utils

`numqi.utils.hf_num_state_to_num_qubit(num_state, kind='exact')`

convert the number of states to the number of qubits

Parameters:

Name	Type	Description	Default
`num_state`	`int`	number of states	required
`kind`	`str`	'exact', 'ceil', 'floor'	`'exact'`

Returns:

Name	Type	Description
`ret`	`int`	number of qubits

`numqi.utils.hf_complex_to_real(x)`

convert a complex matrix to a real matrix

complex -> [[real,-imag], [imag,real]]

\[ A\in\mathbb{C}^{m\times n}\mapsto\begin{bmatrix} \Re[A] & -\Im[A]\\ \Im[A] & \Re[A] \end{bmatrix}\in\mathbb{R}^{2m\times2n} \]

Parameters:

Name	Type	Description	Default
`x`	`(ndarray, Tensor)`	a complex matrix, shape=(...,m,n), support batch	required

Returns:

Name	Type	Description
`ret`	`(ndarray, Tensor)`	a real matrix, shape=(...,2m,2n)

`numqi.utils.hf_real_to_complex(x)`

convert a real matrix to a complex matrix

[[real,-imag], [imag,real]] -> complex

Parameters:

Name	Type	Description	Default
`x`	`(ndarray, Tensor)`	a real matrix, shape=(...,2dim0,2dim1), support batch	required

Returns:

Name	Type	Description
`ret`	`(ndarray, Tensor)`	a complex matrix, shape=(...,dim0,dim1)

`numqi.utils.partial_trace(rho, dim, keep_index)`

partial trace of a density matrix

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	a density matrix, shape=(dim,dim)	required
`dim`	`tuple[int]`	shape of the density matrix	required
`keep_index`	`set[int]`	the indices to keep	required

Returns:

Name	Type	Description
`ret`	`ndarray`	the partial trace of the density matrix, shape=(dim[keep_index], dim[keep_index])

`numqi.utils.get_fidelity(rho0, rho1)`

get the fidelity of two density matrices or pure states

Parameters:

Name	Type	Description	Default
`rho0`	`(ndarray, Tensor)`	pure state (ndim=1) or density matrix (ndim=2)	required
`rho1`	`(ndarray, Tensor)`	pure state or density matrix	required

Returns:

Name	Type	Description
`ret`	`(float, Tensor)`	the fidelity of the two states

`numqi.utils.get_von_neumann_entropy(rho, _torch_logm='eigen')`

get the von Neumann entropy of a density matrix wiki-link

Parameters:

Name	Type	Description	Default
`rho`	`(ndarray, Tensor)`	a density matrix, shape=(dim,dim)	required
`_torch_logm`	`(str, tuple)`	'eigen' or ('pade',num_sqrtm,pade_order), 'pade' is used only when requires_grad	`'eigen'`

Returns:

Name	Type	Description
`ret`	`float`	the von Neumann entropy of the density matrix

`numqi.utils.get_Renyi_entropy(rho, alpha)`

get the Renyi entropy of a density matrix

Parameters:

Name	Type	Description	Default
`rho`	`(ndarray, Tensor)`	a density matrix, shape=(dim,dim)	required
`alpha`	`float`	the order of the Renyi entropy	required

Returns:

Name	Type	Description
`ret`	`float`	the Renyi entropy of the density matrix

`numqi.utils.get_purification(rho, dimR=None, seed=None)`

get the purification of a density matrix. all purification are connected by Stiefel manifold

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	a density matrix, shape=(dim,dim)	required
`dimR`	`(int, None)`	the dimension of the purification, dimR>=dim, if None, dimR=dim	`None`
`seed`	`(int, None)`	random seed	`None`

Returns:

Name	Type	Description
`ret`	`ndarray`	a purification of the density matrix, shape=(dim,dimR)

`numqi.utils.get_trace_distance(rho, sigma)`

get the trace distance of two density matrices. abs is not a good choice for loss function, so no torch-version

Parameters:

Name	Type	Description	Default
`rho`	`ndarray`	a density matrix, shape=(dim,dim)	required
`sigma`	`ndarray`	a density matrix, shape=(dim,dim)	required

Returns:

Name	Type	Description
`ret`	`float`	the trace distance of the density matrices

`numqi.utils.get_purity(rho)`

get the purity of a density matrix

Parameters:

Name	Type	Description	Default
`rho`	`(ndarray, Tensor)`	a density matrix, shape=(dim,dim)	required

Returns:

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

`numqi.utils.get_relative_entropy(rho, sigma, tr_rho_log_rho=None, _torch_logm=('pade', 6, 8))`

get the relative entropy of two density matrices wiki-link

\[ S(\rho,\sigma) = \mathrm{Tr}(\rho \log\rho - \rho \log\sigma) \]

Parameters:

Name	Type	Description	Default
`rho`	`(ndarray, Tensor)`	a density matrix, shape=(dim,dim)	required
`sigma`	`(ndarray, Tensor)`	a density matrix, shape=(dim,dim)	required
`tr_rho_log_rho`	`(float, None)`	tr(rho log(rho)), if None, calculate it	`None`
`_torch_logm`	`(str, tuple)`	'eigen' or ('pade',num_sqrtm,pade_order), 'pade' is used only when requires_grad	`('pade', 6, 8)`

Returns:

Name	Type	Description
`ret`	`(float, Tensor)`	the relative entropy of the density matrices

`numqi.utils.get_tetrahedron_POVM(num_qubit=1)`

Tetrahedron POVM

wiki-link

Parameters:

Name	Type	Description	Default
`num_qubit`	`int`	number of qubits	`1`

Returns:

Name	Type	Description
`ret`	`ndarray`	shape=(N, m, m) where `N=4num_qubit`, `m=2num_qubit`

`numqi.utils.is_positive_semi_definite(np0, shift=0.0, hermitian_eps=None)`

check whether a matrix is positive semi-definite. Cholesky decomposition is used to check the positive semi-definite property.

Determining whether a symmetric matrix is positive-definite stackexchange-link

A practical way to check if a matrix is positive-definite stackexchange-link

Parameters:

Name	Type	Description	Default
`np0`	`ndarray`	matrix, must be Hermitian	required
`shift`	`float`	shift the matrix by a scalar, e.g. `shift=1e-10` or `shift=-1e-10`	`0.0`
`hermitian_eps`	`float \| None`	threshold for the Hermitian property, raise AssertionError if fail. if `None`, then skip the check	`None`

Returns:

Name	Type	Description
`tag`	`bool`	whether the matrix is positive semi-definite

`numqi.utils.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`