PPT criteria

Positive Partial Transpose (PPT)

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

`numqi.entangle.get_ppt_boundary(dm, dim, dm_norm=None, within_dm=True)`

get the PPT (positive partial transpose) boundary of a density matrix

Parameters:

Name	Type	Description	Default
`dm`	`ndarray`	density matrix (support batch)	required
`dim`	`tuple[int]`	tuple of integers, must be of length 2	required
`dm_norm`	`(float, NoneType)`	norm of the density matrix. if None, then calculate it internally	`None`
`within_dm`	`bool`	if `True`, then the boundary is within the density matrix space	`True`

Returns:

Name	Type	Description
`beta_pt_l`	`float`	minimum value of beta
`beta_pt_u`	`float`	maximum value of beta

`numqi.entangle.is_ppt(rho, dim, eps=-1e-07)`

Positive Partial Transpose (PPT)

wiki/entanglement-witness

wiki/Peres-Horodecki-criterion

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	density matrix	required
`dim`	`tuple[int]`	tuple of integers	required
`eps`	`float`	threshold for the eigenvalues, if min(eig(X))>=eps, then X is positive semi-definite	`-1e-07`

Returns:

Name	Type	Description
`tag`	`bool`	whether rho is PPT

`numqi.entangle.is_generalized_ppt(rho, dim, return_info=False)`

Generalized Positive Partial Transpose (PPT)

The generalized partial transposition criterion for separability of multipartite quantum states doi-link

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	density matrix	required
`dim`	`tuple[int]`	tuple of integers	required
`return_info`	`bool`	whether to return the list of nuclear norms	`False`

Returns:

Name	Type	Description
`tag`	`bool`	whether rho is generalized PPT (superset of SEP)
`info`	`list[float]`	list of nuclear norms

`numqi.entangle.get_generalized_ppt_boundary(dm, dim, threshold=1e-10, xtol=1e-05)`

get boundary according to Generalized Positive Partial Transpose (PPT) criterion

Parameters:

Name	Type	Description	Default
`dm`	`ndarray`	density matrix	required
`dim`	`tuple[int]`	tuple of integers	required
`threshold`	`float`	threshold for the nuclear norm	`1e-10`
`xtol`	`float`	tolerance for the root finding	`1e-05`

Returns:

Name	Type	Description
`beta`	`float`	boundary in Gell-Mann space

`numqi.entangle.get_ppt_numerical_range(op_list, direction, dim, return_info=False, use_tqdm=True)`

get the PPT (positive partial transpose) numerical range of a list of operators

TODO bug, this is cross-section, not numerical range

\[ \max\;\beta \]

\[ s.t.\;\begin{cases} \rho\succeq 0\\ \mathrm{Tr}[\rho]=1\\ \rho^{\Gamma}\succeq 0\\ \mathrm{Tr}[\rho A_{i}]=\beta\hat{n}_{i} & i=1,\cdots,m \end{cases} \]

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
`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`

`numqi.entangle.get_ppt_ree(rho, dimA, dimB, return_info=False, sqrt_order=3, pade_order=3, use_tqdm=True)`

`numqi.entangle.get_dm_cross_section_boundary(op0, op1, num_point=101, dim=None, tag_eig=False, tag_ppt=True, tag_gppt=False)`

Get the boundary of the cross section spanned by two Hermitian operators.

Parameters:

Name	Type	Description	Default
`op0`	`ndarray`	Hermitian operator, `ndim=2`	required
`op1`	`ndarray`	Hermitian operator, `ndim=2`	required
`num_point`	`int`	number of points to sample the boundary	`101`
`dim`	`None \| tuple[int]`	the dimension of bipartite system, Required if `tag_ppt` or `tag_gppt` is `True`	`None`
`tag_eig`	`bool`	whether to calculate the eigenvalues of the interpolated density matrix	`False`
`tag_ppt`	`bool`	whether to calculate the PPT boundary	`True`
`tag_gppt`	`bool`	whether to calculate the generalized PPT boundary	`False`

Returns:

Name	Type	Description
`ret`	`dict`	a dictionary containing the following keys: theta_list (np.ndarray): the angles of the boundary points beta_dm (np.ndarray): the boundary points of the cross section theta_op (float): the angle of the plane spanned by `op0` and `op1` eig_dm (np.ndarray): the eigenvalues of the interpolated density matrix beta_ppt (np.ndarray): the boundary points of the PPT beta_gppt (np.ndarray): the boundary points of the generalized PPT

`numqi.entangle.plot_dm_cross_section(beta_dm, theta_op=None, label=None, dim=None, ax=None, tag_show_legend=True, **kwargs)`

Plot the boundary of the cross section spanned by two Hermitian operators.

see numqi.entangle.get_dm_cross_section_boundary

Parameters:

Name	Type	Description	Default
`beta_dm`	`ndarray`	the boundary length of the density matrix in the cross section	required
`theta_op`	`float \| None`	the angle of the plane spanned by `op0` and `op1`	`None`
`label`	`tuple[str] \| None`	the label of the plane and the boundary	`None`
`dim`	`int \| None`	the dimension of the bipartite system, if provided, then the inscribed circle and the circumscribed circle will be plotted	`None`
`ax`	`None \| AxesSubplot`	the axes to plot, if `None`, then create a new figure	`None`
`tag_show_legend`	`bool`	whether to show the legend	`True`
`kwargs`	`dict`	additional boundary points to plot, the key will be used as label, value can be a 1d array or a dictionary of 1d arrays	`{}`

Returns:

Name	Type	Description
`fig`	`Figure`	the figure
`ax`	`AxesSubplot`	the axes

numqi.entangle.cvx_matrix_xlogx TODO

numqi.entangle.cvx_matrix_mlogx TODO