Difference between revisions of "IsPPT"

	IsPPT
	Determines whether or not a matrix has positive partial transpose
Other toolboxes required	none
Related functions	IsPSD; PartialTranspose
Function category	Entanglement and separability

Latest revision as of 15:05, 22 September 2014

IsPPT is a 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

PPT = IsPPT(X)
PPT = IsPPT(X,SYS)
PPT = IsPPT(X,SYS,DIM)
PPT = IsPPT(X,SYS,DIM,TOL)
[PPT,WIT] = IsPPT(X,SYS,DIM,TOL)

Argument descriptions

Input arguments

X: A square matrix.
SYS (optional, default 2): A scalar or vector indicating which subsystem(s) the transpose should be applied on.
DIM (optional, default has X living on two subsystems of equal size): A vector containing the dimensions of the (possibly more than 2) subsystems on which X lives.
TOL (optional, default sqrt(eps)): 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 -TOL.

Output arguments

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

Examples

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:

>> IsPPT(eye(9))

ans =

     1

Notes

Do not request the WIT output argument unless you need it. If WIT is not requested, positive semidefiniteness is determined by attempting a Cholesky decomposition of X, which is both faster and more accurate than computing its minimum eigenvalue/eigenvector pair.

Source code

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

@@ Line 3: / Line 3: @@
 |desc=Determines whether or not a matrix has [[positive partial transpose]]
 |rel=[[IsPSD]]<br />[[PartialTranspose]]
+|cat=[[List of functions#Entanglement_and_separability|Entanglement&nbsp;and&nbsp;separability]]
 |upd=November 20, 2012
-|v=1.00}}
+|v=0.50}}
 <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.