Definition
Let G be a finite group of square order n^2.
A nice error basis is a set of unitary nxn matrices D(g) parametrized by the index group G such that
- D(1) is the identity matrix,
- trace D(g) = 0 for all nonidentity elements g of G,
- D(g)D(h) = w(g,h) D(gh) where w(g,h) is a phase factor.
We only list the matrices corresponding to generators of the group G.
The nice error bases for this index group G can be constructed as follows.
- Construct the matrix group H generated by the matrices D(e), where the e's are generators of G.
- Take a transversal with respect to the center Z(H) of H
Then all unitary error bases with index group G are projectively equivalent
to one of the bases constructed by the above method.
Recall that two nice error bases D(g) and E(g) are said to be
projectively equivalent if and only if there exists
an invertible matrix T and a complex-valued function b such that
- D(g)=b(g) T^-1 E(g)T holds for all g in G.
Notation: E(d) denotes a primitive d-th root of unity.
The following example illustrates the main points of this definition.
An entry from the Catalogue of Nice Error Bases, a joint project of
Andreas Klappenecker and
Martin Rötteler.
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