Optimal quantum tomography with constrained elementary measurements arising from unitary bases

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