c3_otimes_c3

bound_entangled.c3_otimes_c3.chessboard

https://arxiv.org/abs/quant-ph/9911056

D. Bruß, A. Peres, "Construction of quantum states with bound entanglement", Phys. Rev. A 61, 030301(R) (2000). Named "chessboard states" because their 8x8 matrix form looks like a chessboard (see also https://arxiv.org/abs/2010.08372, Appendix C2).

chessboard(a, b, c, d, m, n, s=None, t=None)

A chessboard state.

Parameters:

Name	Type	Description	Default
a	complex	free real parameter.	required
b	complex	free real parameter.	required
c	complex	free real parameter.	required
d	complex	free real parameter.	required
m	complex	free real parameter.	required
n	complex	free real parameter.	required
s	Optional[complex]	defaults to c/n if None.	None
t	Optional[complex]	defaults to a*d/m if None.	None

Returns:

Type	Description
ndarray	np.ndarray: a chessboard quantum state.

chessboard_extremal_PPT()

Extremal PPT chessboard state.

The state from the chessboard family with m = n = b = -3/5, a = 3/5, c = -d = 6/5, shown to be an extremal point of the PPT-entangled states in https://arxiv.org/abs/2010.08372 (Fig. 2 / Fig. 9, page 25).

Returns:

Type	Description
ndarray	np.ndarray: the extremal PPT chessboard state.

bound_entangled.c3_otimes_c3.cross_hatch

https://arxiv.org/abs/1705.09261

J. Lockhart, O. Gühne, S. Severini, "Entanglement properties of quantum grid states", Phys. Rev. A 97, 062340 (2018).

cross_hatch()

The 3x3 "cross-hatch" grid state.

A bound entangled example of a quantum grid state, detected by the CCNR (realignment) criterion.

Returns:

Type	Description
	np.ndarray: the cross-hatch grid state.

bound_entangled.c3_otimes_c3.steering

https://arxiv.org/abs/1405.0262

T. Moroder, O. Gittsovich, M. Huber, O. Gühne, "Steering Bound Entangled States: A Counterexample to the Stronger Peres Conjecture", Phys. Rev. Lett. 113, 050404 (2014).

steering_state(m1, m2)

A 3x3 bound entangled state that is steerable, parametrized by m1, m2.

Counterexample to the (stronger) Peres conjecture that bound entangled states cannot be used to demonstrate EPR steering.

Parameters:

Name	Type	Description	Default
m1	float	first free real parameter.	required
m2	float	second free real parameter (with m12 + m22 <= 1).	required

Returns:

Type	Description
ndarray	np.ndarray: the steerable bound entangled state.

bound_entangled.c3_otimes_c3.upb.tiles_UPB

https://arxiv.org/abs/quant-ph/9808030

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).

tiles_upb()

The bound entangled state built from the "Tiles" UPB on C^3 (x) C^3.

Returns:

Type	Description
ndarray	np.ndarray: the bound entangled state on the orthogonal complement of
ndarray	the Tiles unextendible product basis (toqito.states.tile).

bound_entangled.c3_otimes_c3.upb.pyramid_UPB

https://arxiv.org/abs/quant-ph/9808030

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).

pyramid_basis()

The five product vectors of the "Pyramid" UPB on C^3 (x) C^3.

Returns:

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

pyramid_upb()

The bound entangled state built from the Pyramid UPB.

Returns:

Type	Description
ndarray	np.ndarray: the bound entangled state on the orthogonal complement of
ndarray	the Pyramid UPB.

bound_entangled.c3_otimes_c3.upb.parametrized_UPB

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 IV A. A six-parameter family of unextendible product bases on C^3 (x) C^3 that contains the Pyramid and Tiles UPBs (pyramid_upb, tiles_upb) as special cases.

parametrized_basis(gamma_a, teta_a, phi_a, gamma_b, teta_b, phi_b)

The five product vectors of the six-parameter family of UPBs on C^3 (x) C^3.

Reduces to the Pyramid UPB for phi_a = phi_b = 0 and teta_a = teta_b = gamma_a = gamma_b = arccos((sqrt(5) - 1) / 2), and to the Tiles UPB for phi_a = phi_b = 0 and teta_a = teta_b = gamma_a = gamma_b = 3*pi/4.

Parameters:

Name	Type	Description	Default
gamma_a	float	Alice's gamma angle. Requires cos(gamma_a) != 0 and sin(gamma_a) != 0 for the basis to be unextendible.	required
teta_a	float	Alice's theta angle. Requires cos(teta_a) != 0 and sin(teta_a) != 0 for the basis to be unextendible.	required
phi_a	float	Alice's phase angle.	required
gamma_b	float	Bob's gamma angle, subject to the same restriction as gamma_a.	required
teta_b	float	Bob's theta angle, subject to the same restriction as teta_a.	required
phi_b	float	Bob's phase angle.	required

Returns:

Type	Description
list[ndarray]	list[np.ndarray]: the five normalized product vectors |a_i> (x) |b_i>
list[ndarray]	forming the unextendible product basis, as column vectors.

parametrized_upb(gamma_a, teta_a, phi_a, gamma_b, teta_b, phi_b)

The bound entangled state built from the six-parameter family of UPBs.

Parameters:

Name	Type	Description	Default
gamma_a	float	Alice's gamma angle. Requires cos(gamma_a) != 0 and sin(gamma_a) != 0 for the basis to be unextendible.	required
teta_a	float	Alice's theta angle. Requires cos(teta_a) != 0 and sin(teta_a) != 0 for the basis to be unextendible.	required
phi_a	float	Alice's phase angle.	required
gamma_b	float	Bob's gamma angle, subject to the same restriction as gamma_a.	required
teta_b	float	Bob's theta angle, subject to the same restriction as teta_a.	required
phi_b	float	Bob's phase angle.	required

Returns:

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

bound_entangled.c3_otimes_c3.horodecki

https://arxiv.org/abs/quant-ph/9703004

P. Horodecki, "Separability criterion and inseparable mixed states with positive partial transposition", Phys. Lett. A 232, 333 (1997), Section 4.1. One of the first known families of bound entangled states: PPT for every a_param in [0, 1], separable only at the endpoints a_param = 0 or 1.

horodecki(a_param)

The 3x3 Horodecki state.

Parameters:

Name	Type	Description	Default
a_param	float	free real parameter in [0, 1]. The state is PPT for every value in this range, and separable only at a_param = 0 or 1.	required

Returns:

Type	Description
ndarray	np.ndarray: the 3x3 Horodecki density matrix.