Hilbert Space Bases for Distinguishing Pure Quantum States in Low Dimensions
In a Hilbert space of arbitrary dimension, at least four orthonormal vector bases are needed to distinguish all pure quantum states. This Demonstration constructs the four bases using orthogonal polynomials (except for the case , in which only 3 are needed, and for , for which the exact lower bound is an open problem). Since the one-dimensional case is trivial and five or more dimensions are hard to compute, the dimensions considered in this Demonstration are 2, 3 and 4. You can also adjust the complex phase parameter α used in the construction, choose an analytic or numeric calculation, choose a family of orthogonal polynomials with which the bases are constructed and transform the bases with a unitary matrix to make one of the bases canonical. Chebyshev I and Chebyshev II denote Chebyshev polynomials of the first and the second kind.
Let be a -dimensional Hilbert space with orthonormal bases . Since the bases are of the form where are the corresponding basis vectors, let denote the projection on the vector of the basis . Using this notation, the bases cannot distinguish pure states; and if and only if . It is shown in chapter 5 of  that for a system of four special bases, this condition implies —that is, the states and are essentially the same state. Thus, four bases can distinguish all pure states in any finite-dimensional Hilbert space. It is also proven in  that if , only three orthonormal bases are needed.
The four bases are constructed in the following way:
Let denote an orthogonal polynomial of order . For basis 1, the basis vectors are where denotes the roots of . Basis 2 is similar, except the are now the roots of and the last basis vector is . Bases 3 and 4 are obtained from 1 and 2 simply by transforming each into , where is not a rational multiple of . All four bases can then be easily normalized. It is also important to note here that all of the bases are constructed using the same family of orthogonal polynomials. Curiously, when , bases 2 and 4 become identical. This method of construction is based on a paper by Jaming  and is also discussed in chapter 5 of .
 C. Carmeli, T. Heinosaari, J. Schultz and A. Toigo, "How Many Orthonormal Bases are Needed to Distinguish All Pure Quantum States?," The European Physical Journal D, 69(179), 2015. doi:10.1140/epjd/e2015-60230-5.
 P. Jaming, "Uniqueness Results in an Extension of Pauli's Phase Retrieval Problem," Applied and Computational Harmonic Analysis, 37(3), 2014 pp. 413–441. doi:10.1016/j.acha.2014.01.003.