Measuring the distance between quantum many-body wave functions

Abstract

We study the distance of two wave functions under chaotic time evolution. The two initial states are differed only by a local perturbation. To be entitled "chaos" the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance $d^2(t)$ by investigating the difference of two reduced density matrices of the subsystem $A$ that is spatially separated from the local perturbation. This distance $d^2(t)$ grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates $d^2(t)$ to the square of commutator $C(t)$ (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem $A$. Among various one-dimensional spin-$\frac{1}{2}$ models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in $d^2(t)$ when the local perturbation and subsystem $A$ are well separated. This regime is absent in the spin-$\frac{1}{2}$ chain with local interaction only. After sufficiently long time evolution, $d^2(t)$ relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that $d^2(t)$ is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.

27 pages, 10 figures

References in corpus (4)

• Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
• Testing whether all eigenstates obey the Eigenstate Thermalization Hypothesis
• Strong and weak thermalization of infinite non-integrable quantum systems
• Statistical distribution of quantum entanglement for a random bipartite state

Cited by in corpus (11)

• Programmable Quantum Simulations of Spin Systems with Trapped Ions
• Scrambling and entanglement spreading in long-range spin chains
• Accessing scrambling using matrix product operators
• Probing scrambling using statistical correlations between randomized measurements
• Quantum chaos dynamics in long-range power law interaction systems
• Emergent locality in systems with power-law interactions
• Measuring the single-particle density matrix for fermions and hard-core bosons in an optical lattice
• Out-of-time-ordered correlators in short-range and long-range hard-core boson models and in the Luttinger-liquid model
• Out of Time Order Correlations in the Quasi-Periodic Aubry-André model
• Genuine Quantum Chaos and Physical Distance Between Quantum States
• Out-of-time-order correlators of nonlocal block-spin and random observables in integrable and nonintegrable spin chains