entangle

numqi.entangle

`numqi.entangle.hf_interpolate_dm(rho, alpha=None, beta=None, dm_norm=None)`

interpolate the density matrix between rho and the maximally mixed state

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	density matrix, `ndim=2`	required
`alpha`	`(float, None)`	`rho0(1-alpha) + rhoalpha`, return `rho` itself if `alpha=1`, and maximally mixed state if `alpha=0`, if None, then use `beta`	`None`
`beta`	`(float, None)`	`rho0 + beta*unit(rho0)`, `beta` reflects the vector length in Gell-Mann basis, if None, then use `alpha`	`None`
`dm_norm`	`(float, None)`	norm of the density matrix. if None, then calculate it internally	`None`

Returns:

Name	Type	Description
`ret`	`ndarray`	interpolated density matrix, `ndim=2`

WARNING: rename to numqi.utils.hf_interpolate_dm

`numqi.entangle.get_dm_numerical_range(op0, op1, num_point=100)`

return the joint algebra numerical range of two Hermitian operators (projection)

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	`100`

Returns:

Name	Type	Description
`ret`	`ndarray`	joint algebra numerical range, `ndim=2`, `shape=(num_point,2)`

`numqi.entangle.get_negativity(rho, dim)`

return the negativity of the density matrix wiki-link

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	density matrix, `ndim=2`	required
`dim`	`tuple[int]`	dimension of the density matrix, `(dimA,dimB)`	required

Returns:

Name	Type	Description
`ret`	`float`	negativity of the density matrix

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

return whether the density matrix passes the reduction criterion quantiki-link

weaker than PPT

TODO, can positive linear map be parameterized

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	density matrix, `ndim=2`	required
`dim`	`tuple[int]`	dimension of the density matrix, `(dimA,dimB,dimC,...)`	required
`eps`	`float`	threshold for the reduction witness. if min(eig(X))>eps, then we say X is positive	`-1e-07`

Returns:

Name	Type	Description
`ret`	`bool`	whether the density matrix passes the reduction criterion

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

return whether the density matrix passes the swap witness criterion

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	density matrix, `ndim=2`	required
`eps`	`float`	threshold for the swap witness	`-1e-07`

Returns:

Name	Type	Description
`ret`	`bool`	whether the density matrix passes the swap witness criterion

`numqi.entangle.get_density_matrix_plane(op0, op1)`

return the plane spanned by two Hermitian operators

Parameters:

Name	Type	Description	Default
`op0`	`ndarray`	Hermitian operator, `ndim=2`, no need to be unit trace	required
`op1`	`ndarray`	Hermitian operator, `ndim=2`, no need to be unit trace	required

Returns:

Name	Type	Description
`theta1`	`float`	angle between op0 and op1 in the Gell-Mann basis
`hf0`	`function`	function to generate the density matrix on the plane. function signature: `hf0(theta_or_xy:float\|tuple[float], norm:float=1) -> np.ndarray`. if `theta_or_xy` is a float, then return the density matrix with the given angle and `norm`. if `theta_or_xy` is a tuple, then return the density matrix with the given x,y in the plane.

`numqi.entangle.get_density_matrix_boundary(dm, dm_norm=None)`

return the boundary of the density matrix

Parameters:

Name	Type	Description	Default
`dm`	`ndarray`	density matrix, 2d array (support batch)	required
`dm_norm`	`(float, NoneType)`	norm of the density matrix. if None, then calculate it internally	`None`

Returns:

Name	Type	Description
`beta_l`	`float`	minimum value of beta
`beta_u`	`float`	maximum value of beta

`numqi.entangle.get_dm_cross_section_moment(op0, op1, order=1, quad_epsrel=1e-08, dim=None, kind='dm')`

return the moment of the cross section of the density matrix or PPT boundary spanned by two Hermitian operators

Parameters:

Name	Type	Description	Default
`op0`	`ndarray`	traceless Hermitian operators, `ndim=2`	required
`op1`	`ndarray`	traceless Hermitian operators, `ndim=2`	required
`order`	`int`	order of the moment, for dimension=3/4/5, order=1 is enough. larger order might be needed for larger dimension	`1`
`quad_epsrel`	`float`	relative tolerance for the quadrature	`1e-08`
`dim`	`None \| tuple[int]`	dimension of the density matrix, `(dimA,dimB)`, only used when `kind='ppt'`	`None`
`kind`	`str`	'dm' for density matrix, 'ppt' for PPT boundary	`'dm'`

Returns:

Name	Type	Description
`ret`	`ndarray`	moment of the cross section of the density matrix or PPT boundary, `ndim=1`

`numqi.entangle.is_dm_cross_section_similar(moment0, moment1, zero_eps=0.001)`

return whether two moments are similar

Parameters:

Name	Type	Description	Default
`moment0`	`ndarray`	moment of the cross section of the density matrix, `ndim=1`	required
`moment1`	`ndarray`	moment of the cross section of the density matrix, `ndim=1`	required
`zero_eps`	`float`	threshold for zero	`0.001`

Returns:

Name	Type	Description
`ret`	`bool`	whether two moments are similar

`numqi.entangle.group_dm_cross_section_moment(moment, zero_eps=0.0001)`

group the moments of the cross section of the density matrix

Parameters:

Name	Type	Description	Default
`moment`	`ndarray`	moments of the cross section of the density matrix, `ndim=2`	required
`zero_eps`	`float`	threshold for zero	`0.0001`

Returns:

Name	Type	Description
`group_list`	`list[list[int]]`	list of groups of indices