hi, Cardy. In today’s lesson we are going to talk about systems of two qubits. For useful QCs we will need systems with at least two qubits. How do we describe the states of those systems? What happens when we make measurements on the individual qubit states? As we shall see, systems with two (or more) qubits have properties that are both weird and useful as computational resources.

Before we talk about quantum states, let’s think about general systems with two (or more) parts. We know that quantum states carry information about probabilities of results of various measurements. From probability theory, we know that if two events are independent then the probability of event $1$ and event $2$ happening is given by the product of the probabilities of each of the events $P(1~\text{AND}~2) = P(1)P(2)$. For example, if we have a system that consists of a standard six-sided dice and a fair coin, the probability of getting a $5$ on the dice $\text{AND}$ getting heads on the coin is

$$P(5~\text{AND heads}) = P(5)P(\text{heads}) = 1/6*1/2 = 1/12.$$