TY - JOUR

T1 - Probing Multipartite Entanglement Through Persistent Homology

AU - Hamilton, Gregory A.

AU - Leditzky, Felix

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

PY - 2024/5

Y1 - 2024/5

N2 - We propose a study of multipartite entanglement through persistent homology, a tool used in topological data analysis. In persistent homology, a 1-parameter filtration of simplicial complexes called a persistence complex is used to reveal persistent topological features of the underlying data set. This is achieved via the computation of homological invariants that can be visualized as a persistence barcode encoding all relevant topological information. In this work, we apply this technique to study multipartite quantum systems by interpreting the individual systems as vertices of a simplicial complex. To construct a persistence complex from a given multipartite quantum state, we use a generalization of the bipartite mutual information called the deformed total correlation. Computing the persistence barcodes of this complex yields a visualization or ‘topological fingerprint’ of the multipartite entanglement in the quantum state. The barcodes can also be used to compute a topological summary called the integrated Euler characteristic of a persistence complex. We show that in our case this integrated Euler characteristic is equal to the deformed interaction information, another multipartite version of mutual information. When choosing the linear entropy as the underlying entropy, this deformed interaction information coincides with the n-tangle, a well-known entanglement measure. The persistence barcodes thus provide more fine-grained information about the entanglement structure than its topological summary, the n-tangle, alone, which we illustrate with examples of pairs of states with identical n-tangle but different barcodes. Furthermore, a variant of persistent homology computed relative to a fixed subset yields an interesting connection to strong subadditivity and entropy inequalities. We also comment on a possible generalization of our approach to arbitrary resource theories.

AB - We propose a study of multipartite entanglement through persistent homology, a tool used in topological data analysis. In persistent homology, a 1-parameter filtration of simplicial complexes called a persistence complex is used to reveal persistent topological features of the underlying data set. This is achieved via the computation of homological invariants that can be visualized as a persistence barcode encoding all relevant topological information. In this work, we apply this technique to study multipartite quantum systems by interpreting the individual systems as vertices of a simplicial complex. To construct a persistence complex from a given multipartite quantum state, we use a generalization of the bipartite mutual information called the deformed total correlation. Computing the persistence barcodes of this complex yields a visualization or ‘topological fingerprint’ of the multipartite entanglement in the quantum state. The barcodes can also be used to compute a topological summary called the integrated Euler characteristic of a persistence complex. We show that in our case this integrated Euler characteristic is equal to the deformed interaction information, another multipartite version of mutual information. When choosing the linear entropy as the underlying entropy, this deformed interaction information coincides with the n-tangle, a well-known entanglement measure. The persistence barcodes thus provide more fine-grained information about the entanglement structure than its topological summary, the n-tangle, alone, which we illustrate with examples of pairs of states with identical n-tangle but different barcodes. Furthermore, a variant of persistent homology computed relative to a fixed subset yields an interesting connection to strong subadditivity and entropy inequalities. We also comment on a possible generalization of our approach to arbitrary resource theories.

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U2 - 10.1007/s00220-024-04953-4

DO - 10.1007/s00220-024-04953-4

M3 - Article

AN - SCOPUS:85192530344

SN - 0010-3616

VL - 405

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 5

M1 - 125

ER -