HorodeckiState

HorodeckiState
Produces a Horodecki state
Other toolboxes required	none
Related functions	BreuerState ChessboardState
Function category	Special states, vectors, and operators

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].

Examples

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:

>> rho = HorodeckiState(0.5)

rho =

    0.1000         0         0         0    0.1000         0         0         0    0.1000
         0    0.1000         0         0         0         0         0         0         0
         0         0    0.1000         0         0         0         0         0         0
         0         0         0    0.1000         0         0         0         0         0
    0.1000         0         0         0    0.1000         0         0         0    0.1000
         0         0         0         0         0    0.1000         0         0         0
         0         0         0         0         0         0    0.1500         0    0.0866
         0         0         0         0         0         0         0    0.1000         0
    0.1000         0         0         0    0.1000         0    0.0866         0    0.1500

>> IsPPT(rho)

ans =

     1

>> IsSeparable(rho)
Determined to be entangled via the realignment criterion. Reference:
K. Chen and L.-A. Wu. A matrix realignment method for recognizing entanglement. Quantum Inf. Comput., 3:193-202, 2003.

ans =

     0

A (2 ⊗ 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:

>> rho = HorodeckiState(0.5,[2,4])

rho =

    0.1111         0         0         0         0    0.1111         0         0
         0    0.1111         0         0         0         0    0.1111         0
         0         0    0.1111         0         0         0         0    0.1111
         0         0         0    0.1111         0         0         0         0
         0         0         0         0    0.1667         0         0    0.0962
    0.1111         0         0         0         0    0.1111         0         0
         0    0.1111         0         0         0         0    0.1111         0
         0         0    0.1111         0    0.0962         0         0    0.1667

>> IsPPT(rho,2,[2,4])

ans =

     1

>> IsSeparable(rho,[2,4])
Determined to be entangled by not having a 2-copy PPT symmetric extension. Reference:
A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. A complete family of separability criteria. Phys. Rev. A, 69:022308, 2004.

ans =

     0

Source code

Click here to view this function's source code on github.