toqito.states.singlet
- toqito.states.singlet(dim)[source]
Produce a generalized singlet state acting on two n-dimensional systems [Gsinglet].
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
toqitoas follows:>>> from toqito.states import singlet >>> dim = 2 >>> singlet(dim) [[ 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) [[ 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
[Gsinglet]Cabello, Adan, “N-particle N-level singlet states: Some properties and applications”. Phys. Rev. Lett., 89 (2002): 100402.
- Parameters:
dim – The dimension of the generalized singlet state.
- Returns:
The singlet state of dimension dim.