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 [1] 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 [1] 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 [2] and is also discussed in chapter 5 of [1].

[1] 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.

[2] 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.