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We prove an analytic positive lower bound for the geometric distance betweenentangled positive partial transpose (PPT) states of a broad class and anyprivate state that delivers one secure key bit. Our proof holds for any Hilbertspace of finite dimension. Although our result is proven for a specific classof PPT states, we show that our bound nonetheless holds for all known entangledPPT states with non-zero distillable key rates whether or not they are in ourspecial class.

On the geometric distance between quantum states with positive partial transposition and private states