What is the procedure to generate a random qutrit quantum state?

That is, I want to generate a $3 \times 1$ vector $u$ such that $u$ is generated by a Haar random $3 \times 3$ unitary $U$ multiplied with the $3 \times 1$ column vector $$ \begin{pmatrix} 1 \\ 0 \\ 0\\ \end{pmatrix}. $$


1 Answer 1


There are two ways I can see (so made this an answer!):

  1. Generate a random unitary matrix and take the first column (or row)

For this, you're looking for CircularUnitaryMatrixDistribution[3] for the Haar distribution on unitary matrices. To get an actual matrix from this distribution, use


Check out the Background & Context section of the documentation:

Probabilistically, the circular unitary matrix distribution represents a uniform distribution over the unitary square matrices, while mathematically it is a so-called Haar measure on the unitary group $U(n)$.

Note that for efficiency, you can simply take the First element instead of dotting with {1,0,0}.

  1. Generate a random point on the 6D sphere

The unit sphere in $\mathbb{C}^3$ under the standard (physical) inner product can be viewed as the unit sphere in $\mathbb{R}^6$.

Then we have

Complex @@@ Partition[RandomPoint[Sphere[6]], 2]

as a random (representative of a) qutrit directly.

(One potential issue is that quantum mechanical states are not actually in correspondence with points on the unit sphere in $\mathbb{C}^3$, but with orbits of such points under the action of multiplication by phase. However, since each orbit has the same induced measure, choosing a random point on the sphere is equivalent to choosing a random orbit.)


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