toqito.perms.symmetric_projection

toqito.perms.symmetric_projection(dim: int, p_val: int = 2, partial: bool = False) → [<class 'numpy.ndarray'>, <class 'scipy.sparse._lil.lil_matrix'>]

Produce the projection onto the symmetric subspace [CJKLZ14].

For a complex Euclidean space \(\mathcal{X}\) and a positive integer \(n\), the projection onto the symmetric subspace is given by

\[\frac{1}{n!} \sum_{\pi \in S_n} W_{\pi}\]

where \(W_{\pi}\) is the swap operator and where \(S_n\) is the symmetric group on \(n\) symbols.

Produces the orthogonal projection onto the symmetric subspace of p_val copies of dim-dimensional space. If partial = True, then the symmetric projection (PS) isn’t the orthogonal projection itself, but rather a matrix whose columns form an orthonormal basis for the symmetric subspace (and hence the PS * PS’ is the orthogonal projection onto the symmetric subspace).

This function was adapted from the QETLAB package.

Examples

The \(2\)-dimensional symmetric projection with \(p=1\) is given as \(2\)-by-\(2\) identity matrix

\[\begin{split}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.\end{split}\]

Using toqito, we can see this gives the proper result.

>>> from toqito.perms import symmetric_projection
>>> symmetric_projection(2, 1).todense()
[[1., 0.],
 [0., 1.]]

When \(d = 2\) and \(p = 2\) we have that

\[\begin{split}\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1/2 & 1/2 & 0 \\ 0 & 1/2 & 1/2 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.\end{split}\]

Using toqito we can see this gives the proper result.

>>> from toqito.perms import symmetric_projection
>>> symmetric_projection(dim=2).todense()
[[1. , 0. , 0. , 0. ],
 [0. , 0.5, 0.5, 0. ],
 [0. , 0.5, 0.5, 0. ],
 [0. , 0. , 0. , 1. ]]

References

[CJKLZ14]

J. Chen, Z. Ji, D. Kribs, N. Lütkenhaus, and B. Zeng. “Symmetric extension of two-qubit states”. Physical Review A 90.3 (2014): 032318. https://arxiv.org/abs/1310.3530 E-print: arXiv:1310.3530 [quant-ph]

Parameters:

• dim – The dimension of the local systems.
• p_val – Default value of 2.
• partial – Default value of 0.

Returns:

Projection onto the symmetric subspace.