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arXiv:2108.02225

Entanglement Entropy Transitions with Random Tensor Networks

Ryan Levy, Bryan K. Clark

Published Aug 4, 2021• 8 citations
arXivHTMLPDF

Abstract

Entanglement is a key quantum phenomena and understanding transitions between phases of matter with different entanglement properties are an interesting probe of quantum mechanics. We numerically study a model of a 2D tensor network proposed to have an entanglement entropy transition first considered by Vasseur et al.[Phys. Rev. B 100, 134203 (2019)]. We find that by varying the bond dimension of the tensors in the network we can observe a transition between an area and volume phase with a logarithmic critical point around $D\approx 2$. We further characterize the critical behavior measuring a critical exponent using entanglement entropy and the tripartite quantum mutual information, observe a crossover from a `nearly pure' to entangled area law phase using the the distributions of the entanglement entropy and find a cubic decay of the pairwise mutual information at the transition. We further consider the dependence of these observables for different Rényi entropy. This work helps further validate and characterize random tensor networks as a paradigmatic examples of an entanglement transition.

DOI
10.48550/arxiv.2108.02225

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The authors explore critical properties of entanglement phase transitions in random tensor networks and monitored quantum circuits with Clifford tensors/gates for on-site Hilbert space dimensions which are powers of a prime number ppp. Exact mappings to replica spin models and characterizations of their symmetry groups predict that all universal properties will depend only on ppp. These predictions are confirmed with extensive numerical simulations. The authors also establish multifractal scaling of the purity, reflected in a continuous spectrum of critical exponents, while the typical exponent is the prefactor of the logarithm in the entanglement entropy.
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A new class of quantum entanglement transitions separating phases with different entanglement entropy scaling has been observed in recent numerical studies. Despite the numerical efforts, an analytical understanding of such transitions has remained elusive. Here, the authors propose a theory for the area-law to volume-law entanglement transition in many-body systems that undergo both random unitary evolutions and projective measurements. Using the replica method, the authors map analytically this entanglement transition to an ordering transition in a classical statistical mechanics model. They derive the general entanglement scaling properties at the transition and show a solvable limit where this transition can be mapped onto two-dimensional percolation.
A random quantum circuit is a minimally structured model to study entanglement dynamics of many-body quantum systems. We consider a one-dimensional quantum circuit with noisy Haar-random unitary gates using density matrix operator and tensor contraction methods. It is shown that the entanglement evolution of the random quantum circuits is properly characterized by the logarithmic entanglement negativity. By performing exact numerical calculations, we find that, as the physical error rate is decreased below a critical value p c ≈ 0.056, the logarithmic entanglement negativity changes from the area law to the volume law, giving rise to an entanglement transition. The critical exponent of the correlation length can be determined from the finite-size scaling analysis, revealing the universal dynamic property of the noisy intermediate-scale quantum devices.