The dynamical $\alpha$-Rényi entropies of local Hamiltonians grow at most linearly in time

Abstract

We consider a generic one dimensional spin system of length $L$, arbitrarily large, with strictly local interactions, for example nearest neighbor, and prove that the dynamical $\alpha$-Rényi entropies, $0<\alpha \le 1$, of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical $\alpha$-Rényi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of $\alpha$ close enough to $1$, and moreover to initial pure states with low entanglement, of order $\log L$, that are typically represented by critical states. We establish that low entanglement states have an efficient MPS representation that persists at least up to times of order $\log L$. The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on $\alpha$-Rényi entropies and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.

v4: minor improvements throughout, accepted by PRX

References in corpus (25)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Matrix product states represent ground states faithfully
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Entanglement negativity in quantum field theory
• Entropy scaling and simulability by Matrix Product States
• Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field
• Aspects of generic entanglement
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• General entanglement scaling laws from time evolution
• The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
• Entanglement negativity after a global quantum quench
• On entropy growth and the hardness of simulating time evolution
• Some general properties of unified entropies
• Upper bounds on entangling rates of bipartite Hamiltonians
• Speed limits and locality in many-body quantum dynamics
• Entanglement Negativity and Mutual Information after a Quantum Quench: Exact Link from Space-Time Duality
• Bounds in Nonequilibrium Quantum Dynamics
• An area law for 2D frustration-free spin systems
• Prethermalization and the local robustness of gapped systems
• An Area Law for the Bipartite Entanglement of Disordered Oscillator Systems
• Disordered Lieb-Robinson bounds in one dimension
• Slow propagation of information on the random XXZ quantum spin chain
• Measuring out quasi-local integrals of motion from entanglement
• Tightening continuity bounds for entropies and bounds on quantum capacities

Cited by in corpus (4)

• Breaking global symmetries with locality-preserving operations
• The Magic Barrier before Thermalization
• Quantum speed limits based on Jensen-Shannon and Jeffreys divergences for general physical processes
• Entanglement Growth from Entangled States: A Unified Perspective on Entanglement Generation and Transport