The previous chapters focused on the properties and manipulation of a single qubit. While concepts like superposition and unitary evolution are fundamental, a single qubit offers limited computational utility. To perform complex algorithms, we must combine multiple qubits into a single system. This requires a mathematical framework capable of describing the growth of the state space as bits are added.

In this module, we introduce the tensor product, denoted as $\otimes$. This operation allows us to combine independent vector spaces into a larger composite space. For two qubits, the basis states expand from the standard $|0\rangle$ and $|1\rangle$ to four combinations: $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$. You will see how the state vector size scales according to $2^n$, where $n$ is the number of qubits.

With multiple qubits, we can introduce multi-qubit gates. The specific focus here is the Controlled-NOT (CNOT) gate. Unlike single-qubit gates which rotate a state vector, the CNOT gate introduces conditional logic, flipping a target qubit only if the control qubit acts as a $|1\rangle$. This interaction is the mechanism used to generate entanglement.

We define entanglement as a scenario where the quantum state of a composite system cannot be factored into independent states for each qubit. When qubits are entangled, the measurement of one correlates with the measurement of another. You will examine this mathematically through the creation of Bell states, which are specific, maximally entangled pairs.

By the end of this chapter, you will be able to:

• Calculate the state vector of composite systems using tensor products.
• Apply CNOT gates to control interactions between qubits.
• Mathematically verify if a state is entangled or separable.
• Write Python code to generate and measure Bell states.