Optimal quantum tomography with constrained measurements arising from unitary bases

Article Type

Research Article

Publication Title

Reviews in Mathematical Physics


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 d2 of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). Finally, we give a combination of the tensor product and constrained elementary measurement techniques to deal with all d, though with more overlaps or angles depending on the factorization of d as a product of primes or their powers like d=∏ kj=1 dj with d dj=pjsj,p1 < p2 < ⋯ < pk, all primes, sj ≥ 1 for 1 ≤ j ≤ k, or other types. A comparison is drawn for different forms of unitary bases for the Hilbert space factors of the tensor product like L2(ft) or L2(ℤu), where ℱt is the Galois field of size t = ps and ℤu is the ring of integers modulo u. Even though as Hilbert spaces they are isomorphic, but quantum mechanical system-wise, these tensor products are different. In the process, we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081-7094], 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.



Publication Date



Open Access, Green

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