states.horodecki¶
Horodecki states are bound entangled states.
These states are entangled, but no pure entangled states can be extracted from these states through local operations and classical communication (LOCC).
Functions¶
Module Contents¶
- states.horodecki.horodecki(a_param, dim=None)¶
Produce a Horodecki state [1, 2].
Returns the Horodecki state in either \((3 \otimes 3)\)-dimensional space or \((2 \otimes 4)\)-dimensional space, depending on the dimensions in the 1-by-2 vector
dim.The Horodecki state was introduced in [1] which serves as an example in \(\mathbb{C}^3 \otimes \mathbb{C}\) or \(\mathbb{C}^2 \otimes \mathbb{C}^4\) of an entangled state that is positive under partial transpose (PPT). The state is PPT for all \(a \in [0, 1]\) and separable only for
a_param = 0ora_param = 1.These states have the following definitions:
\[\begin{split}\begin{equation} \rho_a^{3 \otimes 3} = \frac{1}{8a + 1} \begin{pmatrix} a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\ 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \left( 1 + a \right) & 0 & \frac{1}{2} \sqrt{1 - a^2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \sqrt{1 - a^2} & 0 & \frac{1}{2} \left(1 + a \right) \\ \end{pmatrix}, \end{equation}\end{split}\]\[\begin{split}\begin{equation} \rho_a^{2 \otimes 4} = \frac{1}{7a + 1} \begin{pmatrix} a & 0 & 0 & 0 & 0 & a & 0 & 0 \\ 0 & a & 0 & 0 & 0 & 0 & a & 0 \\ 0 & 0 & a & 0 & 0 & 0 & 0 & a \\ 0 & 0 & 0 & a & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2} \left(1 + a\right) & 0 & 0 & \frac{1}{2}\sqrt{1 -a^2} \\ a & 0 & 0 & 0 & 0 & a & 0 & 0 \\ 0 & a & 0 & 0 & 0 & 0 & a & 0 \\ 0 & 0 & a & 0 & \frac{1}{2}\sqrt{1 - a^2} & 0 & 0 & \frac{1}{2}\left(1 +a \right) \end{pmatrix}. \end{equation}\end{split}\]Note
Refer to [2] (specifically equations (1) and (2)) for more information on this state and its properties. The 3x3 Horodecki state is defined explicitly in Section 4.1 of [1] and the 2x4 Horodecki state is defined explicitly in Section 4.2 of [1].
Examples
The following code generates a Horodecki state in \(\mathbb{C}^3 \otimes \mathbb{C}^3\)
>>> from toqito.states import horodecki >>> horodecki(0.5, [3, 3]) array([[0.1 , 0. , 0. , 0. , 0.1 , 0. , 0. , 0. , 0.1 ], [0. , 0.1 , 0. , 0. , 0. , 0. , 0. , 0. , 0. ], [0. , 0. , 0.1 , 0. , 0. , 0. , 0. , 0. , 0. ], [0. , 0. , 0. , 0.1 , 0. , 0. , 0. , 0. , 0. ], [0.1 , 0. , 0. , 0. , 0.1 , 0. , 0. , 0. , 0.1 ], [0. , 0. , 0. , 0. , 0. , 0.1 , 0. , 0. , 0. ], [0. , 0. , 0. , 0. , 0. , 0. , 0.15 , 0. , 0.08660254], [0. , 0. , 0. , 0. , 0. , 0. , 0. , 0.1 , 0. ], [0.1 , 0. , 0. , 0. , 0.1 , 0. , 0.08660254, 0. , 0.15 ]])
The following code generates a Horodecki state in \(\mathbb{C}^2 \otimes \mathbb{C}^4\).
>>> from toqito.states import horodecki >>> horodecki(0.5, [2, 4]) array([[0.11111111, 0. , 0. , 0. , 0. , 0.11111111, 0. , 0. ], [0. , 0.11111111, 0. , 0. , 0. , 0. , 0.11111111, 0. ], [0. , 0. , 0.11111111, 0. , 0. , 0. , 0. , 0.11111111], [0. , 0. , 0. , 0.11111111, 0. , 0. , 0. , 0. ], [0. , 0. , 0. , 0. , 0.16666667, 0. , 0. , 0.09622504], [0.11111111, 0. , 0. , 0. , 0. , 0.11111111, 0. , 0. ], [0. , 0.11111111, 0. , 0. , 0. , 0. , 0.11111111, 0. ], [0. , 0. , 0.11111111, 0. , 0.09622504, 0. , 0. , 0.16666667]])
References
[1] (1,2,3,4)Pawel Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Physics Letters A, 232(5):333–339, Aug 1997. URL: http://dx.doi.org/10.1016/S0375-9601(97)00416-7, doi:10.1016/s0375-9601(97)00416-7.
[2] (1,2,3)Dariusz Chruściński and Andrzej Kossakowski. On the symmetry of the seminal horodecki state. Physics Letters A, 375(3):434–436, Jan 2011. URL: https://arxiv.org/abs/1009.4385, doi:10.1016/j.physleta.2010.11.069.
- Parameters:
a_param (float)
dim (list[int])
- Return type:
numpy.ndarray