What is Hilbert Space?
In quantum mechanics, Hilbert spaces provide the mathematical framework used to describe quantum systems. Quantum states are represented as elements of a Hilbert space, while the dimension of the Hilbert space relates to the number of possible configurations of the system. For instance, an \(N \) -qubit system will live on a \( 2^N \) -dimensional Hilbert space, while continuous-variable quantum computing (using for instance quantum harmonic oscillators) will operate on an infinite-dimensional Hilbert space.
Mathematical definition:
Formally, a Hilbert space is a vector space (typically defined over the complex numbers), equipped with an inner product \( \langle\cdot\mid\cdot\rangle \) . This implies that quantum states are represented as vectors.
The inner product is used to compute the overlap between states. The physical interpretation of the inner product in quantum mechanics relates to the measurement probabilities. If a system is in the state \( \ket{\psi} \) , the probability of measuring the state \( \ket{\phi} \) is:
$$P\left[\phi \mid \psi \right] = \left|\langle \phi \mid \psi \rangle\right|^2$$
The inner product also defines the norm (length) of a vector. Valid quantum states are vectors that are normalized, i.e. have norm 1. This condition ensures that if a quantum system prepared in a state \( \ket {\psi} \), a measurement designed to detect this state will find it with certainty.
In addition to the existence of the inner product and its induced metric, Hilbert spaces are also required to be complete. Formally, this means that sequences where vectors get arbitrarily close as the sequence unfolds (i.e. Cauchy sequences) always converge to a limit. This ensures that arbitrary superpositions of quantum states, including infinite superpositions, exist as elements of the same Hilbert space.
Examples:
Qubit space
The elementary Hilbert space used in quantum information is the qubit space, which is a complex Hilbert space of dimension 2. Each qubit state is then defined as a 2-dimensional complex vector:
$$ \begin{pmatrix} \alpha\\ \beta\end{pmatrix} = \alpha \ket{0} + \beta\ket{1}$$
The overlap of two states \( \alpha_1\ket{0} + \beta_1\ket{1}$ and $\alpha_2\ket{0} + \beta_2\ket{1} \) is defined as:
$$ \begin{pmatrix} \alpha_1^* & \beta_1^*\end{pmatrix} \begin{pmatrix} \alpha_2\\ \beta_2\end{pmatrix} = \alpha_1^* \alpha_2 + \beta_1^* \beta_2$$
Similarly, the norm of a vector is defined as:
$$ \left|\left|\begin{pmatrix} \alpha \\ \beta\end{pmatrix}\right|\right| ^2 = \begin{pmatrix} \alpha^* & \beta^*\end{pmatrix} \begin{pmatrix} \alpha\\ \beta\end{pmatrix} = \left|\alpha\right|^2 + \left|\beta\right|^2 $$
Then a vector is a valid qubit state if \( \left|\alpha\right|^2 + \left|\beta\right|^2 = 1 \) (i.e. is normalized)
Finally, since the qubit space is finite-dimensional, it is automatically complete.
Fock spaces
In photonics, another important Hilbert space is the bosonic Fock space. The intuition is to keep track of how many particles, in our case photons, are in each mode (e.g. optical fiber). Now, a photonic system can contain an arbitrary number of photons. Therefore, the associated Hilbert space, the Fock space, combines the Hilbert spaces for different particle numbers. Mathematically, given a base Hilbert space \( \mathcal{H} \), the associated Fock space is given by:
$$\mathcal{F}(\mathcal{H}) = \bigoplus_{n=0}^\infty S\left(\mathcal{H}^{\otimes n}\right)$$
Where \(S\) symmetrizes the tensors in \(\mathcal{H}^{\otimes n} \). Indeed, since the particles are indistinguishable, exchanging two particles will result in the same state. More specifically, the symmetrization of the vector \(v_1\otimes v_2 \otimes \ldots \otimes v_n \) is:
$$S\left(v_1\otimes v_2 \otimes \ldots \otimes v_n \right) = \frac{1}{\sqrt{n!}}\sum_{\sigma} v_{\sigma_1} \otimes v_{\sigma_2} \otimes \ldots \otimes v_{\sigma_n}$$
Where \( \sigma \) runs over all possible permutations.
Fock spaces are examples of infinite-dimensional Hilbert spaces. A convenient choice of basis is given by the Fock states, namely states of the form \( \ket{n_1, n_2, \ldots, n_m} \), which represent the state containing \( n_1 \) particles in the first mode, \(n_2\) particles in the second mode, etc. Arbitrary states in the Fock space are then superpositions of Fock states. In particular, coherent states are defined on a single mode as:
$$ \ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}\ket{n} $$
And they are allowed by the completeness of the Fock space, and describe the classical light emitted by a laser.