Hilbert Space

A Hilbert space is a vector space equipped with an inner product that makes it complete: every Cauchy sequence of vectors converges to a vector in the space. In quantum mechanics, every quantum system is associated with a Hilbert space, and every physically realizable state of the system corresponds to a unit vector (or ray) in that space.

Structure and Inner Product

The inner product on a quantum Hilbert space is written in bra-ket notation as <phi|psi> and satisfies three axioms: conjugate symmetry (<phi|psi> = <psi|phi>*), linearity in the second argument, and positive definiteness (<psi|psi> >= 0 with equality only for the zero vector). These axioms ensure that probabilities are well-defined and always non-negative.

For a qubit, the Hilbert space is the two-dimensional complex vector space C^2. An n-qubit system lives in the tensor product of n copies of C^2, giving a 2^n-dimensional Hilbert space. This exponential growth of the Hilbert space dimension with qubit number is one reason quantum simulation of large systems is classically hard, and also one reason quantum computers could have an advantage.

Operators on Hilbert Space

Physical quantities (observables) are represented by linear operators on the Hilbert space. An observable must be Hermitian (equal to its own conjugate transpose) so that its eigenvalues (the possible measurement outcomes) are real numbers. Quantum gates are represented by unitary operators, which preserve the inner product structure and thus preserve probability.

The spectral theorem guarantees that every Hermitian operator on a finite-dimensional Hilbert space can be diagonalized in an orthonormal basis of eigenvectors. The corresponding eigenvalues are the measurement outcomes, and the eigenvectors are the states with definite values of that observable.

Infinite-Dimensional Hilbert Spaces

In continuous-variable quantum mechanics and quantum field theory, Hilbert spaces are infinite-dimensional. The Hilbert space of a single harmonic oscillator, for example, is spanned by an infinite ladder of number states |0>, |1>, |2>, … Bosonic quantum computing and photonic quantum computing work in such infinite-dimensional spaces, though practical computations truncate them to finite subspaces.

Tensor Products and Composite Systems

When two quantum systems with Hilbert spaces H_A and H_B are combined, the composite system lives in the tensor product H_A x H_B. This tensor product structure is what allows entanglement to exist: states in H_A x H_B that cannot be written as products of individual states in H_A and H_B are entangled. The dimension of the tensor product space is dim(H_A) times dim(H_B), which is why the description of multi-qubit systems grows exponentially.

Relevance to Quantum Computing

Every concept in quantum computing has a precise interpretation in terms of Hilbert space: quantum states are unit vectors, gates are unitary operators, measurements are projections onto orthogonal subspaces, and entanglement is a property of vectors in tensor product spaces that cannot be factored. Understanding Hilbert space is the foundation for reading quantum computing research literature and understanding why quantum systems behave as they do.