Hilbert Space Bases for Distinguishing Pure Quantum States in Low Dimensions

dimension

2

3

4

α

0.1

polynomial

Chebyshev I

Chebyshev II

Hermite

Legendre

numeric

transform

base

array

real part

imaginary

1

2 3	- 1 3
2 3	1 3

0	0
0	0

2

1	0
0	1

0	0
0	0

3

0.816497	-0.574466
0.816497	0.574466

0.	-0.0576388
0.	0.0576388

In a Hilbert space

H

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

dim(H)=2

, in which only 3 are needed, and for

dim(H)=4

, 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.