Quantum Entanglement Growth Under Random Unitary Dynamics (1608.06950v1)

Abstract: Characterizing how entanglement grows with time in a many-body system, for example after a quantum quench, is a key problem in non-equilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time--dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the entanglement tsunami' in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar--Parisi--Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like $(\text{time})^{1/3}$ and are spatially correlated over a distance $\propto (\text{time})^{2/3}$. We derive KPZ universal behaviour in three complementary ways, by mapping random entanglement growth to: (i) a stochastic model of a growing surface; (ii) aminimal cut' picture, reminiscent of the Ryu--Takayanagi formula in holography; and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple minimal cut' picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of thevelocity' of entanglement growth in the 1D `entanglement tsunami'. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder.

• The paper reveals that entanglement growth in 1D systems exhibits KPZ universality with linear trends and power-law fluctuations.
• It employs a trifaceted theoretical approach including stochastic growth, minimal cut analogies, and hydrodynamic operator spreading to validate predictions.
• Numerical simulations of random unitary circuits confirm the KPZ scaling exponents, offering insights for quantum simulations and information spread.

Quantum Entanglement Growth Under Random Unitary Dynamics

The paper "Quantum Entanglement Growth Under Random Unitary Dynamics" presents a thorough analysis of entanglement dynamics in many-body quantum systems subjected to random unitary evolution. This research addresses a central question in non-equilibrium quantum physics: how entanglement evolves over time, particularly after a quantum quench. The paper highlights the behavior of entanglement growth both with time-dependent noise and through random quantum circuits, uncovering a universal pattern in noisy entanglement evolutions.

Summary of Findings

1. KPZ Universality Class:
   • The paper reveals that in one-dimensional systems, entanglement growth conforms to the Kardar--Parisi--Zhang (KPZ) universality class, a paradigmatic model for surface growth in classical statistical mechanics.
   • The entanglement entropy exhibits linear growth on average, accompanied by fluctuations following a power law $(\text{time})^{1/3}$ and spatial correlations extending over distances scaling as $(\text{time})^{2/3}$.
2. Trifaceted Theoretical Approach:
   • The paper derives KPZ behavior from three models: (i) a stochastic surface growth analogy, (ii) a minimal cut analogous to Ryu--Takayanagi's holographic principle, and (iii) a hydrodynamic perspective based on operator spreading dynamics.
   • These mappings not only confirm KPZ universality but also bridge different conceptual frameworks in physics, enriching our understanding of entanglement dynamics.
3. Numerical Simulations:
   • Detailed simulations using random unitary circuits confirm the theoretical predictions of KPZ scaling in entanglement entropy. Both the mean behavior and fluctuations match the expected critical exponents.
4. Generalizations to Higher Dimensions:
   • The paper extends its analysis to higher dimensions, depicting the growth of entanglement as a membrane pinned by disorder, and discusses the potential dimensional crossovers in scaling behavior.
   • In higher dimensions, the paper predicts a broader range of possible scaling behaviors, potentially leading to different universal characteristics.
5. Implications for Non-Noisy Hamiltonian Dynamics:
   • While the focus is on noisy dynamics, the paper conjectures extension to pure Hamiltonian systems. It argues that the deterministic features, captured by a "minimal membrane" in spacetime, may remain applicable.

Implications and Future Directions

The implications of this work stretch beyond theoretical interest, crossing into practical applications.

• Quantum Simulations and Computation: Understanding entanglement dynamics underpins future efforts to use and manipulate entangled states in quantum devices robust against noise.
• Thermalization and Quantum Chaos: The insights into entanglement spreading velocities contribute to our understanding of thermalization processes in non-equilibrium quantum states.
• Expanding Beyond Noisy Systems: The exploration of noisy systems provides a comparative ground for studying Hamiltonian systems, opening up investigations into whether similar dynamical universality might manifest in the absence of stochastic elements.

Conclusion

The paper makes a significant contribution by identifying the universal properties of entanglement growth under random unitary dynamics and proposing a physical model extending across traditional quantum dynamics frameworks. The use of random dynamics as a theoretical laboratory allows for the derivation of universal scaling laws that elucidate the many-body dynamics of complex quantum systems. This line of research is expected to spur further inquiries into both random and non-random quantum dynamics, potentially uncovering new universal principles governing quantum information spread in many-body systems.