spheres.symmetrization.symmetrized_basis

spheres.symmetrization.symmetrized_basis(n, d=2)

Constructs a symmetrized basis set for n systems in d dimensions.

Parameters

• n (int) – The number of systems to symmetrize.

• d (int or list) – Either an integer representing the dimensionality of the individual subsystems, in which case, we work in the computational basis; or else a list of basis states for the individual systems.

Returns

sym_basis – sym_basis["labels"] is a list of labels for the symmetrized basis states. Each element of the list is a tuple whose length is the dimensionality of the individual subsystems, with an integer counting the number of subsystems in that basis state.

sym_basis["basis"] is a dictionary mapping labels to symmetrized basis states.

sym_basis["map"] is a linear transformation from the permutation symmetric subspace to the full tensor product of the n systems.

The dimensionality of the symmetric subspace corresponds to the number of ways of distributing \(n\) elements in \(d\) boxes, where \(n\) is the number of systems and \(d\) is the dimensionality of an individual subsytems. In other words, the dimensionality \(s\) of the permutation symmetric subspace is \(\binom{d+n-1}{n}\).

So sym_basis["map"] is a map from \(\mathbb{C}^{s} \rightarrow \mathbb{C}^{d^{n}}\).

Return type

dict