Boundary time crystals exhibit measurement-induced phase transitions in their steady-state entanglement, with critical behavior that depends on the particular unraveling of the Lindblad dynamics. In this work, we investigate another key measure of quantum complexity -- nonstabilizerness (or ``magic'') -- and show that it follows a markedly different pattern. Importantly, in contrast to entanglement, for large system sizes, nonstabilizerness remains invariant under different unraveling schemes -- a property we attribute to the inherent permutational symmetry of the model. Although the steady-state stabilizer entropy does not display a genuine phase transition, it exhibits a singular derivative (a cusp) at the mean-field critical point. Furthermore, we demonstrate that finite-size simulations of the average Lindblad evolution fail to capture the asymptotic behavior of nonstabilizerness in the time-crystal phase, while quantum trajectory unravelings correctly reveal its extensive scaling with system size. These findings offer insights into how different quantum resources manifest in open systems.
Boundary time crystals exhibit measurement-induced phase transitions in their steady-state entanglement, with critical behavior that depends on the particular unraveling of the Lindblad dynamics. In this work, we investigate another key measure of quantum complexity -- nonstabilizerness (or ``magic'') -- and show that it follows a markedly different pattern. Importantly, in contrast to entanglement, for large system sizes, nonstabilizerness remains invariant under different unraveling schemes -- a property we attribute to the inherent permutational symmetry of the model. Although the steady-state stabilizer entropy does not display a genuine phase transition, it exhibits a singular derivative (a cusp) at the mean-field critical point. Furthermore, we demonstrate that finite-size simulations of the average Lindblad evolution fail to capture the asymptotic behavior of nonstabilizerness in the time-crystal phase, while quantum trajectory unravelings correctly reveal its extensive scaling with system size. These findings offer insights into how different quantum resources manifest in open systems.
We investigate the entanglement dynamics along quantum trajectories during the seeding of time-crystalline order in a boundary time crystal (BTC). Specifically, how entanglement spreads among different spin ensembles when a BTC attempts to seed its time-crystalline behavior onto otherwise static spin ensembles, through a collective dissipative channel. We analyse both the dynamical growth of entanglement in time and the steady-state properties of the system. Our results reveal two fundamentally distinct regimes. In the seeded BTC phase, the steady-state entanglement entropy between the ensembles grows with system size N, accompanied by macroscopic fluctuations along the trajectories. In contrast, in the non-seeded static phase, both the steady-state entanglement and its fluctuations decay exponentially with N. The model thus features a measurement-induced phase transition (MIPT) driven by the seeding mechanism. Furthermore, these findings establish dissipative seeding as a powerful mechanism for controlling quantum correlations in open many-body systems, with direct experimental relevance to this class of model without a postselection barrier.
Nonstabilizerness, also known as "magic," quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this Letter, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer Rényi entropy (SRE). Leveraging a phenomenological description based on the ℓ-bit model, we analytically and numerically demonstrate that interactions profoundly influence nonstabilizerness spreading, inducing a power-law growth of SRE that markedly contrasts with the rapid saturation observed in ergodic systems. We validate our theoretical predictions through numerical simulations of the disordered transverse-field Ising model, showing excellent agreement across various disorder strengths, system sizes, and initial states. Additionally, we uncover a universal relationship between SRE and entanglement entropy, revealing their common scaling in the MBL regime independent of disorder strength and system size. Our results offer critical insights into the interplay of disorder, interactions, and complexity in quantum many-body systems.
Magic, or nonstabilizerness, is a crucial quantum resource, yet its dynamics in open quantum systems remain largely unexplored. We investigate magic in the open XXZ spin chain under either boundary gain and loss or bulk dephasing using the stabilizer Rényi entropy M 2 . To enable scalable simulations of large systems, we develop a unique, highly efficient algorithm for computing M 2 within the matrix product state formalism while maintaining constant bond dimension—an advancement over existing methods. For boundary driving, we uncover universal scaling laws, M 2 ( t ) ∼ t 1 / z , linked to the dynamical exponent z for several distinct universality classes. We also disentangle classical and quantum contributions to magic by introducing a mean-field approximation for magic, thus emphasizing the prominent role of quantum-critical fluctuations in nonstabilizerness. For bulk dephasing, dissipation can transiently enhance magic before suppressing it and drive it to a nontrivial steady-state value. These findings position magic as a powerful diagnostic tool for probing universality and dynamics in open quantum systems.
We demonstrate the existence of a dynamical quantum phase transition (DQPT) in a dissipative collective-spin model that exhibits the boundary time crystal (BTC) phase. We initialize the system in the ground state of the Hamiltonian in either the BTC or the non-BTC phase, and drive it across the BTC transition. The driving is done by an abrupt quench or by a finite-time linear ramp of a Hamiltonian control parameter under Markovian Lindblad dynamics. We diagnose DQPTs through zeros of the fidelity-based Loschmidt echo between the initial state and the evolving mixed state, which induce nonanalytic cusp-like features in the associated rate function. For quenches into the BTC phase, the Loschmidt echo exhibits repeated zeros due to the emergent time-periodic steady state, whereas for quenches into the non-BTC phase, the overlap vanishes and remains zero once the dynamics relaxes to a stationary state. We further show that the DQPT persists under the ramp protocol followed by unitary evolution with the final Hamiltonian. Finally, we analyze the finite-size scaling of the first critical time and find convergence to a constant in the thermodynamic limit, with distinct power-law approaches for the quench and the ramp protocols.
Boundary time crystals are a class of exotic dissipative quantum phases that spontaneously break continuous time-translation symmetry in the thermodynamic limit of open quantum systems. In finite-size systems, the long-time evolution of boundary time crystals exhibits decaying oscillations that cannot be captured by widely used mean-field theory. To address this issue, we develop an effective approach called the stroboscopic rotating wave approximation, which provides a well-approximated state for the long-time evolution of boundary time crystals in the strongly driven regime. In this approach, the order parameter exhibits both a long-time decaying envelope governed by an effective Lindblad superoperator and short-time oscillations dominated by a reduced quantum dynamical semigroup. Our results reveal that the competition among dephasing processes along three directions induces persistent oscillations, marking the emergence of the boundary-time-crystal phase. We obtain the analytical expressions for the steady-state density operator, the oscillation period, and the decay rate of the order parameter when the coherent energy splitting exceeds the dissipation rate. Our work provides a beyond-mean-field tool for studying the dynamics of periodically driven open quantum systems and understanding the formation of time crystals.
Complex quantum systems -- composed of many, interacting particles -- are intrinsically difficult to model. When a quantum many-body system is subject to disorder, it can undergo transitions to regimes with varying non-ergodic and localized behavior, which can significantly reduce the number of relevant basis states. It remains an open question whether such transitions are also directly related to an abrupt change in the system's complexity. In this work, we study the transition from chaotic to integrable phases in several paradigmatic models, the power-law random banded matrix model, the Rosenzweig--Porter model, and a hybrid SYK+Ising model, comparing three complementary complexity markers -- fractal dimension, von Neumann entanglement entropy, and stabilizer Rényi entropy. For all three markers, finite-size scaling reveals sharp transitions between high- and low-complexity regimes, which, however, can occur at different critical points. As a consequence, while in the ergodic and localized regimes the markers align, they diverge significantly in the presence of an intermediate fractal phase. Additionally, our analysis reveals that the stabilizer Rényi entropy is more sensitive to underlying many-body symmetries, such as fermion parity and time reversal, than the other markers. As our results show, different markers capture complementary facets of complexity, making it necessary to combine them to obtain a comprehensive diagnosis of phase transitions. The divergence between different complexity markers also has significant consequences for the classical simulability of chaotic many-body systems.