Density Matrix

Helpers for diagonalizing dense or sparse reduced density matrices and for building projectors from their dominant eigenvectors.

Density-matrix helpers used by entanglement and LBO workflows.

edlgt.entanglement.density_matrix.diagonalize_density_matrix(rho)[source]

Diagonalize a dense or sparse Hermitian density matrix.

edlgt.entanglement.density_matrix.get_projector_for_efficient_density_matrix(rho_eigvals, rho_eigvecs, threshold)[source]

Build a projector from the dominant eigenvectors of a density matrix.

edlgt.entanglement.density_matrix.flux_block_projector(rho, flux_sig, truncation, flux_tol=1e-09)[source]

Diagonalize a density matrix within symmetry blocks and keep dominant modes.

Used by the plaquette-block LBO: group the RDM basis rows by a conserved “signature” (for lattice gauge theories, the block’s boundary electric flux – a superselection label, so the RDM is block-diagonal in it), diagonalize each block separately, and keep the eigenmodes above a probability threshold. Because every kept eigenvector lives inside a single signature block, any operator diagonal in the signature (e.g. the boundary electric-field generators) stays diagonal after projection – so link symmetries survive the truncation.

Parameters:

  • rho (numpy.ndarray or scipy.sparse matrix) – (D, D) Hermitian reduced density matrix (densified if sparse).

  • flux_sig (numpy.ndarray) – (D, L) real signatures; rows with identical signatures form one symmetry block. L = 0 (no signature) puts every row in one block.

  • truncation (float) – Keep eigenvectors with eigenvalue (probability) strictly above this.

  • flux_tol (float, optional) – Quantization tolerance when grouping rows by signature.

Returns:

  • proj (numpy.ndarray) – (D, k) complex column projector, columns ordered by decreasing eigenvalue; each column is supported on a single signature block.

  • info (dict) – Diagnostics: kept_dim (k), subsys_dim (D), n_flux_blocks, kept_eigvals, discarded_weight (the truncation error, 1 - sum(kept)), and the off-block residual of rho (offblock_residual_abs / _rel) measuring how well the signature is conserved – it must be ~machine-eps for the block-wise eigendecomposition (and discarded_weight) to be exact.