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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?

can you elaborate more on: "Thus, a column of $V$, let's call is $|\chi\rangle$, has again the property that $|\chi\rangle$ and $U|\chi\rangle$ for any $U$ are "equally likely", that is, $|\chi\rangle$ is distributed Haar random." Why is $|\chi\rangle$ and $U|\chi\rangle$ for any $U$ "equally likely"? – Quantum Guy 123 – 2021-01-19T16:16:50.310

given a column vector, does that map to a unique unitary? not sure how to show that. – Quantum Guy 123 – 2021-01-19T16:17:53.223

@QuantumGuy123 No, obviously not, just by parameter counting. – Norbert Schuch – 2021-01-19T17:36:01.923

@QuantumGuy123 With regard to the first question: If the probability of picking a unitary U or V*U are the same, then also the probability of picking their respective first columns, which are chi (for U, that's how I define chi) and U*chi, are the same. – Norbert Schuch – 2021-01-19T17:41:50.457

ok thanks. also can you provide a reference/source for the two claims you stated:

and 2) On the other hand, a Haar random unitary $V$ is defined the same way: "$UV$ is just as likely as $V$, for any $U$." – Quantum Guy 123 – 2021-01-19T20:32:24.280

the only reason I ask is because i find the definitions of Haar measures I've seen before are quite difficult to understand. your definitions are easy to understand though – Quantum Guy 123 – 2021-01-19T20:34:29.327

an explanation of those claims would be helpful as well. thanks a bunch for the answers. – Quantum Guy 123 – 2021-01-19T20:47:47.187

What definitions have you seen? – Norbert Schuch – 2021-01-19T20:53:41.120