Entanglement measures constructed from two positive, but not completely positive, maps on density operators are used as constraints in placing bounds on the entanglement of formation, the tangle, and the concurrence of
4N mixed states. The maps are the partial transpose map and the phi map introduced by Breuer [H.-P. Breuer, Phys. Rev. Lett. 97, 080501 (2006)]. The norm-based entanglement measures constructed from these two maps, called negativity and phi negativity, respectively, lead to two sets of bounds on the entanglement of formation, the tangle, and the concurrence. We compare these bounds and identify the sets of
4N density operators for which the bounds from one constraint are better than the bounds from the other. In the process, we present a derivation of the already known bound on the concurrence based on the negativity. We compute bounds on the three measures of entanglement using both the constraints simultaneously. We demonstrate how such doubly constrained bounds can be constructed. We discuss extensions of our results to bipartite states of higher dimensions and with more than two constraints.
More than two multipartite orthogonal states cannot always be discriminated if only local operations and classical communication (LOCC) are allowed. We show that four Bell states cannot be discriminated by LOCC, even probabilistically, using the separability properties of a four-party unlockable bound entangled state. Using an existing inequality among the measures of entanglement, we show that any three Bell states cannot be discriminated with certainty by LOCC. Exploiting the inequality, we calculate the distillable entanglement of a certain class of 4 multiply sign in circle 4 mixed states.
We develop MIQCR-FR, a framework for exact convex reformulation from facially reduced semidefinite relaxations of equality-constrained binary quadratic problems. Standard mixed-integer quadratic convex reformulation (MIQCR) uses dual multipliers of the original relaxation. Facial reduction (FR) reduces the semidefinite matrix order, but recovery of the original dual multipliers is not guaranteed in general. For this class of problems, we identify how retaining the linear equalities and their products with each binary variable enables this recovery. An established dual recovery formula gives an explicit extension of every feasible solution of the reduced dual to the original dual, preserving its bound. For approximate multiplier estimates, we give a spectral shift restoring dual feasibility and two constructions optimizing different bounds within a restricted multiplier family. Maximizing the dual bound yields a trust-region subproblem; maximizing the continuous MIQCR bound requires only a reduced eigenvalue calculation. The latter construction enforces convexity directly, rather than full dual feasibility. We implement it using multiplier estimates from the alternating direction method of multipliers (ADMM). On 431 instances from six problem families, the complete method certifies 296 optima, compared with 156 for the conic-bundle method MIQCR-CB and 187 for direct Gurobi under a common time budget.