# AbsPPTConstraints

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 Other toolboxes required AbsPPTConstraints Builds the eigenvalue matrices that determine whether or not a state is absolutely PPT none IsAbsPPT Ball of separability

AbsPPTConstraints is a function that computes the matrices of eigenvalues that determine whether or not a given density matrix $\rho$ is "absolutely PPT" (that is whether or not $U\rho U^\dagger$ has positive partial transpose for all unitary matrices $U$). The state $\rho$ is absolutely PPT if and only if every matrix that this function produces as output is positive semidefinite. The matrices that characterize when a density matrix is absolutely PPT were derived in .

## Syntax

• L = AbsPPTConstraints(LAM,DIM)
• L = AbsPPTConstraints(LAM,DIM,ESC_IF_NPOS)
• L = AbsPPTConstraints(LAM,DIM,ESC_IF_NPOS,LIM)

## Argument descriptions

• LAM: The eigenvalues of a bipartite density matrix.
• DIM: A specification of the dimensions of the subsystems that the density matrix acts on. DIM can be provided in one of two ways:
• If DIM is a scalar, it is assumed that the first subsystem has dimension DIM and the second subsystem has dimension length(LAM)/DIM.
• DIM can be a 1-by-2 vector containing the dimensions of the local subsystems.
• ESC_IF_NPOS (optional, default 0): A flag, either 1 or 0, indicating that the function should stop computing constraint matrices as soon as it finds one that is not positive semidefinite (i.e., as soon as it finds proof that the density matrix in question is not absolutely PPT).
• LIM (optional, default 0): A non-negative integer specifying how many constraint matrices should be computed. If this equals 0 then all constraint matrices will be computed.

## Examples

### Qutrit-qutrit states

It was noted in  that, in the qutrit-qutrit case, the absolutely PPT property is determined by whether or not two 3-by-3 matrices of eigenvalues are positive semidefinite. We can compute these two matrices for a given set of eigenvalues as follows:

>> rho = RandomDensityMatrix(9);
>> lam = eig(rho);
>> L = AbsPPTConstraints(lam,3) % compute the two constraint matrices

L =

[3x3 double]    [3x3 double]

>> L{1}

ans =

0.0013   -0.3100   -0.2579
-0.3100    0.0679   -0.1159
-0.2579   -0.1159    0.2102

>> L{2}

ans =

0.0013   -0.3100   -0.2381
-0.3100    0.0282   -0.1159
-0.2381   -0.1159    0.2102

>> IsPSD(L{1})

ans =

0 % L{1} is not positive semidefinite, so rho is not absolutely PPT