[Paper Review] Dynamics of Renyi entanglement entropy in local quantum circuits with charge conservation

Yichen Huang|arXiv (Cornell University)|Feb 3, 2019

Quantum many-body systems30 references56 citations

TL;DR

This paper proves that in local quantum circuits with charge conservation, the Rényi entanglement entropy Rα (for α > 1) grows at most as O(√t ln t) when charge transport is diffusive. The result is derived via a Schmidt rank-based analysis and the Eckart-Young theorem, showing that Rényi entropy growth is a probe of transport dynamics—contrasting with the linear growth of von Neumann entropy. The bound is tight, as recent numerical results confirm saturation up to sub-logarithmic corrections.

ABSTRACT

In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_\alpha$. We rigorously prove that $R_\alpha$ with Renyi index $\alpha>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.

Motivation & Objective

• To establish rigorous upper bounds on Rényi entanglement entropy Rα (α > 1) in local quantum circuits with charge conservation.
• To demonstrate that the growth of Rα serves as a probe of quantum transport, particularly under diffusive dynamics.
• To extend the bound to sub- and super-diffusive transport regimes via scaling analysis.
• To connect theoretical bounds with recent numerical results showing saturation of the bound in random local circuits.

Proposed method

• Uses a bipartite spin chain with subsystem A at the center, initialized in random product states in the σx basis.
• Applies a projection P onto states with |00⟩ at the central cut, exploiting the diffusive spread of charge to bound the overlap with the initial state.
• Constructs a modified circuit V(t,0) that preserves entanglement structure while approximating the original U(t,0) with error bounded by e−Ω(m²/t).
• Employs the Eckart-Young theorem to bound the largest Schmidt coefficient λ1 of the evolved state, linking it to the Rényi entropy via R∞(ρA) = −ln Λ1.
• Uses Markov’s inequality to show that with high probability (≥1−1/p(t)), the state remains close to the projected subspace, ensuring the bound holds for most initial states.
• Optimizes the region size m = O(√t ln t) to minimize the entropy bound, balancing error and entanglement growth.

Experimental results

Research questions

• RQ1 What is the upper bound on Rényi entanglement entropy Rα (α > 1) in local quantum circuits with charge conservation under diffusive transport?
• RQ2 How does the growth of Rα relate to the underlying transport properties of the system?
• RQ3 Can the bound be extended to sub- or super-diffusive transport regimes?
• RQ4 Is the derived upper bound tight, and does it match numerical observations?

Key findings

• The Rényi entanglement entropy Rα (α > 1) is bounded above by O(√t ln t) when charge transport is diffusive, with high probability ≥1−1/p(t).
• The bound is saturated (up to sub-logarithmic corrections) in random local quantum circuits with charge conservation, as confirmed by recent numerical results.
• The growth of Rα is fundamentally tied to transport: it grows sub-linearly (O(√t ln t)) under diffusive transport, contrasting with the linear growth of von Neumann entropy.
• For sub- or super-diffusive transport with scaling distance ∼tz (0 < z < 1), the bound generalizes to O(tz poly ln t).
• The proof relies on the Schmidt decomposition and the Eckart-Young theorem to bound the largest Schmidt coefficient, which controls the min-entropy and hence Rα.
• The result holds under general diffusive transport, without requiring randomness in the local unitaries, making it robust to circuit structure.