toqito.states.isotropic
- toqito.states.isotropic(dim, alpha)[source]
Produce a isotropic state [HH99].
Returns the isotropic state with parameter
alphaacting on (dim-by-dim)-dimensional space. The isotropic state has the following form\[\begin{equation} \rho_{\alpha} = \frac{1 - \alpha}{d^2} \mathbb{I} \otimes \mathbb{I} + \alpha |\psi_+ \rangle \langle \psi_+ | \in \mathbb{C}^d \otimes \mathbb{C}^2 \end{equation}\]where \(|\psi_+ \rangle = \frac{1}{\sqrt{d}} \sum_j |j \rangle \otimes |j \rangle\) is the maximally entangled state.
Examples
To generate the isotropic state with parameter \(\alpha=1/2\), we can make the following call to
toqitoas>>> from toqito.states import isotropic >>> isotropic(3, 1 / 2) [[0.22222222, 0. , 0. , 0. , 0.16666667, 0. , 0. , 0. , 0.16666667], [0. , 0.05555556, 0. , 0. , 0. , 0. , 0. , 0. , 0. ], [0. , 0. , 0.05555556, 0. , 0. , 0. , 0. , 0. , 0. ], [0. , 0. , 0. , 0.05555556, 0. , 0. , 0. , 0. , 0. ], [0.16666667, 0. , 0. , 0. , 0.22222222, 0. , 0. , 0. , 0.16666667], [0. , 0. , 0. , 0. , 0. , 0.05555556, 0. , 0. , 0. ], [0. , 0. , 0. , 0. , 0. , 0. , 0.05555556, 0. , 0. ], [0. , 0. , 0. , 0. , 0. , 0. , 0. , 0.05555556, 0. ], [0.16666667, 0. , 0. , 0. , 0.16666667, 0. , 0. , 0. , 0.22222222]]
References
[HH99]Horodecki, Michał, and Paweł Horodecki. “Reduction criterion of separability and limits for a class of distillation protocols.” Physical Review A 59.6 (1999): 4206.
- Parameters:
dim – The local dimension.
alpha – The parameter of the isotropic state.
- Returns:
Isotropic state of dimension
dim.