Environment-parametrized channels: trade-off between secret-key capacity and quantum Fisher information

Quantum channels serve as the fundamental carriers of quantum information. Under suitable conditions, they allow for informational-theoretical secure communication. For the effective characterization and engineering of quantum channels, determining the secret-key capacity (SKC) is highly beneficial. Here, we characterize models of dissipative and dephasing quantum channels obtained from an interaction with a sub-Ohmic, Ohmic, and super-Ohmic environment. We estimate a lower bound on the SKC using the relative entropy of entanglement (REE) of the associated Choi matrix. In some cases, for example if the channel is teleportation covariant, the lower bound becomes an equality. This approach is also applied to estimate the quantum Fisher information (QFI) for the estimation of the parameters that characterize the channel, e.g., the cutoff frequency of the channel environment. By comparing the REE and the QFI of the Choi matrix, we establish an analytical link between the two quantities, revealing an inverse relationship. This trend is observed for both dissipative and dephasing channels and across the Ohmic, sub-Ohmic, and super-Ohmic regimes. With increasing noise in the quantum channel, more information is lost into the environment, leading to reduced SKC. In turn, more information from the environment enters into the channel, allowing to estimate the environmental parameters with higher precision, thus increasing the associated QFI.