Exact entanglement cost of quantum states and channels under positive-partial-transpose-preserving operations

This paper establishes single-letter formulas for the exact entanglement cost of simulating quantum channels under free operations that completely preserve positivity partial transpose (PPT). First, we introduce $\ensuremath{\kappa}$-entanglement measure point-to-point channels, based on idea $\ensuremath{\kappa}$ bipartite states, and establish several fundamental properties it, including amortization collapse, monotonicity PPT superchannels, additivity, normalization, faithfulness, nonconvexity. Second, solve in both parallel sequential settings, along with assistance PPT-preserving operations. In particular, cases is given by same formula, a channel. We further show this equal to largest can be shared or generated sender receiver formula calculable semidefinite program, thus allowing an efficiently computable solution general channels. Noting regime more powerful than regime, another notable implication our result regimes have power channel simulation, when superchannels are free. For basic Gaussian Holevo-Werner [Holevo Werner, Phys. Rev. A 63, 032312 (2001)], giving operational meaning quantity these