Arbitrarily large link key, zero repeater key: two PPT barriers in quantum networks

Two quantum links can each supply arbitrarily much secret key and still supply none to their remote endpoints through an untrusted repeater. We prove that this obstruction is exact whenever the two links are independent and have positive partial transpose (PPT), even with collective processing of all copies, two-way public communication and postselection. Every positive-probability endpoint output remains at least 1/5 in unhalved trace norm from every nonempty private state, independently of block length, local memory and shield dimension. Explicit link families therefore realise KD(ρr) → ∞ while Krep(ρr, ρr) = 0. For a network of independent edge states, let dP(a,b) count the fewest PPT edges along any terminal path. The condition dP(a,b) ≥ 2 is necessary and sufficient for the PPT labels alone to force zero endpoint key for every compatible resource assignment. The threshold is sharp: one fixed pair of eight-dimensional link resources gives positive key exactly when dP(a,b) ≤ 1. The proof transfers the four-effect inequality for local PPT maps to complete repeater operations and reduces every blocked network to two PPT boundary bundles. Combined with the known failure of the PPT-square conjecture, the result separates entanglement swapping from private connectivity: an entangled endpoint state can survive two PPT links, while secret key remains impossible. Exact Markov sources and normal local operations on separable Hilbert spaces obey corresponding extensions.