Gell-Mann matrix

numqi.gellmann

`numqi.gellmann.gellmann_matrix(i, j, d)`

get the Gell-Mann matrix wiki-link/Gell-Mann-matrices wiki-link/Generalizations-of-Pauli-matrices github-link/CQuIC/pysme arxiv-link/0806.1174

normalization condition: Tr(Mi Mj)=2 delta_ij

Parameters:

Name	Type	Description	Default
`i`	`int`	row indices of Gell-Mann matrix, 0<=i<d	required
`j`	`int`	column indices of Gell-Mann matrix, 0<=j<d	required
`d`	`int`	dimension of Gell-Mann matrix	required

Returns:

Name	Type	Description
`ret`	`ndarray`	Gell-Mann matrix. if `i<j`, Pauli-X like; if `i>j`, Pauli-Y like; if `i=j=0`, identity; if `i=j>0`, Pauli-Z like

`numqi.gellmann.all_gellmann_matrix(d, /, tensor_n=1, with_I=True)`

get all Gell-Mann matrices

Parameters:

Name	Type	Description	Default
`d`	`int`	dimension of Gell-Mann matrix	required
`tensor_n`	`int`	tensor product of Gell-Mann matrices	`1`
`with_I`	`bool`	include identity matrix or not. if `True`, the last item is identity matrix.	`True`

Returns:

Name	Type	Description
`ret`	`ndarray`	Gell-Mann matrices. Ordering: PauliX, PauliY, PauliZ, I. if `tensor_n=1`, shape=(d2, d, d); if `tensor_n>1`, shape=(d(2n), dn, dn)

`numqi.gellmann.matrix_to_gellmann_basis(A)`

convert a matrix to Gell-Mann basis

ordering: PauliX, PauliY, PauliZ, I

Parameters:

Name	Type	Description	Default
`A`	`(ndarray, Tensor)`	matrix, support batch, `shape=(...,d,d)`	required

Returns:

Name	Type	Description
`ret`	`(ndarray, Tensor)`	vector in Gell-Mann basis, `shape=(...,d**2)`

`numqi.gellmann.gellmann_basis_to_matrix(vec)`

convert a vector in Gell-Mann basis to a matrix

ordering: PauliX, PauliY, PauliZ, I

Parameters:

Name	Type	Description	Default
`vec`	`(ndarray, Tensor)`	vector in Gell-Mann basis, `shape=(...,d**2)`	required

Returns:

Name	Type	Description
`ret`	`(ndarray, Tensor)`	matrix, `shape=(...,d,d)`

`numqi.gellmann.dm_to_gellmann_basis(dm, with_rho0=False)`

convert density matrix to a vector in Gell-Mann basis

Parameters:

Name	Type	Description	Default
`dm`	`(ndarray, Tensor)`	density matrix, 2d array (support batch)	required
`with_rho0`	`bool`	include rho0 or not. if `True`, the last item is not included.	`False`

Returns:

Name	Type	Description
`ret`	`(ndarray, Tensor)`	real vector in Gell-Mann basis, `shape=(...,d2-1)` if `with_rho0=False`, otherwise `shape=(...,d2)`

`numqi.gellmann.gellmann_basis_to_dm(vec)`

convert vector in Gell-Mann basis to a density matrix

Parameters:

Name	Type	Description	Default
`vec`	`(ndarray, Tensor)`	vector in Gell-Mann basis, `shape=(...,d**2-1)`	required

Returns:

Name	Type	Description
`ret`	`(ndarray, Tensor)`	density matrix, `shape=(...,d,d)`

`numqi.gellmann.dm_to_gellmann_norm(dm)`

get the norm of a density matrix in Gell-Mann basis, For traceless hermitian matrix, the norm is the Frobenius norm divided by sqrt(2)

Parameters:

Name	Type	Description	Default
`dm`	`ndarray`	density matrix, 2d array (support batch)	required

Returns:

Name	Type	Description
`ret`	`ndarray`	norm of dm

`numqi.gellmann.get_density_matrix_distance2(rho, sigma)`

Get the distance between two density matrices in Gell-Mann basis. Equivalent to the Frobenius distance over 2.

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	density matrix, 2d array	required
`sigma`	`ndarray`	density matrix, 2d array	required

Returns:

Name	Type	Description
`ret`	`float`	distance