states.isotropic¶

Isotropic state is a bipartite quantum state.

These states are separable for α ≤ 1/(d+1), but are otherwise entangled.

Functions¶

isotropic(dim, alpha)

Produce a isotropic state [1].

Module Contents¶

states.isotropic.isotropic(dim, alpha)¶

Produce a isotropic state [1].

Returns the isotropic state with parameter alpha acting on (dim-by-dim)-dimensional space. The isotropic state has the following form

\[\begin{equation} \rho_{\alpha} = \frac{1 - \alpha}{d^2} \mathbb{I} \otimes \mathbb{I} + \alpha |\psi_+ \rangle \langle \psi_+ | \in \mathbb{C}^d \otimes \mathbb{C}^2 \end{equation}\]

where \(|\psi_+ \rangle = \frac{1}{\sqrt{d}} \sum_j |j \rangle \otimes |j \rangle\) is the maximally entangled state.

Examples

To generate the isotropic state with parameter \(\alpha=1/2\), we can make the following call to toqito as

>>> from toqito.states import isotropic
>>> isotropic(3, 1 / 2)
array([[0.22222222, 0.        , 0.        , 0.        , 0.16666667,
        0.        , 0.        , 0.        , 0.16666667],
       [0.        , 0.05555556, 0.        , 0.        , 0.        ,
        0.        , 0.        , 0.        , 0.        ],
       [0.        , 0.        , 0.05555556, 0.        , 0.        ,
        0.        , 0.        , 0.        , 0.        ],
       [0.        , 0.        , 0.        , 0.05555556, 0.        ,
        0.        , 0.        , 0.        , 0.        ],
       [0.16666667, 0.        , 0.        , 0.        , 0.22222222,
        0.        , 0.        , 0.        , 0.16666667],
       [0.        , 0.        , 0.        , 0.        , 0.        ,
        0.05555556, 0.        , 0.        , 0.        ],
       [0.        , 0.        , 0.        , 0.        , 0.        ,
        0.        , 0.05555556, 0.        , 0.        ],
       [0.        , 0.        , 0.        , 0.        , 0.        ,
        0.        , 0.        , 0.05555556, 0.        ],
       [0.16666667, 0.        , 0.        , 0.        , 0.16666667,
        0.        , 0.        , 0.        , 0.22222222]])

References

[1] (1,2)

Michal Horodecki and Pawel Horodecki. Reduction criterion of separability and limits for a class of protocols of entanglement distillation. 1998. arXiv:quant-ph/9708015.

Parameters:

dim (int) – The local dimension.
alpha (float) – The parameter of the isotropic state.

Returns:

Isotropic state of dimension dim.

Return type:

numpy.ndarray