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The purpose of this paper is to introduce techniques of obtaining optimalways to determine a d-level quantum state or distinguish such states. Itentails designing constrained elementary measurements extracted from maximalabelian subsets of a unitary basis U for the operator algebra B(H) of a Hilbertspace H of finite dimension d > 3 or, after choosing an orthonormal basis forH, for the *-algebra Md of complex matrices of order d > 3. Illustrations aregiven for the techniques. It is shown that the Schwinger basis U of unitaryoperators can give for d, a product of primes p and a, the ideal number d^2 ofrank one projectors that have a few quantum mechanical overlaps (or, for thatmatter, a few angles between the corresponding unit vectors). We also give acombination of the tensor product and constrained elementary measurementtechniques to deal with all d. A comparison is drawn for different forms ofunitary bases for the Hilbert space and also for different Hilbert spacefactors of the tensor product. In the process we also study the equivalencerelation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34(2001) 7081], connect it to local operations on maximally entangled vectorsbases, find an invariant for equivalence classes in terms of certain commutingsystems, called fan representations, and, relate it to mutually unbiased basesand Hadamard matrices. Illustrations are given in the context of latin squaresand projective representations as well.

Optimal quantum tomography with constrained elementary measurements arising from unitary bases