Symmetric extension criteria

Accessible in both numqi.entangle (recommended) and numqi.entangle.symext

`numqi.entangle.get_ABk_symmetric_extension_ree(rho, dim, kext, use_ppt=False, use_boson=False, return_info=False, sqrt_order=3, pade_order=3, use_tqdm=False)`

get the relative entropy of entanglement of k-symmetric extension on B-party

Parameters:

Name	Type	Description	Default
`rho`	`(ndarray, list)`	density matrix, or list of density matrices (3d array)	required
`dim`	`tuple(int`	tuple of length 2, dimension of A-party and B-party	required
`kext`	`int`	number of copies of symmetric extension	required
`use_ppt`	`bool`	if True, use PPT (positive partial transpose) constraint	`False`
`use_boson`	`bool`	if True, use bosonic symmetry	`False`
`return_info`	`bool`	if True, return information of the SDP solver	`False`
`sqrt_order`	`int`	the order of sqrtm approximation	`3`
`pade_order`	`int`	the order of Pade approximation	`3`
`use_tqdm`	`bool`	if True, use tqdm to show progress bar	`False`

Returns:

Name	Type	Description
`ret0`	`(float, array)`	`ret` is a float indicates relative entropy of entanglement. If `rho` is list of density matrices, `ret` is a 1d `np.array` of float
`ret1`	`list[dict]`	if `return_info` is `True`, then `ret1` is a list of information of the SDP solver.

`numqi.entangle.is_ABk_symmetric_ext(rho, dim, kext, use_ppt=False, use_boson=False, use_tqdm=False, return_info=False)`

check if rho has symmetric extension of kext copies on B-party

Parameters:

Name	Type	Description	Default
`rho`	`(ndarray, list)`	density matrix, or list of density matrices (3d array)	required
`dim`	`tuple(int`	tuple of length 2, dimension of A-party and B-party	required
`kext`	`int`	number of copies of symmetric extension	required
`use_ppt`	`bool`	if True, use PPT (positive partial transpose) constraint	`False`
`use_boson`	`bool`	if True, use bosonic symmetry	`False`
`use_tqdm`	`bool`	if True, use tqdm to show progress bar	`False`
`return_info`	`bool`	if True, return information of the SDP solver	`False`

Returns:

Name	Type	Description
`ret`	`bool`	If `return_info=False` and rho is single density matrix, `ret` is a bool indicates if rho has symmetric extension. If `return_info=False` and rho is list of density matrices, `ret` is a 1d `np.array` of bool If `return_info=True` and `rho` is single density matrix, `ret` is a tuple of (bool, info) where `info` is a list of information of the SDP solver. If `return_info=True` and `rho` is list of density matrices, `ret` is a tuple of (np.array, info) where `info` is a dict of information of the SDP solver.

`numqi.entangle.get_ABk_symmetric_extension_boundary(rho, dim, kext, use_ppt=False, use_boson=False, use_tqdm=False, return_info=False)`

get the boundary (in Euclidean space) of k-ext symmetric extension on B-party along rho direction

Parameters:

Name	Type	Description	Default
`rho`	`(ndarray, list)`	density matrix, or list of density matrices (3d array)	required
`dim`	`tuple(int`	tuple of length 2, dimension of A-party and B-party	required
`kext`	`int`	number of copies of symmetric extension	required
`use_ppt`	`bool`	if True, use PPT (positive partial transpose) constraint	`False`
`use_boson`	`bool`	if True, use bosonic symmetry	`False`
`use_tqdm`	`bool`	if True, use tqdm to show progress bar	`False`
`return_info`	`bool`	if True, return information of the SDP solver	`False`

Returns:

Name	Type	Description
`beta`	`(float, array)`	`beta` is a float indicates Euclidean distance. if `rho` is list of density matrices, `beta` is a 1d `np.array` of float
`vecA`	`ndarray`	`vecA` is a 1d `np.ndarray` of float indicates the position of the boundary. if `rho` is list of density matrices, `vecA` is a 2d `np.ndarray`
`vecN`	`ndarray`	`vecN` is a 1d `np.ndarray` of float indicates the normal vector of the boundary. if `rho` is list of density matrices, `vecA` is a 2d `np.ndarray`

`numqi.entangle.get_ABk_extension_numerical_range(op_list, direction, dim, kext, use_ppt=False, use_boson=False, use_tqdm=True, return_info=False)`

get the symmetric extension numerical range of a list of operators

\[ \max\;\beta \]

\[ s.t.\;\begin{cases} \rho_{AB^{k}}\succeq0\\ \mathrm{Tr}[\rho_{AB^{k}}]=1\\ P_{B_{i}B_{j}}\rho_{AB^{k}}P_{B_{i}B_{j}}=\rho_{AB^{k}}\\ \mathrm{Tr}\left[\mathrm{Tr}_{B^{k-1}}\left[\rho\right]A_{i}\right]=\beta\hat{n}_{i} \end{cases} \]

Parameters:

Name	Type	Description	Default
`op_list`	`list`	a list of operators, each operator is a 2d numpy array	required
`direction`	`ndarrray`	the boundary along the direction will be calculated, if 2d, then each row is a direction	required
`dim`	`tuple[int]`	the dimension of the density matrix, e.g. (2,2) for 2 qubits, must be of length 2	required
`kext`	`int`	the number of copies of symmetric extension	required
`use_ppt`	`bool`	if `True`, then use PPT (positive partial transpose) constraint in the pre-image	`False`
`use_boson`	`bool`	if `True`, then use bosonic symmetrical extension	`False`
`return_info`	`bool`	if `True`, then return the boundary and the boundary's normal vector	`False`
`use_tqdm`	`bool`	if `True`, then use tqdm to show the progress	`True`

Returns:

Name	Type	Description
`beta`	`ndarray`	the distance from the origin to the boundary along the direction. If `direction` is 2d, then `beta` is 1d array.
`boundary`	`ndarray`	the boundary along the direction. only returned if `return_info` is `True`
`normal_vector`	`ndarray`	the normal vector of the boundary. only returned if `return_info` is `True`