toqito.state_ops.schmidt_decomposition

toqito.state_ops.schmidt_decomposition(rho, dim=None, k_param=0)

Compute the Schmidt decomposition of a bipartite vector [WikSD].

Examples

Consider the \(3\)-dimensional maximally entangled state .. math:

u = \frac{1}{\sqrt{3}} \left( |000 \rangle + |111 \rangle + |222 \rangle \right)

We can generate this state using the toqito module as follows. >>> from toqito.states import max_entangled >>> max_entangled(3) [[0.57735027],

[0. ], [0. ], [0. ], [0.57735027], [0. ], [0. ], [0. ], [0.57735027]]

Computing the Schmidt decomposition of \(u\), we can obtain the corresponding singular values of \(u\) as .. math:

\frac{1}{\sqrt{3}} \left[1, 1, 1 \right]^{\text{T}}.

>>> from toqito.states import max_entangled
>>> from toqito.state_ops import schmidt_decomposition
>>> singular_vals, u_mat, vt_mat = schmidt_decomposition(max_entangled(3))
>>> singular_vals
[[0.57735027]
 [0.57735027]
 [0.57735027]]
>>> u_mat
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
>>> vt_mat
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]

References

[WikSD]

Wikipedia: Schmidt decomposition https://en.wikipedia.org/wiki/Schmidt_decomposition

Raises:

ValueError – If matrices are not of equal dimension.

Parameters:

• rho – A bipartite quantum state to compute the Schmidt decomposition of.

• dim – An array consisting of the dimensions of the subsystems (default gives subsystems equal dimensions).

• k_param – How many terms of the Schmidt decomposition should be computed (default is 0).

Returns:

The Schmidt decomposition of the rho input.