states.singlet

Generalized singlet state.

Module Contents

Functions

singlet(dim)

Produce a generalized singlet state acting on two n-dimensional systems [1].

states.singlet.singlet(dim)

Produce a generalized singlet state acting on two n-dimensional systems [1].

Examples

For \(n = 2\) this generates the following matrix

\[\begin{split}S = \frac{1}{2} \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]

which is equivalent to \(|\phi_s \rangle \langle \phi_s |\) where

\[|\phi_s\rangle = \frac{1}{\sqrt{2}} \left( |01 \rangle - |10 \rangle \right)\]

is the singlet state. This can be computed via toqito as follows:

>>> from toqito.states import singlet
>>> dim = 2
>>> singlet(dim)
array([[ 0. ,  0. ,  0. ,  0. ],
       [ 0. ,  0.5, -0.5,  0. ],
       [ 0. , -0.5,  0.5,  0. ],
       [ 0. ,  0. ,  0. ,  0. ]])

It is possible for us to consider higher dimensional singlet states. For instance, we can consider the \(3\)-dimensional Singlet state as follows:

>>> from toqito.states import singlet
>>> dim = 3
>>> singlet(dim)
array([[ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.16666667,  0.        , -0.16666667,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.16666667,  0.        ,  0.        ,
         0.        , -0.16666667,  0.        ,  0.        ],
       [ 0.        , -0.16666667,  0.        ,  0.16666667,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.16666667,  0.        , -0.16666667,  0.        ],
       [ 0.        ,  0.        , -0.16666667,  0.        ,  0.        ,
         0.        ,  0.16666667,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
        -0.16666667,  0.        ,  0.16666667,  0.        ],
       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,
         0.        ,  0.        ,  0.        ,  0.        ]])

References

[1] (1,2)

Adán Cabello. $n$-particle $n$-level singlet states: some properties and applications. Phys. Rev. Lett., 89:100402, Aug 2002. URL: https://arxiv.org/abs/quant-ph/0203119.

Parameters:

dim (int) – The dimension of the generalized singlet state.

Returns:

The singlet state of dimension dim.

Return type:

numpy.ndarray