Bounds on action of local quantum channels

We derive an upper bound on the action of a direct product of two quantum maps (channels) acting on bi-partite quantum states. We assume that the individual channels Λj affect single-particle states so that for an arbitrary input ρj, the distance Dj(Λj[ρj], ρj) between the input ρj and the output Λj[ρj] of the channel is less than ε. Given this assumption we show that for an arbitrary separable two-partite state ρ12, the distance between the input ρ12 and the output Λ1 x Λ2[ρ12] fulfils the bound D12(Λ1 x Λ2 [ρ12], ρ2) ≤ √(2 + 2√(91-1/d1)(1-1/d2))) where d1 and d2 are the dimensions of the first and second quantum system respectively. In contrast, entangled states are transformed in such a way that the bound on the action of the local channels is D12(Λ1 x Λ2 [ρ12], ρ12) ≤ 2√(2-1/d)ε, where d is the dimension of the smaller of the two quantum systems passing through the channels. Our results show that the fundamental distinction between the set of separable and the set of entangled states results in two different bounds which in turn can be exploited for discrimination between the two sets of states. We generalize our results to multi-partite channels