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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.
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