Hilbert Space

The mathematical space where quantum states live: a complete complex vector space with an inner product.

A Hilbert space is the mathematical arena for quantum mechanics. Every quantum state is a vector in a Hilbert space, and quantum operations are linear transformations on this space.

The Basics

For quantum computing purposes, think of a Hilbert space as:

• A complex vector space (vectors have complex number components)
• Equipped with an inner product $⟨\varphi |\psi ⟩$
• Complete (no “holes” or missing points)

Dimension

The dimension of the Hilbert space determines how many basis states exist:

The exponential growth ($2^n$) is both the power and the challenge of quantum computing.

Key Properties

Inner Product

For any two states $|\varphi ⟩$ and $|\psi ⟩$: $$⟨\varphi |\psi ⟩=\text{complex number}$$

Properties:

• $⟨\psi |\psi ⟩\ge 0$ (and equals zero only if $|\psi ⟩=0$)
• $⟨\varphi |\psi ⟩=⟨\psi |\varphi {⟩}^{\ast }$ (conjugate symmetry)

Orthonormality

Basis states are orthonormal: $$⟨i|j⟩={\delta }_{ij}=\{\begin{array}{ll}1& i=j\\ 0& i\ne j\end{array}$$

Normalization

Physical quantum states have unit norm: $$⟨\psi |\psi ⟩=1$$

Tensor Products

When combining quantum systems, Hilbert spaces combine via tensor products:

$${\mathcal{H}}_{AB}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$$

Dimension multiplies: if ${\mathcal{H}}_A$ has dimension $d_A$ and ${\mathcal{H}}_B$ has dimension $d_B$, then ${\mathcal{H}}_{AB}$ has dimension $d_A\cdot d_B$.

For Working Quantum Scientists

In practice, you usually don’t need to think deeply about Hilbert space theory. Just know:

1. States are vectors: $|\psi ⟩\in \mathcal{H}$
2. Operators are matrices: $U:\mathcal{H}\to \mathcal{H}$
3. Inner products give probabilities: $|⟨\varphi |\psi ⟩{|}^2$
4. Dimension grows exponentially with qubit count

Infinite-Dimensional Hilbert Spaces

Continuous-variable quantum computing and quantum field theory use infinite-dimensional Hilbert spaces. These require more mathematical care but the basic ideas remain similar.

See also: Quantum State, Bra-Ket Notation, Qubit