Persistent homology of quantum entanglement

TL;DR

This paper applies persistent homology, a topological data analysis method, to study the structure of quantum entanglement entropy in various spin models, revealing new insights and potential applications in understanding spacetime emergence.

Contribution

It introduces the use of persistent homology to analyze quantum entanglement entropy, exploring different coefficients, boundary conditions, and potential links to spacetime emergence.

Findings

Persistent homology captures entanglement structure in quantum states.

Analysis of ground and excited states reveals topological features.

Potential for future applications in quantum physics and spacetime theories.

Abstract

Structure in quantum entanglement entropy is often leveraged to focus on a small corner of the exponentially large Hilbert space and efficiently parameterize the problem of finding ground states. A typical example is the use of matrix product states for local and gapped Hamiltonians. We study the structure of entanglement entropy using persistent homology, a relatively new method from the field of topological data analysis. The inverse quantum mutual information between pairs of sites is used as a distance metric to form a filtered simplicial complex. Both ground states and excited states of common spin models are analyzed as an example. Furthermore, the effect of homology with different coefficients and boundary conditions is also explored. Beyond these basic examples, we also discuss the promising future applications of this modern computational approach, including its connection to the question of how spacetime could emerge from entanglement.