SymmetricExtension

	SymmetricExtension
	Determines whether or not an operator has a symmetric extension
Other toolboxes required	CVX
Related functions	IsPPT; IsSeparable; SymmetricInnerExtension; SymmetricProjection
Function category	Entanglement and separability

SymmetricExtension is a function that determines whether or not a given positive semidefinite operator has a symmetric extension. This function is extremely useful for showing that quantum states are entangled (see the Examples section). Various types of symmetric extensions (such as Bosonic and/or PPT extensions) can be looked for by specifying optional arguments in the function.

Syntax

EX = SymmetricExtension(X)
EX = SymmetricExtension(X,K)
EX = SymmetricExtension(X,K,DIM)
EX = SymmetricExtension(X,K,DIM,PPT)
EX = SymmetricExtension(X,K,DIM,PPT,BOS)
EX = SymmetricExtension(X,K,DIM,PPT,BOS,TOL)
[EX,WIT] = SymmetricExtension(X,K,DIM,PPT,BOS,TOL)

Argument descriptions

Input arguments

X: A positive semidefinite operator.
K (optional, default 2): The number of copies of the second subsystem in the desired symmetric extension.
DIM (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the two subsystems that X acts on.
PPT (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have positive partial transpose.
BOS (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must be Bosonic (i.e., have its range contained within the symmetric subspace).
TOL (optional, default eps^(1/4)): The numerical tolerance used throughout this script. It is recommended that this is left at the default value unless numerical problems arise and the script has difficulty determining whether or not X has a symmetric extension.

Output arguments

EX: A flag (either 1 or 0) indicating that X does or does not have a symmetric extension of the desired type.
WIT (optional): A witness that verifies that the answer provided by EX is correct. If EX = 1 (i.e., X has a symmetric extension) then WIT is such a symmetric extension. If EX = 0 (i.e., no symmetric extension exists) then WIT is an entanglement witness with trace(WIT*X) = -1 but trace(WIT*Y) >= 0 for all symmetrically extendable Y.

Examples

2-qubit symmetric extension

It is known^[1] that a 2-qubit state $\rho_{AB}$ has a (not necessarily PPT) symmetric extension if and only if ${\rm Tr}(\rho_B^2) \geq {\rm Tr}(\rho_{AB}^2) - 4\sqrt{\det(\rho_{AB})}$. The following code verifies that one such state does indeed have a symmetric extension.

>> rho = [1 0 0 -1;0 1 1/2 0;0 1/2 1 0;-1 0 0 1];
>> [trace(PartialTrace(rho)^2), trace(rho^2) - 4*sqrt(det(rho))] % if the first number is >= the second number, rho has a symmetric extension
 
ans =
 
    8.0000    6.5000
 
>> SymmetricExtension(rho) % verify that rho has a symmetric extension
 
ans =
 
     1

Notes

If your goal is to detect entanglement in an operator, then it is always in your best interest to set the optional arguments PPT and BOS to be 1. Setting BOS = 1 increases the effectiveness of the entanglement test without any computational overhead at all. Setting PPT = 1 slows down the computation quite a bit, but increases the effectiveness as an entanglement test considerably.

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

%%  SYMMETRICEXTENSION    Determines whether or not an operator has a symmetric extension

%   This function has one required argument:

```
%     X: a positive semidefinite matrix
```
```
%
```

%   EX = SymmetricExtension(X) is either 1 or 0, indicating that X does or

%   does not have a 2-copy positive partial transpose Bosonic symmetric

%   extension. The extension is always taken on the second subsystem of X.

```
%
```

%   This function has five optional arguments:

```
%     K (default 2)
```

%     DIM (default has both subsystems of equal dimension)

```
%     PPT (default 0)
```
```
%     BOS (default 0)
```
```
%     TOL (default eps^(1/4))
```
```
%
```

%   [EX,WIT] = SymmetricExtension(X,K,DIM,PPT,BOS,TOL) determines whether

%   or not X has a symmetric extension and provides a witness WIT that

%   verifies the answer. If a symmetric extension of X exists

%   (i.e., EX = 1) then WIT is such a symmetric extension. If no symmetric

%   extension exists (i.e., EX = 0) then WIT is an entanglement witness

%   with trace(WIT*X) = -1 but trace(WIT*Y) >= 0 for all symmetrically

```
%   extendable Y.
```
```
%
```

%   K is the desired number of copies of the second subsystem. DIM is a

%   1-by-2 vector containing the dimensions of the subsystems on which X

%   acts. PPT is a flag (either 0 or 1) indicating whether the desired

%   symmetric extension must have positive partial transpose. BOS is a flag

%   (either 0 or 1) indicating whether the desired symmetric extension must

%   be Bosonic (i.e., be supported on the symmetric subspace). TOL is the

%   numerical tolerance used when determining whether or not a symmetric

```
%   extension exists.
```
```
%
```

%   URL: http://www.qetlab.com/SymmetricExtension

```
 
```

%   requires: cvx (http://cvxr.com/cvx/), IsPPT.m, IsPSD.m, opt_args.m,

%             PartialTrace.m, PartialTranspose.m, PermutationOperator.m,

%             PermuteSystems.m, sporth.m, SymmetricProjection.m

```
%
```

%   author: Nathaniel Johnston (nathaniel@njohnston.ca)

```
%   package: QETLAB
```
```
%   last updated: December 12, 2014
```
```
 
```

function [ex,wit] = SymmetricExtension(X,varargin)

```
 
```
```
lX = length(X);
```
```
 
```

% set optional argument defaults: k=2, dim=sqrt(length(X)), ppt=1, bos=1, tol=eps^(1/4)

[k,dim,ppt,bos,tol] = opt_args({ 2, round(sqrt(lX)), 0, 0, eps^(1/4) },varargin{:});

```
 
```

% allow the user to enter a single number for dim

```
if(length(dim) == 1)
```
```
    dim = [dim,lX/dim];
```

    if abs(dim(2) - round(dim(2))) >= 2*lX*eps

        error('SymmetricExtension:InvalidDim','If DIM is a scalar, it must evenly divide length(X); please provide the DIM array containing the dimensions of the subsystems.');

```
    end
```
```
    dim(2) = round(dim(2));
```
```
end
```
```
 
```

% In certain situations, we don't need semidefinite programming.

if(k == 1 || (lX <= 6 && ppt && nargout <= 1))

    if(k == 1 && ~ppt) % in some cases, the problem is *really* trivial

```
        if(nargout > 1)
```
```
            [ex,wit] = IsPSD(X,tol);
```
```
            if(ex == 1)
```
```
                wit = X;
```
```
            end
```
```
        else
```
```
            ex = IsPSD(X,tol);
```
```
        end
```
```
 
```
```
        return
```
```
    end
```
```
 
```

    % In this case, all they asked for is a 1-copy PPT symmetric extension

    % (i.e., they're asking if the state is PPT).

```
    if(nargout > 1)
```
```
        [ex,wit] = IsPPT(X,2,dim,tol);
```
```
        if(ex)
```
```
            wit = X;
```
```
        else
```

            wit = PartialTranspose(wit*wit');

```
            wit = -wit/trace(wit*X);
```
```
        end
```
```
    else
```

        ex = (IsPPT(X,2,dim,tol) && IsPSD(X,tol));

```
    end
```
```
 
```

% In the 2-qubit case, an analytic formula is known for whether or not a

% state has a (2-copy, non-PPT) symmetric extension that is much

% faster to use than semidefinite programming.

elseif(~isa(X,'cvx') && k == 2 && ~ppt && dim(1) == 2 && dim(2) == 2 && nargout <= 1) % we don't need "&& ~bos" thanks to a lemma of Myhr and Lutkenhaus

    ex = (trace(PartialTrace(X,1)^2) >= trace(X^2) - 4*sqrt(det(X)));

```
 
```

% otherwise, use semidefinite programming to find a symmetric extension

```
elseif(k > 1)
```

    sdp_dim = [dim(1),dim(2)*ones(1,k)];

    % For Bosonic extensions, it suffices to optimize over the symmetric

```
    % subspace, which is smaller.
```
```
    if(bos)
```

        sdp_prod_dim = dim(1)*nchoosek(k+dim(2)-1, dim(2)-1);

```
    else
```
```
        sdp_prod_dim = dim(1)*dim(2)^k;
```
```
    end
```
```
 
```
```
    cvx_begin sdp quiet
```
```
        cvx_precision(tol);
```

        variable rho(sdp_prod_dim,sdp_prod_dim) hermitian

        if(nargout > 1) % don't want to compute the dual solution in general (especially not if this is called within CVX)

```
            dual variable W
```
```
        end
```
```
        subject to
```
```
            rho >= 0;
```

            if(bos) % check for a Bosonic extension (faster *and* more effective)

                P = kron(speye(dim(1)),SymmetricProjection(dim(2),k,1));

```
                if(nargout > 1)
```

                    W : PartialTrace(P*rho*P',3:k+1,sdp_dim) == X;

```
                else
```

                    PartialTrace(P*rho*P',3:k+1,sdp_dim) == X;

```
                end
```
```
                if(ppt)
```

                    PartialTranspose(P*rho*P',1:ceil(k/2)+1,sdp_dim) >= 0;

```
                end
```

            else % check for a non-Bosonic extension (slower but perhaps sometimes useful)

                % It's a bit messy that the code from above is largely

                % repeated here, but the goal is to not slow down the

                % Bosonic optimization at all, so we don't want to

                % introduce a new variable sigma = P*rho*P' in that case.

```
                if(nargout > 1)
```

                    W : PartialTrace(rho,3:k+1,sdp_dim) == X;

```
                else
```

                    PartialTrace(rho,3:k+1,sdp_dim) == X;

```
                end
```
```
                for j = 3:k+1
```

                    PartialTrace(rho,setdiff(2:k+1,j),sdp_dim) == X;

```
                end
```
```
                if(ppt)
```

                    PartialTranspose(rho,1:ceil(k/2)+1,sdp_dim) >= 0;

```
                end
```
```
            end
```
```
    cvx_end
```
```
 
```

    ex = 1-min(cvx_optval,1); % CVX-safe way to map (0,Inf) to (1,0)

    if(~isa(X,'cvx')) % make the output prettier if it's not a CVX input

```
        ex = round(ex);
```
```
 
```

        % Deal with error messages and witnesses.

        if(nargout > 1 && strcmpi(cvx_status,'Solved'))

```
            if(bos)
```
```
                wit = P*rho*P';
```
```
            else
```
```
                wit = rho;
```
```
            end
```

        elseif(strcmpi(cvx_status,'Inaccurate/Solved'))

```
            if(bos)
```
```
                wit = P*rho*P';
```
```
            else
```
```
                wit = rho;
```
```
            end
```

            warning('SymmetricExtension:NumericalProblems','Minor numerical problems encountered by CVX. Consider adjusting the tolerance level TOL and re-running the script.');

        elseif(nargout > 1 && strcmpi(cvx_status,'Infeasible'))

```
            wit = -W;
```

        elseif(strcmpi(cvx_status,'Inaccurate/Infeasible'))

```
            if(nargout > 1)
```
```
                wit = -W;
```
```
            end
```

            warning('SymmetricExtension:NumericalProblems','Minor numerical problems encountered by CVX. Consider adjusting the tolerance level TOL and re-running the script.');

        elseif(strcmpi(cvx_status,'Unbounded') || strcmpi(cvx_status,'Inaccurate/Unbounded') || strcmpi(cvx_status,'Failed'))

            error('SymmetricExtension:NumericalProblems',strcat('Numerical problems encountered (CVX status: ',cvx_status,'). Please try adjusting the tolerance level TOL.'));

```
        end
```
```
    end
```
```
else
```

    error('SymmetricExtension:InvalidK','K must be a positive integer.');

```
end
```

References

↑ J. Chen, Z. Ji, D. Kribs, N. Lütkenhaus, and B. Zeng. Symmetric extension of two-qubit states. E-print: arXiv:1310.3530 [quant-ph]

[1] J. Chen, Z. Ji, D. Kribs, N. Lütkenhaus, and B. Zeng. Symmetric extension of two-qubit states. E-print: arXiv:1310.3530 [quant-ph]

[1]

SymmetricExtension

Contents

Syntax

Argument descriptions

Input arguments

Output arguments

Examples

2-qubit symmetric extension

Notes

Source code

References

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