# 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'>][source]

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. Projection onto the symmetric subspace.