Entanglement growth in monitored systems. In particular we are interested in understanding how measurements during the dynamics can reduce the way in

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Generically, the entanglement entropy of a quantum system grows under unitary dynamics (non interacting or interacting). Measurements of the systems local observables tends to reduce entanglement. It is a natural question to ask how do these two effects compete with one another and what is the resulting dynamics of the systems entanglement. This has been previously studied and is well understood in certain setups (Random unitary circuits with random projective measurements and free fermions with continuous local density monitoring for example) and has been shown to lead to interesting phenomena, e.g. entanglement phase transitions. It had been previously shown that for the free fermion example, the effective description of the dynamics is that of non-interacting bosons evolving with a non-hermitian effective Hamiltonian. Only the infinite time steady state under these dynamics for local measurements of lattice observables has been studied so we are interested in developing a dynamical description of the entanglement growth. We would also like to understand how the entanglement dynamics is different when the observables measured are coarse-grained over a number of lattice sites since the more coarse-grained the observables measured are the less information is gained about the system hence the entanglement is not reduced as much. The effective dynamics are non-interacting hence if the system is initialised in what is known as a Gaussian state, then the description of the dynamics can be reduced to a set of non-linear first order matrix differential equations for the correlation matrix . The entanglement dynamics can be computed using the correlation matrix using a well established tool called the 'quasiparticle picture' .