entanglement_persistent_homology

This is a collection of MATLAB scripts to reproduce the results of the paper

Gergory A. Hamilton, Felix Leditzky: "Probing multipartite entanglement through persistent homology", arXiv:2307.07492.

It computes and compares the persistent homology induced by the linear entropy of two quantum states. There is a special version that directly computes the homology of graph states.

Prerequisites

• Matlab R2022b or newer (the codes uses the dictionary function which was implemented in R2022b).
• Javaplex: http://appliedtopology.github.io/javaplex/
• function TrX from Toby Cubitt's quantinf package: https://www.dr-qubit.org/matlab/TrX.m

How to use this script

To work with graph states, run test_graph_barcodes.m. At the beginning of the script you should call an external script that generates two graphs in variables G1 and G2, respectively. Some examples of pairs of graphs are located in the script files starting with g_. The default version calls g_6vertices.m, which is the example shown in Fig. 2 of the paper.

To work with arbitrary states on n systems of local dimension d, run test_state_barcodes.m. Create the two states to compare in the variables psi1 and psi2, respectively. The default version calls osterloh_siewert.m, which is the example shown in Fig. 3 of the paper.

List of script files

• compute_entropies.m: Computes a vector of all subsystem entropies of a given state, either for a graph state or a generic state.
• compute_persistent_homology.m: Computes the persistent homology of a pair of list of vertices and simplices using the Javaplex package.
• compute_weights.m: Computes the weights of all vertices and simplices of a given state, either for a graph state or a generic state.
• deformed_total_correlation.m: Computes the deformed total correlation functional from eq.(2) in the paper for a given subset A of the n qubits using the entropy function specified in compute_entropies.m.
• graph_state_marginal.m: Computes the eigenvalues of the marginal of a graph state.
• print_weights.m: Auxiliary function to print the weights of simplices.
• subset2string.m: Auxiliary function that converts a subset of {1,...,n} into the corresponding binary string.
• test_graph_barcodes.m: Test script to compute and compare the persistent homology of two given graph states.
• test_state_barcodes.m: Test script to compute and compare the persistent homology of two given states.

List of example files

• g_6vertices.m: Example file with two graphs on 6 qubits with n-tangle 1 (Fig. 2 in the paper).
• g_6vertices2.m: Example file with two graphs on 6 qubits with n-tangle 1.
• g_6vertices_3regular.m: Example file with two graphs on 6 qubits with n-tangle 1.
• g_8vertices.m: Example file with two graphs on 8 qubits with n-tangle 1.
• g_petersen_star.m: Example file with the Petersen graph and a star graph on 10 qubits, each with n-tangle 1.
• osterloh_siewert.m: Example file with the 6-qubit states from Osterloh, Siewert (https://arxiv.org/abs/quant-ph/0506073) and a variation used in Fig. 3 of the paper.