I know only slightly more quantum mechanics than you do and nothing about string theory (except that it involves 10 or 11 dimensions instead of 3 or 4). However, your suggestion is very interesting, here is a slightly more mathematical way of looking at it:

- Non-entangled (vectors) states can be factored into tensor products of lesser dimensional (vectors) states, whereas entangled states cannot.

I'll try to explain: take a pair of electrons each with two possible basis states, say spin up and spin down, e-a(x,y) and e-b(u,v). Each electron is thus in a 2-dimensiona state. The complete pair is thus in a 4-dimension state e2(x,y,u,v). If the electrons are not-entangeled, than e2 can be written e2 = alpha*e-a + beta*e-b, that is a linear combination of the 2 2-dimension states. however, if the pair is entangled, than the 4-dimension e2 cannot be described by any combination (whatsoever!) of 2-dimension states. It is a "purely" [

] 4-dimensional state, wich sounds similar to your idea of entagled being really higher dimensional single objects.