Discrete and Continuous Hilbert Spaces

Tue 06 March 2018
Physics
#quantum

The state of a qubit is a member of a discrete Hilbert space of finite dimension. Namely, two-dimensional:

\begin{align*} |\psi \rangle &= \psi_{0} |0 \rangle + \psi_{1} |1 \rangle, & |\psi_{0}|^{2} + |\psi_{1}|^{2} &= 1 \end{align*}

In a similar way, you can describe the state of a quantum trit as a member of a three-dimensional discrete Hilbert space:

\begin{align*} |\psi \rangle &= \psi_{0} |0 \rangle + \psi_{1} |1 \rangle + \psi_{2} |2 \rangle, & |\psi_{0}|^{2} + |\psi_{1}|^{2} + |\psi_{2}|^{2} & = 1 \end{align*}

The qubit is the quantum version of a member of \(\mathbb{Z}_{2}\), the integers modulo 2. Similarly, the qutrit is the quantum version of a member of \(\mathbb{Z}_{3}\), the integers modulo 3. In a similar way, you can describe the quantum version of a member of \(\mathbb{Z}_{N}\) as a vector in an \(N\)-dimensional discrete Hilbert space:

\begin{align*} |\psi \rangle &= \sum_{n = 0}^{N-1} \psi_{n} | n \rangle, & \sum_{n = 0}^{N-1} |\psi_{n}|^{2} &= 1, & \langle m | n \rangle &= \delta_{mn} \end{align*}

You can also have discrete but infinite-dimensional Hilbert spaces. For example, the quantum version of an integer:

\begin{align*} |\psi \rangle &= \sum_{n = -\infty}^{\infty} \psi_{n} |n \rangle, & \sum_{n = -\infty}^{\infty} |\psi_{n}|^{2} &= 1, & \langle m | n \rangle &= \delta_{mn} \end{align*}

In a similar way you can describe a quantum rational number!

Besides discrete Hilbert spaces, you can also have continuous Hilbert spaces. The (discrete) finite sum is replaced by the integral, and the Kronecker delta is replaced by the Dirac delta. With these you could describe a quantum real number in the circle:

\begin{align*} |\psi \rangle &= \int\limits_{0}^{2\pi} \mathrm{d}\theta \psi(\theta) | \theta \rangle, & \int\limits_{0}^{2\pi} \mathrm{d}\theta |\psi(\theta)|^{2} &= 1, & \langle \theta | \phi \rangle &= \delta(\theta - \phi) \end{align*}

That is, an element of a continuous but compact one-dimensional space. Maybe you have to impose the requirement of periodicity in \(\theta\):

\begin{equation*} |\theta \pm 2\pi \rangle = |\theta \rangle \end{equation*}

This would lead to the periodicity of the coefficient.

And finally, you can have a quantum real number along the real line:

\begin{align*} |\psi \rangle &= \int\limits_{-\infty}^{\infty} \mathrm{d}x \psi(x) | x \rangle, & \int\limits_{-\infty}^{\infty} \mathrm{d}x |\psi(x)|^{2} &= 1, & \langle x | y \rangle &= \delta(x - y) \end{align*}

In contrast with the circle, the real line is non-compact.

The position basis in quantum mechanics is an example of such a continuous basis. Adding more physics brings in the momentum basis, which is the Heisenberg conjugate to position. Both position and momentum are observables. However, it is rare to have bare kets \(|x \rangle\) as the state of a system. In scattering is more common to have momentum basis states \(|p \rangle\) as the in and out states of a scattering process. I guess I never thought of a vector like

\begin{equation*} |p_{1} \rangle + |p_{2} \rangle \end{equation*}

as being a quantum superposition of two real numbers.

I am not sure where I am going with this post. It was meant to point out a way to think about other bases in quantum physics in the same way that the finite 2-dimensional logic basis is thought of in quantum information.