Finite Gaussian assistance protocols and a conic metric for extremizing spacelike vacuum entanglement

In a pure Gaussian tripartition, a range of entanglement between two parties ($AB$) can be purified through classical communication of Gaussian measurements performed within the third ($C$). To begin, this work introduces a direct method to calculate a hierarchic series of projective $C$ measurements for the removal of any $AB$ Gaussian noise, circumventing divergences in prior protocols. Next, a multimode conic framework is developed for pursuing the maximum (Gaussian entanglement of assistance, GEOA) or minimum (Gaussian entanglement of formation, GEOF) pure entanglement that may be revealed or required between $AB$. Within this framework, a geometric necessary and sufficient entanglement condition emerges as a doubly-enclosed conic volume, defining a novel distance metric for conic optimization. Extremizing this distance for spacelike vacuum entanglement in the massless and massive free scalar fields yields (1) the highest known lower bound to GEOA, the first that decays slower than the two-point correlation functions and (2) the lowest known upper bound to GEOF, the first that decays exponentially mirroring the mixed $AB$ negativity. Furthermore, combination of the above with a generalization of previous partially-transposed noise filtering techniques allows calculation of a single $C$ measurement that maximizes the purified $AB$ entanglement. Beyond expectation that these behaviors of spacelike GEOA and GEOF persist in interacting theories, the present measurement and optimization techniques are applicable to physical many-body Gaussian states beyond quantum fields.