states.werner

Werner states.

Werner states are mixtures of projectors onto the symmetric and permutation operator that exchanges the two subsystems.

Functions

werner(dim, alpha)

Produce a Werner state [1].

Module Contents

states.werner.werner(dim, alpha)

Produce a Werner state [1].

A Werner state is a state of the following form

\[\begin{equation} \rho_{\alpha} = \frac{1}{d^2 - d\alpha} \left(\mathbb{I} \otimes \mathbb{I} - \alpha S \right) \in \mathbb{C}^d \otimes \mathbb{C}^d. \end{equation}\]

Yields a Werner state with parameter alpha acting on (dim * dim)- dimensional space. More specifically, \(\rho\) is the density operator defined by \((\mathbb{I} - `alpha\) S)` (normalized to have trace 1), where \(\mathbb{I}\) is the density operator and \(S\) is the operator that swaps two copies of dim-dimensional space (see swap and swap_operator for example).

If alpha is a vector with \(p!-1\) entries, for some integer \(p > 1\), then a multipartite Werner state is returned. This multipartite Werner state is the normalization of I - alpha(1)*P(2) - … - alpha(p!-1)*P(p!), where P(i) is the operator that permutes p subsystems according to the i-th permutation when they are written in lexicographical order (for example, the lexicographical ordering when p = 3 is: [1, 2, 3], [1, 3, 2], [2, 1,3], [2, 3, 1], [3, 1, 2], [3, 2, 1], so P(4) in this case equals permutation_operator(dim, [2, 3, 1]).

Examples

Computing the qutrit Werner state with \(\alpha = 1/2\) can be done in toqito as

>>> from toqito.states import werner
>>> werner(3, 1 / 2)
array([[ 0.06666667,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.13333333,  0.        , -0.06666667,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.13333333,  0.        ,  0.        ,
         0.        , -0.06666667,  0.        ,  0.        ],
       [ 0.        , -0.06666667,  0.        ,  0.13333333,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.06666667,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.13333333,  0.        , -0.06666667,  0.        ],
       [ 0.        ,  0.        , -0.06666667,  0.        ,  0.        ,
         0.        ,  0.13333333,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
        -0.06666667,  0.        ,  0.13333333,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.06666667]])

We may also compute multipartite Werner states in toqito as well.

>>> from toqito.states import werner
>>> werner(2, [0.01, 0.02, 0.03, 0.04, 0.05])
array([[ 0.11286089,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.12729659, -0.00787402,  0.        , -0.00656168,
         0.        ,  0.        ,  0.        ],
       [ 0.        , -0.00918635,  0.1312336 ,  0.        , -0.00918635,
         0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.12860892,  0.        ,
        -0.01049869, -0.00524934,  0.        ],
       [ 0.        , -0.00524934, -0.01049869,  0.        ,  0.12860892,
         0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        , -0.00918635,  0.        ,
         0.1312336 , -0.00918635,  0.        ],
       [ 0.        ,  0.        ,  0.        , -0.00656168,  0.        ,
        -0.00787402,  0.12729659,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.11286089]])

References

[1] (1,2)

Reinhard F. Werner. Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40:4277–4281, Oct 1989. URL: https://link.aps.org/doi/10.1103/PhysRevA.40.4277, doi:10.1103/PhysRevA.40.4277.

Raises:

ValueError – Alpha vector does not have the correct length.

Parameters:

• dim (int) – The dimension of the Werner state.

• alpha (float | list[float]) – Parameter to specify Werner state.

Returns:

A Werner state of dimension dim.

Return type:

numpy.ndarray