matrix space

numerical range

numqi.matrix_space.get_matrix_numerical_range(matA, num_point=100)

get the numerical range of a square matrix A, A could be complex

\[W(A)=\left\{ x^{\dagger}Ax:x\in\mathbb{C}^{n},\lVert x\rVert_{2}=1\right\}\]

see arxiv-link appendix A

Parameters:

Name	Type	Description	Default
matA	ndarray	a square matrix, could be complex	required
num_point	int	number of points to sample the numerical range	100

Returns:

Name	Type	Description
ret	ndarray	a 1d array of length num_point, dtype is complex, the numerical range of matA

numqi.matrix_space.get_matrix_numerical_range_along_direction(matA, alpha, kind='max')

get the numerical range of a square matrix A along a direction alpha

\[W^{\alpha}\left(A\right)=\left\{ x\in\mathbb{R}:xe^{i\alpha}\in W\left(A\right)\right\}\]

This function might give wrong results especially when the numerical range is not smooth, In that case, a warning will be printed.

see arxiv-link appendix A

Parameters:

Name	Type	Description	Default
matA	ndarray	a square matrix, could be complex	required
alpha	float	the direction to sample the numerical range, in radian	required
kind	str	'max' or 'min', the maximum or minimum value along the direction	'max'

Returns:

Name	Type	Description
maximum	float	the maximum value along the direction \(e^{i\alpha}\)
EVC	ndarray	the eigenvector corresponding to the maximum value

numqi.matrix_space.get_real_bipartite_numerical_range(mat, kind='min', method='eigen')

get the real bipartite numerical range of a square matrix A

\[B\in\mathbb{R}^{mn\times mn},W^{x+iy}\left(B\right)=W^{\arctan y/x}\left(\frac{xB+iyB^{\Gamma}}{\sqrt{x^{2}+y^{2}}}\right)\]

where \(W^{\arctan y/x}(A)\) is the numerical range of \(A\) along the direction \(e^{i\arctan y/x}\), see numqi.matrix_space.get_matrix_numerical_range_along_direction()

see arxiv-link

Parameters:

Name	Type	Description	Default
mat	ndarray	4-dimensional numpy array of shape (dimA,dimB,dimA,dimB), must be real	required
kind	str	'min' or 'max', the boundary of numerical range to compute	'min'
method	str	'rotation' or 'eigen'. If 'rotation': use numqi.matrix_space.get_matrix_numerical_range_along_direction() to compute the numerical range. 'rotation' method might give wrong results, especially when numerical range is not smooth. If 'eigen': (see theorem 2 in arxiv-link) use the eigenvalue decomposition to compute the numerical range. the matrix mat must be symmetric. Two methods usually give the same output and one might be faster (TODO benchmark)	'eigen'

Returns:

Name	Type	Description
ret	float	the boundary of the real bipartite numerical range

numqi.matrix_space.get_joint_algebraic_numerical_range(op_list, direction, return_info=False, use_tqdm=True)

get the joint algebraic numerical range (JANR) of a list of operators along a direction

\[ L(A_{1},A_{2},\cdots,A_{r})=\left\{ a\in\mathbb{C}^{r}:\rho\in\mathbb{C}^{d\times d},\rho\succeq0,\mathrm{Tr}[\rho]=1,a_{i}=\mathrm{Tr}[A_{i}\rho]\right\} \]

\[ \max\;\beta \]

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

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
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