Simplest explanation is just a couple of examples:

-- append a column with 6 elements on: sa: append-column[6] |X> |X: 0> + |X: 1> + |X: 2> + |X: 3> + |X: 4> + |X: 5> -- append a column with only 1 element on: sa: random-column[6] |X> |X: 3> -- append a column to all kets, with 3 elements on: sa: append-column[3] (|X> + |Y> + |Z>) |X: 0> + |X: 1> + |X: 2> + |Y: 0> + |Y: 1> + |Y: 2> + |Z: 0> + |Z: 1> + |Z: 2> -- append a column to all kets, with only 1 element on: sa: random-column[3] (|X> + |Y> + |Z>) |X: 2> + |Y: 0> + |Z: 2>Next, observe that extract-category is essentially the inverse of these two operators:

sa: extract-category append-column[20] (|X> + |Y> + |Z>) 20|X> + 20|Y> + 20|Z> sa: extract-category random-column[20] (|X> + |Y> + |Z>) |X> + |Y> + |Z>And I guess that is about it!

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