Good post, thanks.

For people who are puzzled by the reference to "monads", think about Schrödinger's cat. Schrödinger's cat has two classical states, "alive" and "dead". The "monad" of quantum states is basically the vector space generated by this basis. From outside the vector space you can't know whether the cat is alive or dead, and in fact the cat is in a superposition of these two states.

This vector space is a monad because:

• you can "lift" elements of the basis into the space (put a live cat or a dead cat into a box)
• you can "map" functions on the basis to functions on the entire vector space (put a device which kills the cat with certainty into the box)
• and you can "bind" a function which takes classical states to quantum states to the entire vector space (i.e. make a device which randomly chooses whether to kill the cat based on a quantum event and put it into the box).

This is not a complete description; for instance, one needs to introduce a tensor product to account for the fact that independent (disentangled) monads can be effectively separated out. On the other hand this monadic description applies to all vector spaces; it's not specific to quantum mechanics.