states.singlet¶
Generalized singlet state is a singlet state of n qubits in the invariant space of alternating representation.
Functions¶
Module Contents¶
- 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
- Parameters:
dim (int) – The dimension of the generalized singlet state.
- Returns:
The singlet state of dimension dim.
- Return type:
numpy.ndarray