# Parametrization of a two-qubit state

A single-qubit state can be parametrized with real $$\theta$$ and $$\phi$$ as follows: $$|\psi(\theta, \phi)\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i \phi} \sin\frac{\theta}{2}|1\rangle.$$

I would like to know how to parameterize an arbitrary two-qubit state in a similar way.

• Feb 24, 2023 at 6:02

TL;DR: The logic behind the equation $$|\psi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle$$ can be iterated to obtain a real parameterization for states of arbitrary finite dimension.

## Iterative procedure

We parameterize an expansion of $$|\psi\rangle$$ in basis $$|0\rangle,\dots,|N\rangle$$ in $$N$$ iterations. The $$k$$th iteration for $$k=1,\dots,N$$ consists of two steps. First, we introduce $$\theta_k\in[0,\pi]$$ to apportion the unit norm between the magnitudes of the amplitudes of $$|k-1\rangle$$ and the projection $$|\psi_{k\dots N}\rangle$$ of $$|\psi\rangle$$ onto the subspace spanned by $$|k\rangle,\dots,|N\rangle$$. Next, we stick a relative phase $$\alpha_k\in[0,2\pi)$$ on $$|k\rangle$$. Thus, the $$k$$th iteration rewrites $$|\psi_{k-1\dots N}\rangle$$ as $$|\psi_{k-1\dots N}\rangle=\cos\frac{\theta_k}{2}|k-1\rangle + e^{i\alpha_k}\sin\frac{\theta_k}{2}|\psi_{k\dots N}\rangle.\tag1$$ The final result can be simplified by setting $$\phi_k:=\alpha_1+\dots+\alpha_k\mod 2\pi$$.

Each step introduces a single real parameter, for a total of $$2N$$ real parameters, as expected for an $$N+1$$ dimensional system.

## Two-qubit case

For a two-qubit state the above procedure yields \begin{align} |\psi\rangle&=\cos\frac{\theta_1}{2}|00\rangle\\ &+ e^{i\phi_1}\sin\frac{\theta_1}{2}\cos\frac{\theta_2}{2}|01\rangle\\ &+ e^{i\phi_2}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}|10\rangle\\ &+ e^{i\phi_3}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\sin\frac{\theta_3}{2}|11\rangle \end{align}\tag2 where $$\theta_1,\theta_2,\theta_3\in[0,\pi]$$ and $$\phi_1,\phi_2,\phi_3\in[0,2\pi)$$.

This isn't the only way to parameterize a two-qubit state. An alternative approach uses Schmidt decomposition and single-qubit parameterization.