HorodeckiState

From QETLAB
Jump to navigation Jump to search
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 [1] 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 on "expand" to the right to view the MATLAB source code for this function.

References

  1. P. Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A, 232:333, 1997. E-print: arXiv:quant-ph/9703004