bound-entangled

Reference implementations of several families of bound entangled states from the quantum information literature, as mathjs matrices.

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.

This is the TypeScript port of the Python package of the same name. A handful of states are still to come — see TODO.md.

Installation

npm install bound-entangled

Requires Node ≥ 18. ESM only, ships its own type declarations, and depends on mathjs.

Usage

States are organized by the Hilbert space they live in, and each module is its own subpath export. Everything is re-exported from the root as well.

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

Any factory taking more than one parameter takes a single options object, so call sites name what they pass.

The two Horodecki states are both called horodecki on their own subpath, matching the Python package. Since they would collide at the root, they are re-exported there as horodecki3By3 and horodecki2By4.

Vectors and matrices may be given as plain (nested) arrays or as mathjs matrices; every function returns a mathjs Matrix, so .toArray() gets you back to plain data.

import { ketbra, upb, partialTranspose, isPSD, isPPT } from 'bound-entangled';

const bell = [
  [0.5, 0, 0, 0.5],
  [0, 0, 0, 0],
  [0, 0, 0, 0],
  [0.5, 0, 0, 0.5],
];

isPSD(bell);                        // true  — a valid density matrix
isPPT(bell, [2, 2]);                // false — entangled, and NPT
partialTranspose(bell, [2, 2]);     // SWAP / 2

States

c2OtimesC4 — C² ⊗ C⁴

| Factory | State | Reference | |---|---|---| | horodecki | 2×4 Horodecki bound entangled state | quant-ph/9703004 |

c3OtimesC3 — C³ ⊗ C³

| Factory | State | Reference | |---|---|---| | crossHatch | 3×3 "cross-hatch" grid state (CCNR-detected) | 1705.09261 | | horodecki | 3×3 Horodecki bound entangled state | quant-ph/9703004 | | tilesUpb, tilesBasis | State from the Tiles unextendible product basis | quant-ph/9808030 | | steeringState | Steerable bound entangled state (counterexample to the stronger Peres conjecture) | 1405.0262 | | ncomms6297 | Rank-4 PPT entangled state from an explicit spectral decomposition | ncomms6297 | | pyramidUpb, pyramidBasis | State from the Pyramid unextendible product basis | quant-ph/9808030 | | parametrizedUpb, parametrizedBasis | Six-parameter family of UPBs generalizing Tiles / Pyramid | quant-ph/9908070 |

c4OtimesC4 — C⁴ ⊗ C⁴

| Factory | State | Reference | |---|---|---| | pianni | 4×4 Benatti–Floreanini–Piani state | quant-ph/0411095 |

c5OtimesC5 — C⁵ ⊗ C⁵

| Factory | State | Reference | |---|---|---| | sn3GridState | Smallest known Schmidt-number-3 PPT bound entangled state | 2402.12966 |

cdOtimesCd — C^d ⊗ C^d

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

cmOtimesCn — C^m ⊗ C^n

| Factory | State | Reference | |---|---|---| | gridState | Quantum grid states from graph edges | 1705.09261 | | generalizedGridState | Grid states generalized to hyperedges | 2402.12966 | | genTiles2, genTiles2Basis | GenTiles2 UPB generalizing Tiles to m⊗n, n>3, m≥3, n≥m | quant-ph/9908070 |

multipartite

| Factory | State | Reference | |---|---|---| | smolin | Smolin four-party unlockable bound entangled state | quant-ph/0001001 | | generalizedSmolin | Generalized Smolin state on 2n qubits | quant-ph/0411142 | | quasiDs | Quasi-Dicke bound entangled state on n qubits | ncomms6297 |

utils

Building blocks:

| Function | Description | |---|---| | ket(dims, index) | Computational basis vector. ket(3, 1) is |1> in C³; ket([3, 3], [1, 2]) is |12> in C³ ⊗ C³. | | ketbra(a, b?) | The outer product |a><b|, defaulting to |a><a|. | | tensor(...factors) | Kronecker product of any number of vectors, or of any number of matrices. | | normalize(v) | The unit vector |v> / || |v> ||. | | normalizeTrace(rho) | rho / tr(rho), turning a positive operator into a density matrix. | | pauli(index) | A Pauli operator by name or index (0 = I, 1 = X, 2 = Y, 3 = Z); a list gives their tensor product. | | maxEntangled(dim, options?) | (1/√d) Σ\|ii>, or the bare Σ\|ii> with { normalized: false }. | | fourier(dim) | The DFT matrix W[j][k] = ω^(jk)/√d. | | tile(index) | One of the five Tile states on C³ ⊗ C³, index 0–4. | | upb(basis) | The bound entangled state on the orthogonal complement of an unextendible product basis. |

Properties:

| Function | Description | |---|---| | partialTranspose(rho, dims, sys?) | Partial transpose over the chosen subsystems. | | permuteSystems(rho, perm, dims?) | Reorder the subsystems of an operator. | | isPSD(m, tol?) | Whether m is Hermitian with no negative eigenvalue. | | isPPT(rho, dims, sys?, tol?) | Whether the partial transpose of rho is positive semidefinite. |

dims lists the subsystem dimensions ([3, 3] for C³ ⊗ C³, [2, 2, 2, 2] for four qubits) and sys selects the subsystem(s) to transpose, zero-indexed, defaulting to 1 — the second subsystem. Either a single index or an array of them is accepted, so isPPT(rho, [2, 2, 2, 2], [2, 3]) tests the {0,1} vs {2,3} cut.

permuteSystems reads perm positionally and zero-indexed: position i of the new ordering holds the original subsystem perm[i], so on C^A ⊗ C^B ⊗ C^C a perm of [1, 2, 0] gives the operator on C^B ⊗ C^C ⊗ C^A. dims may be omitted when the subsystems are of equal dimension.

isPSD decides positivity from the eigenvalues of a real symmetric representation of the matrix: a Hermitian H = A + iB is PSD exactly when the real symmetric [[A, -B], [B, A]] is, which keeps complex states on mathjs' real symmetric eigensolver. Eigenvalues above -tol (default 1e-8) count as non-negative, which is the slack the boundary states in this library need.

Development

npm install
npm test
npm run typecheck
npm run build

License

MIT