cd_otimes_cd

bound_entangled.cd_otimes_cd.yu_oh

https://arxiv.org/abs/1509.08991

S. Yu, C. H. Oh, "A family of nonlocal bound entangled states", Phys. Rev. A 95, 032111 (2017).

is_valid_yu_oh_input(d, x, y)

Whether (x, y) lie in the valid parameter domain for yu_oh.

Parameters:

Name	Type	Description	Default
d	int	local dimension.	required
x	float	first free parameter.	required
y	float	second free parameter.	required

Returns:

Name	Type	Description
bool	bool	True if x2 + y2 <= 1 and the resulting delta is positive.

phi_k(d, k)

Helper vector phi_k used in the construction of the Yu-Oh state.

Parameters:

Name	Type	Description	Default
d	int	local dimension.	required
k	int	index, 1 <= k < d.	required

Returns:

Type	Description
	np.ndarray: the vector phi_k in C^d (x) C^d.

psi_ij(d, i, j)

The antisymmetric basis vector |ij> - |ji> of C^d (x) C^d.

Parameters:

Name	Type	Description	Default
d	int	local dimension.	required
i	int	first index, 0 <= i < d.	required
j	int	second index, 0 <= j < d.	required

Returns:

Type	Description
	np.ndarray: the (unnormalized) vector |ij> - |ji>.

psi_k(d, k, x, y, z)

Helper vector psi_k used in the construction of the Yu-Oh state.

Parameters:

Name	Type	Description	Default
d	int	local dimension.	required
k	int	index, 1 <= k < d.	required
x	float	first free parameter of the Yu-Oh state.	required
y	float	second free parameter of the Yu-Oh state.	required
z	float	sqrt(1 - x2 - y2).	required

Returns:

Type	Description
	np.ndarray: the vector psi_k in C^d (x) C^d.

teta_d_gen(d)

Generate the family of vectors {theta_p} used to build phi_k, by induction on the dimension.

Parameters:

Name	Type	Description	Default
d	int	local dimension.	required

Returns:

Type	Description
	list[np.ndarray]: the d-1 vectors theta_p in C^d.

yu_oh(d, x, y)

The Yu-Oh nonlocal bound entangled state, for local dimension d >= 3.

A family of PPT entangled states in C^d (x) C^d, parametrized by (x, y), that violate a Bell inequality (hence are nonlocal despite being bound entangled). See is_valid_yu_oh_input for the validity domain of (x, y).

Parameters:

Name	Type	Description	Default
d	int	local dimension, d >= 3.	required
x	float	first free parameter.	required
y	float	second free parameter.	required

Returns:

Type	Description
	np.ndarray: the Yu-Oh bound entangled state.

bound_entangled.cd_otimes_cd.gen_tiles

https://arxiv.org/abs/quant-ph/9908070

D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, B. M. Terhal, "Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement", Commun. Math. Phys. 238, 379 (2003), Section V B, Theorem 5. A "tile" construction generalizing the Tiles UPB (c3_otimes_c3.tiles_upb) to arbitrary even dimension.

gen_tiles1(d)

The bound entangled state built from the GenTiles1 UPB on C^d (x) C^d.

Parameters:

Name	Type	Description	Default
d	int	local dimension of both parties. Must be even and >= 4.	required

Returns:

Type	Description
ndarray	np.ndarray: the bound entangled state on the orthogonal complement of
ndarray	the GenTiles1 unextendible product basis.

gen_tiles1_basis(d)

The product vectors of the GenTiles1 UPB on C^d (x) C^d.

Built from d/2 - 1 "vertical" and d/2 - 1 "horizontal" families of tile states (Eqs. 5.4-5.5) plus a "stopper" state (Eq. 5.6), for a total of d^2 - 2d + 1 states.

Parameters:

Name	Type	Description	Default
d	int	local dimension of both parties. Must be even and >= 4.	required

Returns:

Type	Description
list[ndarray]	list[np.ndarray]: the normalized product vectors forming the
list[ndarray]	GenTiles1 unextendible product basis, as column vectors.

bound_entangled.cd_otimes_cd.badziag_private_singlet

https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.023101 as PPT singlets.

badziag_private_singlet(shield_dim)

Construct the Bädziąg et al. private-singlet bound-entangled state on C^{2d} ⊗ C^{2d}.

A PPT singlet in dimension 2d × 2d built from a shield subsystem of dimension d and a two-qubit pair, following Phys. Rev. Research 3, 023101 (2021). The state is bound entangled for every shield_dim >= 2.

Parameters:

Name	Type	Description	Default
shield_dim	int	shield subsystem dimension d (>= 2).	required

Returns:

Type	Description
ndarray	np.ndarray: (4d²) × (4d²) density matrix of the bound-entangled state.

basis(shield_dim, i, j, k, l)

basis vector |ijkj>

bound_entangled.cd_otimes_cd.orthogonal_singlet

https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.023101 as PPT singlets. ρ_F2 from Phys. Rev. Research 3, 023101 (2021) — the "second family" of PPT singlets, reverse-engineered from Tóth & Vértesi, PRL 120, 020506 (2018). Port of BES_metro.m from the QUBIT4MATLAB package.

basis(shield_dim, i, j, k, l)

basis vector |i j k l> in C^2 ⊗ C^2 ⊗ C^d ⊗ C^d

orthogonal_singlet(shield_dim)

Construct the ρ_F2 (second-family) PPT singlet on C^{2d} ⊗ C^{2d}.

Reverse-engineered from Tóth & Vértesi, PRL 120, 020506 (2018) and presented as ρ_F2 in Phys. Rev. Research 3, 023101 (2021). The state is bound entangled for every valid shield_dim. The returned matrix is in ABA'B' ordering (two-qubit pair AB, shield pair A'B').

Parameters:

Name	Type	Description	Default
shield_dim	int	shield subsystem dimension d. Must be 3 or a power of 2 (equivalently, total local dimension 2d ∈ {4, 6, 8, 16, …}).	required

Returns:

Type	Description
ndarray	np.ndarray: (4d²) × (4d²) density matrix of the ρ_F2 bound-entangled state.

Raises:

Type	Description
ValueError	If shield_dim is not 3 or a power of 2.