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 |
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.
Contents
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:
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 on "expand" to the right to view the MATLAB source code for this function.
%% ISPPT Determines whether or not a matrix has positive partial transpose
% This function has one required argument:
% X: a square matrix
%
% PPT = IsPPT(X) is either 1 or 0, indicating that X does or does not
% have positive partial transpose (within numerical error). X is assumed
% to act on bipartite space.
%
% This function has three optional input arguments:
% SYS (default 2)
% DIM (default sqrt(length(X)))
% TOL (default sqrt(eps))
%
% [PPT,WIT] = IsPPT(X,SYS,DIM,TOL) determines whether or not X has
% positive partial transpose within the tolerance specified by TOL. DIM
% DIM is a vector containing the dimensions of the subsystems on which X
% acts, and SYS is a scalar or vector indicating which subsystems the
% transpose should be applied on. WIT is the eigenvector corresponding to
% the minimal eigenvalue of the partial transpose of X, and thus can
% act as a witness that proves X does not have positive partial transpose
% (i.e., WIT'*PartialTranspose(X,SYS,DIM)*WIT < 0).
%
% URL: http://www.qetlab.com/IsPPT
% requires: IsPSD.m, opt_args.m, PartialTranspose.m
% author: Nathaniel Johnston (nathaniel@njohnston.ca)
% package: QETLAB
% last updated: December 12, 2014
function [ppt,wit] = IsPPT(X,varargin)
% set optional argument defaults: sys=2, dim=sqrt(length(X)), tol=sqrt(eps)
[sys,dim,tol] = opt_args({ 2, round(sqrt(length(X))), sqrt(eps) },varargin{:});
% Allow this function to be called within CVX optimization problems.
if(isa(X,'cvx'))
cvx_begin sdp quiet
subject to
PartialTranspose(X,sys,dim) >= 0;
cvx_end
ppt = 1-min(cvx_optval,1); % CVX-safe way to map (0,Inf) to (1,0)
% If the function is just being called on a non-CVX variable, just check
% the PPT condition normally (which is much faster).
else
if(nargout > 1)
[ppt,wit] = IsPSD(PartialTranspose(X,sys,dim),tol);
else
ppt = IsPSD(PartialTranspose(X,sys,dim),tol);
end
end