I am wondering if a random unitary matrix taken from a Haar measure (as in, it is uniformly sampled at random) can yield a uniformly sampled random state vector.
In section 3 of this paper it says "It is worthwhile mentioning that, although not advantageous, it is possible to use the rows or columns of such a random unitary matrix as random state vector" and also says in the previous section that " Another manner of obtaining samples with similar properties is by using the rows or columns of random unitary matrices, which we shall discuss in the next section."
I am a bit confused by the wording of this paper. Is it explicitly saying that taking a column or row from a random unitary matrix sampled uniformly will in fact give a random state vector with respect to the Haar measure?