Detecting consistency of overlapping quantum marginals by separability

The quantum marginal problem asks whether a set of given density matrices are consistent, i.e., whether they can be the reduced density matrices of a global quantum state. Not many nontrivial analytic necessary (or sufficient) conditions are known for the problem in general. We propose a method to detect consistency of overlapping quantum marginals by considering the separability of some derived states. Our method works well for the $k$-symmetric extension problem in general and for the general overlapping marginal problems in some cases. Our work is, in some sense, the converse to the well-known $k$-symmetric extension criterion for separability.

Figure 1

(a) The polytope of yellow color is characterized by $0\le p_i\le 3/4$ and $1/4\le p_1+p_2+p_3\le 1$ is the condition given by the separability of ${\tilde{\rho }}_{AB}^{\left(2\right)}$. The convex set of red color is given by the necessary and sufficient condition for 2-symmetric extension of Bell-diagonal states. (b) The left view of the same figure.

Figure 2

The two convex sets of $(p_1,p_2,p_3)$ corresponding to the condition given by the separability condition of ${\tilde{\rho }}_{AB}^{\left(2\right)}$ (the polytope of yellow color) and the SSA condition (the convex set of red color).

Figure 3

The green region is the exact condition for two Werner states to be consistent. The pentagon defined by ${\psi }_1^{-}+{\psi }_2^{-}\ge -1$ and $-1\le {\psi }_i^{-}\le 1$ is the condition given by our criterion. (a) The condition given by the CKW entanglement monogamy inequality; (b) the SSA condition.