toqito.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).
Module Contents¶
- toqito.states.horodecki.horodecki(a_param, dim=None)[source]¶
Produce a Horodecki state [@Horodecki_1997_Separability][@Chruscinski_2011_OnTheSymmetry].
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 = 0 or a_param = 1.
These states have the following definitions:
- [
- 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}
]
- [
- 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 + aright) & 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}
]
- !!! Note
Refer to [@Chruscinski_2011_OnTheSymmetry] (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 [@Horodecki_1997_Separability] and the 2x4 Horodecki state is defined explicitly in Section 4.2 of [@Horodecki_1997_Separability].
Examples
The following code generates a Horodecki state in (mathbb{C}^3 otimes mathbb{C}^3)
`python exec="1" source="above" from toqito.states import horodecki print(horodecki(0.5, [3, 3])) `The following code generates a Horodecki state in (mathbb{C}^2 otimes mathbb{C}^4).
`python exec="1" source="above" from toqito.states import horodecki print(horodecki(0.5, [2, 4])) `- Parameters:
a_param (float)
dim (list[int] | None)
- Return type:
numpy.ndarray