I have no idea if this line of thinking is novel, but if I do a simple search for "poetry and information theory", nothing relevant comes up.
An interesting question for me is why people write in poetry, since the structures in poetry usually don't convey any semantic information. Wikipedia claims that poetry uses structural properties of language to "to evoke meanings in addition to, or in place of, the prosaic ostensible meaning". However, I'm not sure if this explanation is very good. If I use a poetic structure like rhyming or alliteration, what impact does it have on the meaning of the text? I think, it has none at all.
Rather, I think that poetry induces a lower-entropy "sub-language", which is a valid subset of the overall language, but which induces structural requirements that limit the information content of the text. For example, if you take these two lines from T.S. Eliot's "Lovesong of J. Alfred Prufrock":
Combing the white hair of the waves blown back
When the wind blows the water white and black
If you remove the final two words "and black" and try to fill in those blanks, you're immediately constrained by two poetic devices. First, the opposition with the word "white" makes the word "black" particularly elegant. There are also many words that begin with a "w" or a "b", so beginning with these supports alliteration. Additionally, the word has to rhyme with "back". Even if it requires some mental energy to think of "and black" as the completion, there's probably no other good way to complete this line.
I think the main purpose of this is to make the text easier to remember. Once you've remembered the poetical structures in a poem and perhaps one or two lines, there probably are only a small number of optimal completions, and coming up with a good novel completion would be rather difficult. This is pretty useful, especially if you don't have a written language or if writing is expensive. I think it also makes it difficult to edit a poem to change its content. Perhaps this was useful in ancient religions?
A few of the structural techniques I'm familiar with in poetry:
-Rhyming: pretty common in western poetry. Creates a slightly long-range dependency, and of course reduces potential word choices at the end of lines by a substantial amount
-Metering: this also limits the types of lines that can be used.
-Alliteration: this can be used throughout a line.
-Shortness: this is an interesting one, which I'm familiar with in Japanese poetry, but not in western poetry. Many Japanese poems are very short, usually 2-3 lines, but have few other constraints.
One thing that I think is interesting is that different types of poetic techniques make some types of editing harder and some easier. For example if I take these lines from Bishop's "The Map":
Shadows, or are they shallows, at its edges
Showing the line of long sea-weeded ledges
There's a rather strong use of alliteration and rhyming (I think there might also be some metering, but I'm not sure). I think it would be rather difficult to make any edits to this poem which preserve its qualities. At the very least, one might need to be a very skilled poet.
On the other hand, consider this poem by Murasaki Shikibu (quoted from memory, so may not be perfect):
I cannot change myself at such a late age
I know that others have
But I cannot
This poem is still easy to remember and perhaps rather constrained because it's so short. However, because it doesn't have many other structural aspects, it lends itself to a lot of possibilities for editing.