Activating entanglement and EPR steering from continuous-variable resources using witness-based measures

We introduce a general witness-based framework for quantifying and operationally activating continuous-variable (CV) resources into discrete-variable (DV) bipartite entanglement or Einstein- Podolsky-Rosen (EPR) steering. For the three standard CV resource theories associated with Wigner negativity (WN), genuine non-Gaussianity (GNG), and standard non-Gaussianity (SNG), we define infinite families of bounded-witness monotones indexed by box constraints on the witness operators. For closed convex free sets, these monotones are faithful, strongly monotonic under free instruments, Lipschitz continuous, and convex. For closed nonconvex free sets, we show that faithfulness requires a two-copy lift and formulate the corresponding strong-monotonicity statement in the lifted theory. We further construct witness-dependent completely positive trace-preserving (CPTP) measure-and-prepare channels whose outputs are two-qubit Werner states. For the representative case n = m = 1, the optimal entanglement and EPR steering attainable within this witness-dependent activation family are exactly proportional to the underlying monotones. We illustrate the framework with odd-parity states, pure-loss single-photon states, and Gottesman- Kitaev-Preskill (GKP) states, and derive explicit lower bounds for pure-state GNG and SNG. More broadly, our results show that closed CV free sets admit witness-based quantifiers with a direct operational interpretation in terms of experimentally accessible DV correlations.