Nathaniel: Created page with "{{Function |name=HorodeckiState |desc=Produces a Horodecki state |rel=BreuerState
ChessboardState |cat=List of functions#Special_states,_vectors,_and_operators|..."

2015-01-14T19:29:19Z

Created page with "{{Function |name=HorodeckiState |desc=Produces a Horodecki state |rel=BreuerState<br />ChessboardState |cat=List of functions#Special_states,_vectors,_and_operators|..."

New page

{{Function
|name=HorodeckiState
|desc=Produces a Horodecki state
|rel=[[BreuerState]]<br />[[ChessboardState]]
|cat=[[List of functions#Special_states,_vectors,_and_operators|Special states, vectors, and operators]]
|upd=December 15, 2014}}
<tt>'''HorodeckiState'''</tt> is a [[List of functions|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 <ref name="Hor97">P. Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. ''Phys. Lett. A'', 232:333, 1997. E-print: [http://arxiv.org/abs/quant-ph/9703004 arXiv:quant-ph/9703004]</ref> and have the following standard basis representation:
: <math>\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}</math>
and
: <math>\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}.</math>

==Syntax==
* <tt>HORO_STATE = HorodeckiState(A)</tt>
* <tt>HORO_STATE = HorodeckiState(A,DIM)</tt>

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

==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:
<syntaxhighlight>
>> 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
</syntaxhighlight>

===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:
<syntaxhighlight>
>> 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
</syntaxhighlight>
{{SourceCode|name=HorodeckiState}}

==References==
<references />

HorodeckiState - Revision history

Nathaniel: Created page with "{{Function |name=HorodeckiState |desc=Produces a Horodecki state |rel=BreuerStateChessboardState |cat=List of functions#Special_states,_vectors,_and_operators|..."

Nathaniel: Created page with "{{Function |name=HorodeckiState |desc=Produces a Horodecki state |rel=BreuerState
ChessboardState |cat=List of functions#Special_states,_vectors,_and_operators|..."