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Probing Multipartite Entanglement Through Persistent Homology

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Abstract

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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MATLAB code to reproduce the numerical results in Section 4 is available at https://github.com/felixled/entanglement_persistent_homology.

Notes

  1. This construction can also be understood in terms of category theory: Define an \({\mathbb {R}}\)-indexed sublevel set filtration functor \(G:{\mathbb {R}} \rightarrow \text {{\textbf {Simp}}}\) by \(G(\varepsilon )_{\rho } = \{J \in \Delta | F(J)_{\rho } \le \varepsilon \} \), which satisfies \(G(\varepsilon )_{\rho } \le G(\varepsilon ')_{\rho }\) when \(\varepsilon \le \varepsilon '\). Here, Simp denotes the category of simplicial complexes. Denoting by \(H_{k}\) the k-th homology functor with coefficients in a field K, and by Vec the category of vector spaces over K, the functor \(P_{k} \equiv H_{k}G:{\mathbb {R}} \rightarrow {{\textbf {Vec}}}\) can be identified with the persistence module defined above.

  2. MATLAB code to reproduce the results shown here is available at https://github.com/felixled/entanglement_persistent_homology.

  3. For example, for the graph \(G_1\) shown in the top row of Fig. 2, there are 5 (finite) 0-dim. barcodes of length 1/4 each, 6 1-dim. barcodes of length 1/4 and 4 1-dim. barcodes of length 1/2, 10 2-dim. barcodes of length 1/2, 4 3-dim. barcodes of length 1/2 and one 3-dim. barcode of length 3/4, and finally one 4-dim. barcode of length 1. It follows that

    $$\begin{aligned} \tau _n = 1 = 5\cdot \frac{1}{4} - 4\cdot \frac{1}{2} - 6\cdot \frac{1}{4}+10 \cdot \frac{1}{2} -4 \cdot \frac{1}{2}- \frac{3}{4}+1. \end{aligned}$$
    (4.36)

  4. A simple heuristic approach to speed up the computation of the simplicial complex \(G(\varepsilon )\) for a given filtration parameter \(\varepsilon \) is the following. Recall that a simplex \(J\subseteq {\mathcal {A}}\) is added to \(G(\varepsilon )\) if \(F(J)\le \varepsilon \), and that F is monotonic under taking partial traces, \(F(J)\le F(K)\) if \(J\subseteq K\). Hence, in order to build the simplicial complex of an n-qubit system (where \(n=|{\mathcal {A}}|\)), it is advantageous to start at the top of the subset lattice and first evaluate F on the n subsets of size \(n-1\), then on the \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) subsets of size \(n-2\), etc. As soon as \(F(K)\le \varepsilon \) for some \(K\subseteq {\mathcal {A}}\), we automatically add all subsets \(J\subseteq K\) to the complex as well and move to the next subset of size |K|.

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Acknowledgements

The authors acknowledge helpful conversations with Henry Adams, Yuliy Baryshnikov, Jacob Beckey, Eric Chitambar, Jens Siewert, Juan Pablo Vigneaux, Michael Walter and Shmuel Weinberger. This research was partially supported through the IBM-Illinois Discovery Accelerator Institute.

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  1. Department of Physics, University of Illinois Urbana-Champaign, Champaign, USA

    Gregory A. Hamilton

  2. Institute for Condensed Matter Theory, University of Illinois Urbana-Champaign, Champaign, USA

    Gregory A. Hamilton

  3. Department of Mathematics, University of Illinois Urbana-Champaign, Champaign, USA

    Felix Leditzky

  4. Illinois Quantum Information Science and Technology Center (IQUIST), University of Illinois Urbana-Champaign, Champaign, USA

    Gregory A. Hamilton & Felix Leditzky

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Hamilton, G.A., Leditzky, F. Probing Multipartite Entanglement Through Persistent Homology. Commun. Math. Phys. 405, 125 (2024). https://doi.org/10.1007/s00220-024-04953-4

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