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Trade--off relations for operation entropy of complementary quantum channels

The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi-Jamiołkowski state, characterizes the coupling of the principal system with the environment. For any quantum channel $Φ$ acting on a state of size $N$ one defines the complementary channel $\tilde Φ$, which sends the input state into the state of the environment after the operation. Making use of subadditivity of entropy we show that for any dimension $N$ the sum of both entropies, $S(Φ)+ S(\tilde Φ)$, is bounded from below. This result characterizes the trade-off between the information on the initial quantum state accessible to the principal system and the information leaking to the environment. For one qubit maps, $N=2$, we describe the interpolating family of depolarising maps, for which the sum of both entropies gives the lower boundary of the region allowed in the space spanned by both entropies.