We study the dynamics of entanglement in the quantum Ising chain with dephasing dissipation in a Lindblad master equation form. We consider two unravelings which preserve the Gaussian form of the state, allowing us to address large system sizes. The first unraveling gives rise to a quantum-state-diffusion dynamics, while the second one describes a specific form of quantum-jump evolution, suitably constructed to preserve Gaussianity. In the first case we find a crossover from area-law to logarithm-law entanglement scaling and draw the related phase diagram. In the second case we only find logarithm-law scaling, remarking on the different entanglement behavior for different unravelings of the same Lindblad equation. Finally, we compare these outcomes with the predictions of a non-Hermitian Hamiltonian evolution, finding conflicting results.
We study the dynamics of entanglement in the quantum Ising chain with dephasing dissipation in a Lindblad master equation form. We consider two unravelings which preserve the Gaussian form of the state, allowing us to address large system sizes. The first unraveling gives rise to a quantum-state-diffusion dynamics, while the second one describes a specific form of quantum-jump evolution, suitably constructed to preserve Gaussianity. In the first case we find a crossover from area-law to logarithm-law entanglement scaling and draw the related phase diagram. In the second case we only find logarithm-law scaling, remarking on the different entanglement behavior for different unravelings of the same Lindblad equation. Finally, we compare these outcomes with the predictions of a non-Hermitian Hamiltonian evolution, finding conflicting results.
Recent theoretical work has shown that the competition between coherent unitary dynamics and stochastic measurements, performed by the environment, along wave function trajectories can give rise to transitions in the entanglement scaling. In this work, complementary to these previous studies, we analyze a situation where the role of Hamiltonian and dissipative dynamics is reversed. We consider an engineered dissipation, which stabilizes an entangled phase of a quantum $\mathrm{spin}\text{\ensuremath{-}}\frac{1}{2}$ chain, while competing single-particle or interacting Hamiltonian dynamics induce a disentangled phase. Focusing on the single-particle unitary dynamics, we find that the system undergoes an entanglement transition from a logarithmic growth with system size to an area law when the competition ratio between the unitary evolution and the nonunitary dynamics increases. We evidence that the transition manifests itself in state-dependent observables at a finite competition ratio for Hamiltonian and measurement dynamics. On the other hand, it is absent in trajectory-averaged steady-state dynamics, governed by a Lindblad master equation: although purely dissipative dynamics stabilizes an entangled state, for any nonvanishing Hamiltonian contribution the system ends up irremediably in a disordered phase. In addition, a single trajectory analysis reveals that the distribution of the entanglement entropy constitutes an efficient indicator of the transition. Complementarily, we explore the competition of the dissipation with coherent dynamics generated by an interacting Hamiltonian, and demonstrate that the entanglement transition also occurs in this second model. Our results suggest that this type of transition takes place for a broader class of Hamiltonians, underlining its robustness in monitored open quantum many-body systems.
Dissipation generally leads to the decoherence of a quantum state. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. While the focus of these works has been primarily on engineering the nonequilibrium steady state, we investigate the buildup of entanglement in the quantum trajectories. Specifically, we analyze the competition between two different dissipation channels arising from two incompatible continuous monitoring protocols. The first protocol locks the phase of neighboring sites upon registering a quantum jump, thereby generating a long-range entanglement through the system, while the second destroys the coherence via a dephasing mechanism. By studying the unraveling of stochastic quantum trajectories associated with the continuous monitoring protocols, we present a transition for the scaling of the averaged trajectory entanglement entropies, from critical scaling to area-law behavior. Our work provides an alternative perspective on the measurement-induced phase transition: the measurement can be viewed as monitoring and registering quantum jumps, offering an intriguing extension of these phase transitions through the long-established realm of quantum optics.
We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivial dynamic scaling behavior, which is realized when the dissipation parameter u (globally controlling the decay rate of the dissipation within the Lindblad framework) scales as the energy difference $\mathrm{\ensuremath{\Delta}}$ of the lowest levels of the Hamiltonian, i.e., $u\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$. However, unlike continuous quantum transitions where $\mathrm{\ensuremath{\Delta}}$ is power-law suppressed, at first-order quantum transitions $\mathrm{\ensuremath{\Delta}}$ is exponentially suppressed with increasing the system size (provided the boundary conditions do not favor any particular phase).
The competition between unitary quantum dynamics and dissipative stochastic effects, as emerging from continuous-monitoring processes, can culminate in measurement-induced phase transitions. Here, a many-body system abruptly passes, when exceeding a critical measurement rate, from a highly entangled phase to a low-entanglement one. We consider a different perspective on entanglement phase transitions and explore whether these can emerge when the measurement process itself is modified, while keeping the measurement rate fixed. To illustrate this idea, we consider a noninteracting fermionic system and focus on diffusive detection processes. Through extensive numerical simulations, we show that, upon varying a suitable unraveling parameter---interpolating between measurements of different quadrature operators---the system displays a transition from a phase with area-law entanglement to one where entanglement scales logarithmically with the system size. Our findings may be relevant for tailoring quantum correlations in noisy quantum devices and for conceiving optimal classical simulation strategies.
The evolution of a system coupled to baths is commonly described by a master equation that, in the long-time limit, yields a steady-state density matrix. However, when the same evolution is unraveled into quantum trajectories, it is possible to observe a transition in the scaling of entanglement within the system as the system-bath coupling increases - a phenomenon that is invisible in the trajectory-averaged reduced density matrix of the system. Here, we go beyond the paradigm of trajectories from master equations and explore whether a qualitatively analogous entanglement-scaling transition emerges in a single unitary evolution of the combined system-bath setup, without monitoring the dynamics of the system. We investigate the scaling of entanglement in a unitary quantum setup composed of a two-dimensional lattice of free fermions, where each site is coupled to a fermionic bath. As the system-bath coupling increases, the logarithmic fermionic negativity reveals an entanglement transition from logarithmic-law to area-law scaling. This occurs while the system's steady-state properties are trivial, highlighting that the signatures of these different scalings are within the bath-bath correlations. Evidence of the transition is also found in the mutual information and the correlations of the full system-bath setup, suggesting that the entanglement transition is underpinned by a change in the spatial structure of quantum information.
We explain how to apply a Gaussian-preserving operator to a fermionic Gaussian state. We use this method to study the evolution of the entanglement entropy of an Ising spin chain undergoing a quantum-jump dynamics with string measurement operators. We find that, for finite-range string operators, the asymptotic entanglement entropy exhibits a crossover from a logarithmic to an area-law scaling with the system size, depending on the Hamiltonian parameters as well as on the measurement strength. For ranges of the string which scale extensively with the system size, the asymptotic entanglement entropy shows a volume-law scaling, independently of the measurement strength and the Hamiltonian dynamics. The same behavior is observed for the measurement-only dynamics, suggesting that measurements may play a leading role in this context.