utils

bound_entangled.utils.ketbra

ketbra(a, b)

The outer product |a><b|.

Parameters:

Name	Type	Description	Default
a	ndarray	the "ket" vector.	required
b	ndarray	the "bra" vector (its complex conjugate is used).	required

Returns:

Type	Description
ndarray	np.ndarray: the rank-1 operator |a><b|.

bound_entangled.utils.upb

upb(basis)

Bound entangled state built from an unextendible product basis (UPB).

The uniform mixture over the orthogonal complement of a UPB is PPT (each |v_i><v_i| being a product-state projector, partial transposition leaves it unchanged) yet has no product vector in its range, hence is entangled: a bound entangled state.

Reference: C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, B. M. Terhal, "Unextendible Product Bases and Bound Entanglement", Phys. Rev. Lett. 82, 5385 (1999), https://arxiv.org/abs/quant-ph/9808030

Parameters:

Name	Type	Description	Default
basis	list[ndarray]	the product vectors (as column vectors) forming the UPB.	required

Returns:

Type	Description
ndarray	np.ndarray: the corresponding bound entangled density matrix.