Bounds on action of local quantum channels

Stelmachovic, Peter; Buzek, Vladimir

Bounds on action of local quantum channels

Authors

Description

We derive an upper bound on the action of a direct product of two quantum maps (channels) acting on multi-partite quantum states. We assume that the individual channels $Λ_j$ affect single-particle states so, that for an arbitrary input $ρ_j$, the distance $D_j (Λ_j [ ρ_j ], ρ_j)$ between the input $ρ_j$ and the output $Λ_j [ ρ_j ]$ of the channel is less than $ε$. Given this assumption we show that for an arbitrary {\em separable} two-partite state $ρ_{12}$ the distance between the input $ρ_{12}$ and the output $Λ_1\otimesΛ_2[ρ_{12} ]$ fulfills the bound $D_{12} (Λ_1 \otimes Λ_2 [ ρ_{12} ], ρ_{12}) \leq \sqrt{2+ 2 \sqrt{(1-1/d_1)(1-1/d_2)}} ε$ where $d_1$ and $d_2$ are dimensions of first and second quantum system respectively. On the contrary, entangled states are transformed in such a way, that the bound on the action of the local channels is $D_{12} (Λ_1 \otimes Λ_2 [ ρ_{12} ], ρ_{12}) \leq 2 \sqrt{2 - 1/d} \: ε$, 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 into two different bounds which in turn can be exploited for a discrimination between the two sets of states. We generalize our results to multi-partite channels.