bound-entangled

Reference implementations of families of bound entangled states from the quantum information literature, as plain NumPy density matrices.

A bound entangled state is entangled, yet no pure entanglement can be distilled from it, however many copies are available. No state with a positive partial transpose (PPT) is distillable, so every PPT entangled state is bound entangled — and the bipartite bound entangled states collected here are all PPT. They are otherwise scattered across papers; this package gathers them into one tested library, each factory returning the density matrix rho of the state.

📖 Documentation — one page per state, with the construction written out, what is known about it, and the papers it comes from, plus a bibliography with BibTeX.

Installation

pip install bound-entangled

Requires Python ≥ 3.11. The only runtime dependency is NumPy; every state is built from the small linear-algebra helpers in bound_entangled.utils. To classify the states — is_separable and the like — use a toolbox such as toqito, which the test suite depends on.

Usage

Packages are organized by the Hilbert space the states live in (c3_otimes_c3 = C³ ⊗ C³, cm_otimes_cn = C^m ⊗ C^n, etc.).

from bound_entangled.c3_otimes_c3 import chessboard_extremal_PPT, tiles_upb
from bound_entangled.cd_otimes_cd import yu_oh
from bound_entangled.c5_otimes_c5 import sn3_grid_state

rho = chessboard_extremal_PPT()          # 9x9 PPT entangled chessboard state
rho = tiles_upb()                        # 9x9 state from the Tiles UPB
rho = yu_oh(full_dim=3, x=0.5, y=0.1)    # Yu–Oh nonlocal bound entangled state
rho = sn3_grid_state()                   # 25x25, Schmidt number 3

The two Horodecki states are both called horodecki, each in the package of the space it lives in: c2_otimes_c4 and c3_otimes_c3.

States

c2_otimes_c4 — C² ⊗ C⁴

c3_otimes_c3 — C³ ⊗ C³

vertesi_brunner was called ncomms6297 up to version 0.2.2.

c4_otimes_c4 — C⁴ ⊗ C⁴

c5_otimes_c5 — C⁵ ⊗ C⁵

cd_otimes_cd — C^d ⊗ C^d

cm_otimes_cn — C^m ⊗ C^n (parametric constructions)

grid_state and generalized_grid_state are general constructors: not every graph gives a bound entangled state, and the documentation says which ones do.

multipartite — multipartite systems

For the two Smolin states the bound entanglement lives in the cuts where they are separable: with the parties grouped that way nothing is distillable, while the state as a whole is multipartite entangled. The doctests on smolin assert both kinds of cut; those on generalized_smolin assert the separable 2|4 cut of the six-qubit state.

utils

Repository layout

Development

pip install -e ".[test]"
pytest

The test extra installs toqito, which only the tests use. Most of the suite lives in the docstrings: every state factory carries an Examples block asserting that what it returns is a valid density matrix, that it is PPT, and how toqito's is_separable classifies it — run as doctests by pytest, so the documented behaviour and the tested behaviour cannot drift apart. A few of them fall through to toqito's semidefinite-programming check and take around ten seconds each. tests/ pins the package's own linear algebra, and the states it used to take from toqito, entrywise against toqito.

The documentation pages are generated from the docstrings: each page is the module docstring of one file under src/ (the construction, its math and its references) followed by the factory's own docstring. The files in docs/ only say which module goes on which page, so to change a page, edit the docstring. To preview the site:

pip install -e ".[docs]"
zensical serve

License

MIT — see LICENSE.