I agree with Geomancer but only up to the last sentence.
Nitrate is not the only issue, as I'll try to explain.
In non-planted tanks, nitrates are a significant indicator of water quality, but only with respect to the nitrification cycle. In natural (low-tech) planted tanks the nitrates should be very low because of the plants' rapid assimilation of ammonia/ammonium primarily, and perhaps nitrite and nitrate. But with the plants grabbing much of the ammonia/ammonium, little goes through the nitrification oxidation to end up as nitrates. And there are bacteria that take up the nitrates. This occurs in non-planted tanks too, but here the nitrifying bacteria are on their own to oxidize ammonia to nitrite to nitrate. So, all else being equal, nitrate would be somewhat higher in non-planted tanks than in planted.
But the "crud" that you cannot see or measure in any manner, is the real issue. Pheromones released by fish that other fish can "read," dissolved waste and urine, and I don't know what all else--this is the stuff that filters can't remove and for which we use plants and do water changes. In planted tanks, the more plants there are, the more of this "crud" they can take up and remove. Which is why we can get away with fewer water changes, or less volume, with live plants. And again, this crud cannot be measured.
So that brings us back to DKRST's initial question. And I agree as I said that faster growing plants will remove this quicker. But of course, there is a limit to the capacity of plants in a closed system. Some time back I read that a 55g heavily-planted tank that was stocked with 6-7 Neon Tetra
would be self-sufficient with respect to this issue, and never require a water change. The plants would handle the "crud" with this minimal a fish load. And no filter either, just the plants doing the job naturally. Diana Walstad, who advocates no water changes, says this works with well planted tanks with what she calls "a small or moderate number of fish." I have considerably more fish than what anyone could call small or moderate stocking, hence I do weekly 50% water changes. But I have no actual data as to how far I could let this go by simply relying on the plants.