Quantum Designs - Definitions
A quantum design is a set D = {P1, ..., Pv} of
orthogonal d x d projection matrices.
- D is said to be regular if there exists an such that
- The degree s of D is given by the cardinality of the set
- A subset of D is an orthogonal class,
if its projections are pairwise orthogonal.
If the sum of all its projections adds up to the identity matrix,
the orthogonal class is said to be complete.
A quantum design is called resolvable if it can be written as a disjoint
union of complete orthogonal classes.
- A regular resolvable quantum design with degree 2
will be called an affine quantum design.
- Let G be an irreducible group of unitary d x d matrices.
Then D is a quantum t-design with respect to the group G
if the following equation holds for all matrices
and all :
For classical=commutative quantum designs (out of pairwise commuting projections) these
definitions correspond to those of combinatorial design theory (the terminology is as in the book
Design Theory, by T.Beth, D.Jungnickel, H.Lenz - external link).
The assignment goes via the 0/1 diagonal entries of the simultaneously diagonalized
projection matrices taken as incidence functions of subsets of a set of
d elements. The appropriate group G is given by the d x d permutation matrices.
Further references on (classical) combinatorial design theorie
(external link).
|
Diese Seite ist nur auf Englisch verfügbar |
|