c4_otimes_c4
bound_entangled.c4_otimes_c4.breuer
https://arxiv.org/abs/quant-ph/0605036
H.-P. Breuer, "Optimal entanglement criterion for mixed quantum states", Phys. Rev. Lett. 97, 080501 (2006). Bound entangled states for even local dimension d >= 4: PPT for any positive weight lam, yet entangled by Breuer's anti-linear map criterion.
breuer(lam)
Construct a Breuer bound-entangled state on C^4 ⊗ C^4.
Wraps :func:toqito.states.breuer with dim=4, which is the smallest
local dimension for which Breuer's construction yields a bound-entangled
state (requires even dim >= 4).
The state is a mixture of the maximally mixed state and a
partially-transposed singlet component weighted by lam. For any
lam > 0 the state is entangled yet PPT, making it bound entangled.
Parameters
lam:
Weight of the singlet component. Must satisfy 0 < lam <= 1.
Returns
np.ndarray 16×16 density matrix of the Breuer state.
bound_entangled.c4_otimes_c4.pianni
https://arxiv.org/abs/quant-ph/0411095
F. Benatti, R. Floreanini, M. Piani, "Non-decomposable quantum dynamical semigroups and bound entangled states", Open Syst. Inf. Dyn. 11, 325 (2004).
Also presented as "the 4x4 bound entangled Piani state" in https://arxiv.org/abs/2010.08372 (Appendix C5).
P_ij(i, j)
The projector onto the generalized Bell state |Psi_ij> = (1 (x) sigma_i (x) sigma_j)|Psi_4^+>.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
i
|
int
|
index of the first Pauli matrix (0..3). |
required |
j
|
int
|
index of the second Pauli matrix (0..3). |
required |
Returns:
| Type | Description |
|---|---|
|
np.ndarray: the rank-1 projector P_ij on C^4 (x) C^4. |
pianni()
The 4x4 bound entangled "Piani" state.
Uniform mixture of the six projections P_ij for (i, j) in {(0,2), (1,1), (2,3), (3,1), (3,2), (3,3)}, indecomposable and PPT under the bipartition AA'|BB' after regrouping the four underlying qubits.
Returns:
| Type | Description |
|---|---|
|
np.ndarray: the 16x16 bound entangled density matrix. |