IsPPT - Revision history

Nathaniel at 15:05, 22 September 2014

2014-09-22T15:05:47Z

Nathaniel: Updated documentation using GeSHi

2014-09-21T20:00:34Z

Updated documentation using GeSHi

Nathaniel at 15:49, 21 November 2012

2012-11-21T15:49:39Z

Nathaniel: Created page with "{{Function |name=IsPPT |desc=Determines whether or not a matrix has positive partial transpose |req=IsPSD, opt_args, PartialTranspose, PermuteSystems |upd=..."

2012-11-21T15:48:24Z

Created page with "{{Function |name=IsPPT |desc=Determines whether or not a matrix has positive partial transpose |req=IsPSD, opt_args, PartialTranspose, PermuteSystems |upd=..."

New page

{{Function
|name=IsPPT
|desc=Determines whether or not a matrix has [[positive partial transpose]]
|req=[[IsPSD]], [[opt_args]], [[PartialTranspose]], [[PermuteSystems]]
|upd=November 20, 2012
|v=1.00}}
<tt>'''IsPPT'''</tt> is a [[List of functions|function]] that determines whether or not a given matrix has [[positive partial transpose]] (PPT), which is a quick and easy [[separability criterion]]. This function works on both full and sparse matrices, and if desired a witness can be provided that verifies that the input matrix is not PPT.

==Syntax==
* <tt>PPT = IsPPT(X)</tt>
* <tt>PPT = IsPPT(X,SYS)</tt>
* <tt>PPT = IsPPT(X,SYS,DIM)</tt>
* <tt>PPT = IsPPT(X,SYS,DIM,TOL)</tt>
* <tt>[PPT,WIT] = IsPPT(X,SYS,DIM,TOL)</tt>

==Argument descriptions==
===Input arguments===
* <tt>X</tt>: A square matrix.
* <tt>SYS</tt> (optional, default 2): A scalar or vector indicating which subsystem(s) the transpose should be applied on.
* <tt>DIM</tt> (optional, default has <tt>X</tt> living on two subsystems of equal size): A vector containing the dimensions of the (possibly more than 2) subsystems on which <tt>X</tt> lives.
* <tt>TOL</tt> (optional, default <tt>sqrt(eps)</tt>): The numerical tolerance used when determining positive semidefiniteness. The matrix will be determined to have positive partial transpose if its partial transpose's minimal eigenvalue is computed to be at least <tt>-TOL</tt>.

===Output arguments===
* <tt>PPT</tt>: A flag (either 1 or 0) indicating that <tt>X</tt> does or does not have positive partial transpose.
* <tt>WIT</tt> (optional): An eigenvector corresponding to the minimal eigenvalue of <tt>PartialTranspose(X)</tt>. When <tt>PPT = 0</tt>, this serves as a witness that verifies that <tt>X</tt> does not have positive partial transpose, since <tt>WIT'*PartialTranspose(X)*WIT < 0</tt>.

==Examples==
Please fill in examples here.

@@ Line 2: / Line 2: @@
 |name=IsPPT
 |desc=Determines whether or not a matrix has [[positive partial transpose]]
-|req=[[IsPSD]],<br />[[opt_args]],<br />[[PartialTranspose]],<br />[[PermuteSystems]]
+|rel=[[IsPSD]]<br />[[PartialTranspose]]
 |upd=November 20, 2012
 |v=1.00}}
@@ Line 26: / Line 26: @@
 ==Examples==
-Please fill in examples here.
+The following code verifies that the 9-by-9 identity operator (thought of as an operator in $M_3 \otimes M_3$) has positive partial transpose:
 ==Notes==
 Do not request the <tt>WIT</tt> output argument unless you need it. If <tt>WIT</tt> is not requested, positive semidefiniteness is determined by attempting a [http://en.wikipedia.org/wiki/Cholesky_decomposition Cholesky decomposition] of <tt>X</tt>, which is both faster and more accurate than computing its minimum eigenvalue/eigenvector pair.