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We study the dynamics of quantum information and of quantum correlationsafter a quantum quench, in transverse field Ising chains subject to genericlinear dissipation. As we show, in the hydrodynamic limit of long times, largesystem sizes, and weak dissipation, entropy-related quantities -- such as thevon Neumann entropy, the R\'enyi entropies, and the associated mutualinformation -- admit a simple description within the so-called quasiparticlepicture. Specifically, we analytically derive a hydrodynamic formula, recentlyconjectured for generic noninteracting systems, which allows us to demonstratea universal feature of the dynamics of correlations in such dissipativenoninteracting system. For any possible dissipation, the mutual informationgrows up to a time scale that is proportional to the inverse dissipation rate,and then decreases, always vanishing in the long time limit. In passing, weprovide analytic formulas describing the time-dependence of arbitrary functionsof the fermionic covariance matrix, in the hydrodynamic limit.

Hydrodynamics of quantum entropies in Ising chains with linear dissipation