  A few simply-stated rules govern the entanglement patterns that can occur in
mutually unbiased basis sets (MUBs), and constrain the combinations of such
patterns that can coexist (ie, the stoichiometry) in full complements of p^N+1
MUBs. We consider Hilbert spaces of prime power dimension (as realized by
systems of N prime-state particles, or qupits), where full complements are
known to exist, and we assume only that MUBs are eigenbases of generalized
Pauli operators, without using a particular construction. The general rules
include the following: 1) In any MUB, a particular qupit appears either in a
pure state, or totally entangled, and 2) in any full MUB complement, each qupit
is pure in p+1 bases (not necessarily the same ones), and totally entangled in
the remaining p^N-p. It follows that the maximum number of product bases is
p+1, and when this number is realized, all remaining p^N-p bases in the
complement are characterized by the total entanglement of every qupit. This
"standard distribution" is inescapable for two qupits (of any p), where only
product and generalized Bell bases are admissible MUB types. This and the
following results generalize previous results for qubits and qutrits. With
three qupits there are three MUB types, and a number of combinations (p+2) are
possible in full complements. With N=4, there are 6 MUB types for p=2, but new
MUB types become possible with larger p, and these are essential to the
realization of full complements. With this example, we argue that new MUB
types, showing new entanglement characteristics, should enter with every step
in N, and when N is a prime plus 1, also at critical p values, p=N-1. Such MUBs
should play critical roles in filling complements.

Quantum Physics