5. Linear Algebra: Hilbert Spaces

Hilbert Spaces

Excellent. You've built all the necessary components: states, operators, measurement. Now we need the container for all of it.

A Hilbert Space ($\mathcal{H}$) is simply the formal, mathematical environment where all the rules of quantum computing live. It is a specific type of vector space with three core properties that make it suitable for quantum mechanics:

1. It is a Vector Space: 🌌 This means it contains all possible linear combinations (superpositions) of the basis states. If $|0\rangle$ and $|1\rangle$ are in the space, then the vector $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ must also be in the space.
2. It has an Inner Product: $\langle \cdot | \cdot \rangle$ This allows us to calculate the length of vectors (normalization, ensuring $|\alpha|^2 + |\beta|^2 = 1$) and the angle between them (orthogonality, ensuring we can distinguish basis states).
3. It is Complete: This is a technical property that ensures every sequence of vectors that "should" converge, does converge to a valid vector within the space. Physically, this means every possible measurement outcome corresponds to a valid quantum state.

The Single-Qubit Space

For a single qubit, the Hilbert space is 2-dimensional ($\mathcal{H} = \mathbb{C}^2$). The Bloch sphere is the visual representation of all the possible vectors in this 2-dimensional complex space.

📈 Scaling Up

The true power of the Hilbert space concept is in describing systems with multiple qubits. To describe $N$ independent qubits, we use a mathematical operation called the tensor product (which we'll cover next) to combine their individual spaces.

For an $N$-qubit system, the dimension of the Hilbert space is:

Your Task:

Based on this scaling rule, what is the dimension of the Hilbert space needed to fully describe the state of a system composed of two qubits?