cm_otimes_cn
bound_entangled.cm_otimes_cn.grid_state
https://arxiv.org/abs/1705.09261
J. Lockhart, O. Gühne, S. Severini, "Entanglement properties of quantum grid states", Phys. Rev. A 97, 062340 (2018).
grid_state(m_n, *edges)
A quantum grid state: the uniform mixture of pure states over a set of graph edges.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
m_n
|
tuple[int, int]
|
grid dimensions (m, n). |
required |
*edges
|
tuple[tuple[int, int], tuple[int, int]]
|
each edge is a pair of vertices ((i, j), (k, l)). |
()
|
Returns:
| Type | Description |
|---|---|
ndarray
|
np.ndarray: the grid state on C^m (x) C^n. |
bound_entangled.cm_otimes_cn.generalized_grid_state
https://arxiv.org/abs/2402.12966
R. Krebs, M. Gachechiladze, "High Schmidt number concentration in quantum
bound entangled states". Generalizes quantum grid states (see
cm_otimes_cn.grid_state) to hyperedges spanning more than two vertices.
generalized_grid_state(m_n, *hyperedges)
A generalized grid state
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
m_n
|
tuple[int, int]
|
grid dimensions (m, n). |
required |
*hyperedges
|
list[tuple[int, int]]
|
each hyperedge is a list of vertices (i, j) it spans. |
()
|
Returns:
| Type | Description |
|---|---|
|
np.ndarray: the generalized grid state on C^m (x) C^n. |
bound_entangled.cm_otimes_cn.random_not_bound_entangled
random_NPT(m_n)
A uniformly random density matrix with a non-positive partial transpose (rejection sampling).
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
m_n
|
tuple[int, int]
|
dimensions (m, n) of the C^m (x) C^n bipartite system. |
required |
Returns:
| Type | Description |
|---|---|
ndarray
|
np.ndarray: a random NPT (hence necessarily entangled) density matrix. |
random_PPT(m_n)
A uniformly random density matrix with a positive partial transpose (rejection sampling).
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
m_n
|
tuple[int, int]
|
dimensions (m, n) of the C^m (x) C^n bipartite system. |
required |
Returns:
| Type | Description |
|---|---|
ndarray
|
np.ndarray: a random PPT density matrix. |
random_PPT_close_to_the_PPT_edge(m_n, precision)
A random PPT state close to the boundary of the PPT set.
Obtained by bisecting the segment between a random PPT and a random NPT state: at each step the midpoint replaces whichever endpoint is on its own side, halving the distance to the PPT/NPT boundary.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
m_n
|
dimensions (m, n) of the C^m (x) C^n bipartite system. |
required | |
precision
|
int
|
number of bisection steps. |
required |
Returns:
| Type | Description |
|---|---|
ndarray
|
np.ndarray: a PPT density matrix close to the PPT boundary. |
bound_entangled.cm_otimes_cn.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 6. A
"tile" construction generalizing the Tiles UPB (c3_otimes_c3.tiles_upb) to
arbitrary m (x) n dimension.
gen_tiles2(m_n)
The bound entangled state built from the GenTiles2 UPB on C^m (x) C^n.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
m_n
|
tuple[int, int]
|
dimensions (m, n) of the C^m (x) C^n bipartite system. Requires n > 3, m >= 3, and n >= m. Note m = n = 3 does not yield a UPB (excluded by n > 3). |
required |
Returns:
| Type | Description |
|---|---|
ndarray
|
np.ndarray: the bound entangled state on the orthogonal complement of |
ndarray
|
the GenTiles2 unextendible product basis. |
gen_tiles2_basis(m_n)
The product vectors of the GenTiles2 UPB on C^m (x) C^n.
Built from m "short" tile states (Eq. 5.9), m * (n - 3) "long" tile states (Eq. 5.10), and a "stopper" state (Eq. 5.11), for a total of m*n - 2m + 1 states.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
m_n
|
tuple[int, int]
|
dimensions (m, n) of the C^m (x) C^n bipartite system. Requires n > 3, m >= 3, and n >= m. Note m = n = 3 does not yield a UPB (excluded by n > 3). |
required |
Returns:
| Type | Description |
|---|---|
list[ndarray]
|
list[np.ndarray]: the normalized product vectors forming the |
list[ndarray]
|
GenTiles2 unextendible product basis, as column vectors. |