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

## TL;DR

This paper establishes a fundamental lower bound on the sum of operation entropies of a quantum channel and its complement, revealing a trade-off between information retained by the system and leaked to the environment.

## Contribution

It introduces a universal lower bound on the sum of entropies for quantum channels and their complements, advancing understanding of information trade-offs in quantum operations.

## Key findings

- Sum of entropies is bounded from below for any dimension.
- Characterizes the trade-off between system information and environmental leakage.
- Describes depolarising maps that achieve the lower boundary for qubit channels.

## Abstract

The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi-Jamio{\l}kowski state, characterizes the coupling of the principal system with the environment. For any quantum channel $\Phi$ acting on a state of size $N$ one defines the complementary channel $\tilde \Phi$, 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, $S(\Phi)+ S(\tilde \Phi)$, 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.