Hilbert Space

A Hilbert space is the complex inner-product vector space where quantum states live; qubits use finite-dimensional versions.

The setting for quantum states

Quantum states are vectors in a Hilbert space: a complex vector space equipped with an inner product and complete with respect to it. For quantum computing the relevant Hilbert spaces are finite-dimensional — a single qubit uses C^2, and n qubits use C^{2^n}.

Ingredients

• Vectors over the complex numbers (states, written as kets)
• An inner product giving overlaps and norms
• Orthonormal bases, e.g. {|0>, |1>} for one qubit
• Completeness: limits of vector sequences stay in the space

The inner product does the work

The inner product is a complex number. Its magnitude squared gives transition probabilities via the Born rule. Two states are orthogonal — perfectly distinguishable — when their inner product is zero. Normalisation, = 1, ensures probabilities sum to one. Nearly all of quantum theory's predictions are inner products of chosen vectors and operators.

Composite systems

When systems combine, their Hilbert spaces combine by the tensor product, so dimensions multiply. Two qubits live in C^2 tensor C^2 = C^4. This multiplicative growth, not additive, is why the state space of a modest register is astronomically large.

Operators

Physical quantities are Hermitian operators on the space (observables); time evolution is by unitary operators. Both are linear maps of the Hilbert space to itself. Infinite-dimensional Hilbert spaces describe continuous systems like a particle's position, but qubit-based computing stays finite-dimensional, which keeps the mathematics to linear algebra over complex matrices.