# Hilbert Space Bases for Distinguishing Pure Quantum States in Low Dimensions

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.

H

dim(H)=2

dim(H)=4