AbsPPTConstraints

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

Argument descriptions

 * LAM: A bipartite density matrix or a vector containing the eigenvalues of a bipartite density matrix.
 * DIM (optional, default has both subsystems of equal dimension): 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</tt> is a scalar, it is assumed that the first subsystem has dimension DIM</tt> and the second subsystem has dimension length(LAM)/DIM</tt>.
 * DIM</tt> can be a 1-by-2 vector containing the dimensions of the local subsystems.
 * ESC_IF_NPOS</tt> (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</tt> (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.

Qutrit-qutrit states
In the qutrit-qutrit case, a state $\rho$ with eigenvalues $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_9 \geq 0$ is absolutely PPT if and only if the following two matrices are positive semidefinite:
 * $$\begin{bmatrix}2\lambda_9 & \lambda_8-\lambda_1 & \lambda_7-\lambda_2 \\ \lambda_8-\lambda_1 & 2\lambda_6 & \lambda_5-\lambda_3 \\ \lambda_7-\lambda_2 & \lambda_5-\lambda_3 & 2\lambda_4\end{bmatrix} \quad \text{and} \quad \begin{bmatrix}2\lambda_9 & \lambda_8-\lambda_1 & \lambda_6-\lambda_2 \\ \lambda_8-\lambda_1 & 2\lambda_7 & \lambda_5-\lambda_3 \\ \lambda_6-\lambda_2 & \lambda_5-\lambda_3 & 2\lambda_4\end{bmatrix}.$$

We can compute these two matrices for a given set of eigenvalues as follows:

Can be used with CVX
This function can be used with CVX in order to perform optimizations over the set of absolutely PPT states. In particular, because each of the absolutely PPT constraints requires that a certain matrix be positive semidefinite, we can use semidefinite programming to maximize or minimize a linear function over the set of eigenvalues of all absolutely PPT states. For example, the following code maximizes the largest eigenvalue of all absolutely PPT states in a $(4 \otimes 4)$-dimensional system:

Indeed, this agrees with Proposition 8.2 of, which says that the largest maximal eigenvalue of absolutely PPT states in a $(k \otimes n)$-dimensional system is $3/(2+nk)$.

IMPORTANT: If you use this function within a CVX optmization problem, make sure that you enforce lam(1) >= lam(2) >= ... >= 0</tt> as one of the constraints in CVX!