UPBSepDistinguishable
UPBSepDistinguishable | |
Determines whether or not a UPB is distinguishable by separable measurements | |
Other toolboxes required | CVX |
---|---|
Related functions | LocalDistinguishability |
Function category | Distinguishing objects |
UPBSepDistinguishable is a function that determines whether or not a given UPB is perfectly distinguishable by separable measurements. This question is interesting because it is known that all UPBs are indistinguishable by LOCC measurements ^{[1]}, and all UPBs are distinguishable by PPT measurements. Separable measurements lie between these two classes.
Contents
Syntax
- DIST = UPBSepDistinguishable(U,V,W,...)
Argument descriptions
- U,V,W,...: Matrices, each with the same number of columns as each other, whose columns are the local vectors of the UPB.
Examples
Qutrit UPBs are distinguishable
It was shown in ^{[2]} that all UPBs in $\mathbb{C}^3 \otimes \mathbb{C}^3$ are distinguishable by separable measurements. We can verify this fact for the "Tiles" UPB as follows:
>> [u,v] = UPB('Tiles'); % generates the "Tiles" UPB >> UPBSepDistinguishable(u,v) ans = 1
The Feng UPB is indistinguishable
It was shown in ^{[3]} that the UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ found by K. Feng is indistinguishable by separable measurements. We can confirm this fact as follows:
>> [u,v] = UPB('Feng4x4'); % generates the "Feng" UPB >> UPBSepDistinguishable(u,v) ans = 0
Source code
Click on "expand" to the right to view the MATLAB source code for this function.
%% UPBSEPDISTINGUISHABLE Determines whether or not a UPB is distinguishable by separable measurements
%
% DIST = UPBSepDistinguishable(U,V,W,...) returns 1 or 0, indicating that
% the UPB specified by the local vectors U, V, W, ... is or is not
% distinguishable by separable measurements. U, V, W, ... should be
% matrices, each with the same number of columns, whose columns are the
% local vectors of a UPB (i.e., each matrix corresponds to one party).
%
% See [1] for a description of how this computation is carried out.
%
% URL: http://www.qetlab.com/UPBSepDistinguishable
%
% References:
% [1] S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Watrous,
% and N. Yu. Limitations on separable measurements by convex
% optimization. E-print: arXiv:1408.6981 [quant-ph], 2014.
% requires: cvx (http://cvxr.com/cvx/), opt_args.m, vec_partitions.m
% author: Nathaniel Johnston (nathaniel@njohnston.ca)
% last updated: October 30, 2014
function dist = UPBSepDistinguishable(varargin)
% First, compute all of the "replacement vectors" for this UPB.
P = upb_replacement_vectors(varargin{:});
% Convert the replacement vectors that were found above into the projection
% onto those vectors.
ct = 1;
for j = 1:length(P)
for k = 1:size(P{j}{1},2)
Q{ct} = P{j}{1}(:,k);
for l = 2:length(P{j})
Q{ct} = kron(Q{ct},P{j}{l}(:,k));
end
Q{ct} = Q{ct}*Q{ct}';
ct = ct + 1;
end
end
dim = length(Q{1});
% Now solve the LP based on these projections that determines whether or
% not this UPB is distinguishable by separable measurements.
cvx_begin sdp quiet
cvx_precision high;
variable lam(ct)
P_sum = 0;
for j = 1:ct-1
P_sum = P_sum + lam(j)*Q{j};
end
minimize norm(P_sum - eye(dim),'fro')
subject to
lam >= 0;
cvx_end
dist = real(cvx_optval);
% Convert the output into a form that the user expects (0 = no, 1 = yes).
if(dist <= sqrt(eps))
dist = 1;
else
dist = 0;
end
end
% This function computes all of the "replacement vectors" of a UPB. That
% is, it removes one vector from the UPB and finds all product vectors
% orthogonal to the remaining UPB vectors.
function P = upb_replacement_vectors(varargin)
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
for j = s:-1:1
P{j} = cell(p,1);
end
% Loop through each of the s states in the UPB, removing them one at a
% time.
for j = s:-1:1
cind = setdiff(1:s,j); % the indices of the states that have not been removed
pt = vec_partitions(cind,p,dim-1); % all possible partitions of the states among the different parties
num_pt = length(pt); % number of partitions to loop through
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.');
num_orth = num_orth*size(orth_state{l},2);
end
if(num_orth > 1)
error('upb_replacement_vectors:NotUPB','The set of local vectors provided do not form a UPB.');
% If the found state really is orthogonal to all others, add it to
% the output.
elseif(num_orth == 1)
% First, make sure that this state isn't already in P{j}.
if(size(P{j}{1},2) > 0)
trow = ones(1,size(P{j}{1},2));
for l = p:-1:1
trow = trow.*(orth_state{l}'*P{j}{l});
end
new_col = (max(abs(trow)) < 1 - eps*size(P{j}{1},2)^p);
else
new_col = 1;
end
% OK, this state really is new.
if(new_col)
for l = p:-1:1
P{j}{l} = [P{j}{l},orth_state{l}];
end
end
end
end
end
end
References
- ↑ C. Bennett, D. DiVincenzo, T. Mor, P. Shor, J. Smolin, and B. Terhal. Unextendible product bases and bound entanglement. Physical Review Letters, 82(26):5385–5388, 1999. E-print: arXiv:quant-ph/9808030
- ↑ D. DiVincenzo, T. Mor, P. W. Shor, J. Smolin, and B. Terhal. Unextendible product bases, uncompletable product bases and bound entanglement. Communications in Mathematical Physics, 238(3):379–410, 2003. E-print: arXiv:quant-ph/9908070
- ↑ S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Watrous, and N. Yu. Limitations on separable measurements by convex optimization. E-print: arXiv:1408.6981 [quant-ph], 2014.