Logarithmic negativity in out-of-equilibrium open free-fermion chains: An exactly solvable case

Abstract

We derive the quasiparticle picture for the fermionic logarithmic negativity in a tight-binding chain subject to gain and loss dissipation. We focus on the dynamics after the quantum quench from the fermionic Néel state. We consider the negativity between both adjacent and disjoint intervals embedded in an infinite chain. Our result holds in the standard hydrodynamic limit of large subsystems and long times, with their ratio fixed. Additionally, we consider the weaklydissipative limit, in which the dissipation rates are inversely proportional to the size of the intervals. We show that the negativity is proportional to the number of entangled pairs of quasiparticles that are shared between the two intervals, as is the case for the mutual information. Crucially, in contrast with the unitary case, the negativity content of quasiparticles is not given by the Rényi entropy with Rényi index 1/2, and it is in general not easily related to thermodynamic quantities.

Logarithmic entanglement scaling in dissipative free-fermion systems

“…However, in the presence of an environment, the global state is mixed, which implies that neither the von Neumann entropy nor the mutual information are proper measures of the entanglement shared between different regions. Still, it has been shown recently that for out-of-equilibrium free-fermion and freeboson models in the presence of quadratic global dissipation the mutual information admits a hydrodynamic description in terms of a quasiparticle picture [24][25][26][27]. Moreover, we numerically checked that our findings remain qualitatively valid using the fermionic entanglement negativity, which, on the other hand, is a proper measure of entanglement [28,29].…”

“…In writing (27) we assumed the matrix K kq := b † k b q to be diagonal and the matrix F kq := b k b q to be zero, which will turn out to be true in our formalism (see Sec. III).…”

Enhanced entanglement negativity in boundary driven monitored fermionic chains

“…On the other hand, Eq. ( 24) does not generally hold for non-unitary evolution, as shown for non-interacting fermionic chains with dephasing noise [102]. Motivated by these discussions, we have probed the validity of Eq.…”

“…Contrary to the latter, the fermionic negativity can be computed efficiently for fermionic Gaussian states [86], while it is also a genuine entanglement monotone [87]. In the context of Lindbladian dynamics of non-interacting fermionic chains, the fermionic negativity has been extensively investigated [101,102], and its behavior has been understood based on a semi-classical quasi-particle picture. Here, we show that qualitative differences arise in the presence of monitoring.…”

“…The formula Eq. (C5) holds for any quadratic fermionic system [102]. For the specific instance considered in the main text, and for the initial condition c † a c † b = 0, the computation of the negativity can be simplified.…”