IsUPB
From QETLAB
IsUPB | |
Determines whether or a set of product vectors form a UPB | |
Other toolboxes required | none |
---|---|
Related functions | MinUPBSize UPB UPBSepDistinguishable |
Function category | Distinguishing objects |
IsUPB is a function that determines whether or not a given set of product vectors forms an unextendible product basis (UPB).
Contents
Syntax
- IU = IsUPB(U,V,W,...)
Argument descriptions
Input arguments
- U,V,W,...: Matrices, each with the same number of columns as each other, whose columns are the local vectors of the supposed UPB.
Output arguments
- IU: A flag (either 1 or 0) specifying that the set of vectors described by U,V,W,... is or is not a UPB.
- WIT (optional): If IU = 0 then this is a cell containing local vectors for a product vector orthogonal to every member of the non-UPB described by U,V,W,... (thus, it acts as a witness to the fact that it really is not a UPB). If IU = 1 then WIT is meaningless.
Examples
The "Tiles" UPB
The following code verifies that the well-known "Tiles" UPB is indeed a UPB:
However, if we remove the fifth vector from this set, then it is no longer a UPB:
Indeed, it is not difficult to verify that wit{1} is orthogonal to u(:,1) and u(:,2), and wit{2} is orthogonal to v(:,3) and v(:,4), so the first four columns of u,v do not specify a UPB.
Source code
Click on "expand" to the right to view the MATLAB source code for this function.
%% ISUPB Determines whether or not a set of product vectors form a UPB
%
% IU = IsUPB(U,V,W,...) returns 1 or 0, indicating that the local states
% specified by U, V, W, ... is or is not an unextendible product basis.
% U, V, W, ... should be matrices, each with the same number of columns,
% whose columns are the local vectors of the supposed UPB (i.e., each
% matrix corresponds to one party).
%
% This function has one optional output argument:
% WIT
%
% [IU,WIT] = IsUPB(U,V,W,...) is as above. If IU = 0 then WIT is a cell
% containing local vectors for a product vector orthogonal to every
% member of the non-UPB specified by U,V,W,.... If IU = 1 then WIT is
% meaningless.
%
% URL: http://www.qetlab.com/IsUPB
% requires: opt_args.m, vec_partitions.m
% author: Nathaniel Johnston (nathaniel@njohnston.ca)
% last updated: October 21, 2014
function [iu,wit] = IsUPB(varargin)
wit = 0;
p = nargin; % number of parties in the UPB
s = size(varargin{1},2); % number of states in the UPB
for j = p:-1:1 % get local dimension of each party
dim(j) = size(varargin{j},1);
end
pt = vec_partitions(1:s,p,dim-1); % all possible partitions of the states among the different parties
num_pt = length(pt); % number of partitions to loop through
if(num_pt == 0) % no partitions, which means that so few vectors were provided that it can't possibly be a UPB
iu = 0;
if(nargout > 1) % generate a witness, if requested
orth_ct = 0;
wit = cell(1,p);
for l = p:-1:1
if(orth_ct >= s) % the witness we have generated is already orthogonal to all local vectors: just pick the first basis state from here on
wit{l} = zeros(dim(l),1);
wit{l}(1) = 1;
else
ct_upd = min(dim(l)-1,s-orth_ct);
wit{l} = null(varargin{l}(:,orth_ct+1:orth_ct+ct_upd).'); % these are the states orthogonal to the local states provided
wit{l} = wit{l}(:,1); % we only need one witness state
orth_ct = orth_ct + ct_upd;
end
end
end
return
end
% There are non-trivial partitions to loop through. Do it!
for k = 1:num_pt % loop through the partitions
num_orth = 1;
for l = p:-1:1 % loop through the parties
V = varargin{l}(:,pt{k}{l}); % these are the local states that we are choosing on the l-th party
orth_state{l} = null(V.'); % these are the states orthogonal to the local states
num_orth = num_orth*size(orth_state{l},2);
if(num_orth >= 1)
orth_state{l} = orth_state{l}(:,1); % we just do this so that the wit variable is prettier
end
end
if(num_orth >= 1) % all parties have codimension at least 1: not a UPB
iu = 0;
wit = orth_state;
return
end
end
iu = 1;