With the goal in mind of designing radars, interferometers and other sensors based on quantum entanglement the virtues
of N00N states, plain M and M states (PMMSs) and linear combinations of M and M states (LCMMS) are considered.
A derivation of the closed form expression for the detection operator that is optimal subject to constraints is provided.
The raising and lowering properties of the detection operator and its square are developed. The expectations of the
optimal detection operator and its square are derived. The expression for the visibility, the maximum expectation of the
optimal detection operator, is developed. From the expectation of the square of the detection operator and the visibility,
the phase error and the minimum phase error for the detection operator are derived. The optimal resolution for the
maximum visibility and minimum phase error are found. For the visibility, comparisons between PMMSs, LCMMS and
N00N states are provided. For the minimum phase error comparisons between LCMMS, PMMSs, N00N states, separate
photon states (SPSs), the shot noise limit (SNL), and the Heisenberg limit (HL) are provided. A representative
collection of computational results illustrating the superiority of LCMMS when compared to PMMSs and N00N states is
given. It is found for a resolution 12 times the classical result LCMMS has visibility 11 times that of N00N states and
four times that of PMMSs. For the same case, the minimum phase error for LCMMS is 10.7 times smaller than that of
PMMS and 29.7 times smaller than that of N00N states.