AbsPPTConstraints

AbsPPTConstraints
Builds the eigenvalue matrices that determine whether or not a state is absolutely PPT
Other toolboxes required	none
Related functions	IsAbsPPT
Function category	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

Notes

For all practical purposed, this function can only compute all constraint matrices when at least one of the local dimensions is 6 or less. When both local dimensions are 7 or higher, the LIM argument should be specified.

Additionally, this function actually computes slightly more constraints than is optimal. It computes the optimal set of constraints when one of the local dimensions is 5 or less, but when the smallest local dimension is 6, it computes 2612 constraint matrices rather than the optimal set of 2608 constraint matrices: the additional 4 matrices are redundant, but do not cause any harm.

Source code

Click here to view this function's source code on github.