toqito.states.werner

Werner states.

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

Module Contents

toqito.states.werner.werner(dim, alpha)

Produce a Werner state [@Werner_1989_QuantumStates].

A Werner state is a state of the following form

[

begin{equation}

rho_{alpha} = frac{1}{d^2 - dalpha} 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

`python exec="1" source="above" from toqito.states import werner print(werner(3, 1 / 2)) `

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

`python exec="1" source="above" from toqito.states import werner print(werner(2, [0.01, 0.02, 0.03, 0.04, 0.05])) `

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