HorodeckiState

HorodeckiState is a function that produces a "Horodecki" bound entangled state in either $M_3 \otimes M_3$ (two-qutrit space) or $M_2 \otimes M_4$. These states were defined in and have the following standard basis representation:
 * $$\rho_a^{3\otimes 3} := \frac{1}{8a+1}\begin{bmatrix}a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\ 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\ 0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \tfrac{1}{2}(1+a) & 0 & \tfrac{1}{2}\sqrt{1-a^2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a & 0 \\ a & 0 & 0 & 0 & a & 0 & \tfrac{1}{2}\sqrt{1-a^2} & 0 & \tfrac{1}{2}(1+a)\end{bmatrix}$$

and
 * $$\rho_a^{2 \otimes 4} := \frac{1}{7a+1}\begin{bmatrix}a & 0 & 0 & 0 & 0 & a & 0 & 0 \\ 0 & a & 0 & 0 & 0 & 0 & a & 0 \\ 0 & 0 & a & 0 & 0 & 0 & 0 & a \\ 0 & 0 & 0 & a & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \tfrac{1}{2}(1+a) & 0 & 0 & \tfrac{1}{2}\sqrt{1-a^2} \\ a & 0 & 0 & 0 & 0 & a & 0 & 0 \\ 0 & a & 0 & 0 & 0 & 0 & a & 0 \\ 0 & 0 & a & 0 & \tfrac{1}{2}\sqrt{1-a^2} & 0 & 0 & \tfrac{1}{2}(1+a)\end{bmatrix}.$$

Syntax

 * HORO_STATE = HorodeckiState(A)
 * HORO_STATE = HorodeckiState(A,DIM)

Argument descriptions

 * A: A real number between 0 and 1 that determines which Horodecki state is produced.
 * DIM (optional, default [3,3]): The dimensions of the subsystems that the state should act on. Must be one of [3,3] or [2,4].

Two-qutrit bound entangled state
The following code generates a two-qutrit Horodecki state and verifies that it is bound entangled by checking that it has positive partial transpose and is not separable:

A (2 &#8855; 4)-dimensional bound entangled state
The following code generates a Horodecki state in $M_2 \otimes M_4$ and verifies that it is bound entangled by checking that it has positive partial transpose and is not separable: