bound-entangled

Reference implementations of several families of bound entangled states from the quantum information literature, in Python and TypeScript.

A bound entangled state is entangled yet has a positive partial transpose (PPT), so no pure entanglement can be distilled from it. This library collects constructions that are otherwise scattered across papers into a single, tested package, 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 code to build it in either language.

Packages

Package	Source	Install
Python, as NumPy arrays	python/	pip install bound-entangled
TypeScript, as mathjs matrices	js/	npm install bound-entangled

The two expose the same states under the same names — snake_case in Python, camelCase in TypeScript — and the TS values are checked entrywise against the Python ones. Each package has its own README with the full API: python/README.md, js/README.md.

Usage

States are organized by the Hilbert space they live in (c3_otimes_c3 = C³ ⊗ C³, cm_otimes_cn = C^m ⊗ C^n, and so on).

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 = yu_oh(full_dim=3, x=0.5, y=0.1)    # Yu-Oh nonlocal bound entangled state

import { crossHatch, pyramidUpb } from "bound-entangled/c3OtimesC3";
import { yuOh } from "bound-entangled/cdOtimesCd";
import { sn3GridState } from "bound-entangled/c5OtimesC5";

const rho = crossHatch();                            // 9x9 PPT-entangled grid state
const sigma = yuOh({ fullDim: 3, x: 0.5, y: 0.1 });  // Yu–Oh nonlocal bound entangled state

States

A — in the TypeScript column marks a state the port still owes the Python package; see js/TODO.md.

C² ⊗ C⁴

Python	TypeScript	State	Reference
horodecki	horodecki (horodecki2By4)	2×4 Horodecki bound entangled state	quant-ph/9703004

C³ ⊗ C³

Python	TypeScript	State	Reference
chessboard, chessboard_extremal_PPT	—	Bruß–Peres chessboard states	quant-ph/9911056
cross_hatch	crossHatch	3×3 "cross-hatch" grid state (CCNR-detected)	1705.09261
horodecki	horodecki (horodecki3By3)	3×3 Horodecki bound entangled state	quant-ph/9703004
ncomms6297	ncomms6297	Rank-4 PPT entangled state from an explicit spectral decomposition	ncomms6297
steering_state	steeringState	Steerable bound entangled state (counterexample to the stronger Peres conjecture)	1405.0262
tiles_upb	tilesUpb, tilesBasis	State from the Tiles unextendible product basis	quant-ph/9808030
pyramid_upb	pyramidUpb, pyramidBasis	State from the Pyramid unextendible product basis	quant-ph/9808030
parametrized_upb	parametrizedUpb, parametrizedBasis	Six-parameter family of UPBs generalizing Tiles / Pyramid	quant-ph/9908070

C⁴ ⊗ C⁴

Python	TypeScript	State	Reference
piani	piani	4×4 Benatti–Floreanini–Piani state	quant-ph/0411095

C⁵ ⊗ C⁵

Python	TypeScript	State	Reference
sn3_grid_state	sn3GridState	Smallest known Schmidt-number-3 PPT bound entangled state	2402.12966

C^d ⊗ C^d

Python	TypeScript	State	Reference
yu_oh, is_valid_yu_oh_input	yuOh, isValidYuOhInput	Yu–Oh family of nonlocal bound entangled states	1509.08991
gen_tiles1	genTiles1, genTiles1Basis	GenTiles1 UPB generalizing Tiles to d⊗d, even d≥4	quant-ph/9908070
badziag_private_singlet	badziagPrivateSinglet	Bądziąg et al. private-singlet state on C^2d ⊗ C^2d	PRResearch 3, 023101
orthogonal_singlet	orthogonalSinglet	ρ_F2, the second family of PPT singlets	PRResearch 3, 023101
horodecki_2_by_d_generalized	horodecki2ByDGeneralized	C² ⊗ C⁴ Horodecki state generalized to C² ⊗ C^d	1203.3711

C^m ⊗ C^n

Python	TypeScript	State	Reference
grid_state	gridState	Quantum grid states from graph edges	1705.09261
generalized_grid_state	generalizedGridState	Grid states generalized to hyperedges	2402.12966
gen_tiles2	genTiles2, genTiles2Basis	GenTiles2 UPB generalizing Tiles to m⊗n, n>3, m≥3, n≥m	quant-ph/9908070
random_NPT, random_PPT, random_PPT_close_to_the_PPT_edge	—	Random density matrices by PPT class (rejection sampling)	—

Multipartite

Python	TypeScript	State	Reference
smolin	smolin	Smolin four-party unlockable state — separable across 2|2 cuts, NPT across 1|3	quant-ph/0001001
generalized_smolin	generalizedSmolin	Generalized Smolin state on 2n qubits — separable across 2|n−2 cuts, NPT across 1|n−1 and n/2|n/2	quant-ph/0411142
quasi_ds	quasiDs	Quasi-DS (diagonal symmetric) state on an odd number of qubits n ≥ 5 — PPT across every bipartition	1706.09423

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 cuts listed above are asserted by the doctests on each factory.

Repository layout

Directory	Contents
python/	The Python package and its pytest suite, published to PyPI on a v* tag.
js/	The TypeScript package and its vitest suite.
docs/	The React site behind the documentation link, deployed to GitHub Pages on every push to main.

Testing

js

cd js
npm test

python

cd python
pip install -e ".[test]"
pytest -v

License

MIT