  The Bell state measurement (BSM) is the projection of two qubits onto four
orthogonal maximally entangled states. Here, we first propose how to
appropriately define more general BSMs, that have more than four possible
outcomes, and then study whether they exist in quantum theory. We observe that
non-projective BSMs can be defined in a systematic way in terms of equiangular
tight frames of maximally entangled states, i.e.~a set of maximally entangled
states, where every pair is equally, and in a sense maximally, distinguishable.
We show that there exists a five-outcome BSM through an explicit construction,
and find that it admits a simple geometric representation. Then, we prove that
there exists no larger BSM on two qubits by showing that no six-outcome BSM is
possible. We also determine the most distinguishable set of six equiangular
maximally entangled states and show that it falls only somewhat short of
forming a valid quantum measurement. Finally, we study the non-projective BSM
in the contexts of both local state discrimination and entanglement-assisted
quantum communication. Our results put forward natural forms of non-projective
joint measurements and provide insight on the geometry of entangled quantum
states.
