filter_functions.superoperator module

This module provides some functions related to superoperators and quantum maps.

Functions

liouville_representation()

Calculate the Liouville representation of a unitary with respect to a basis

liouville_to_choi()

Convert from Liouville to Choi matrix representation.

liouville_is_CP()

Check if superoperator in Liouville representation is completely positive.

liouville_is_cCP()

Check if superoperator in Liouville representation is conditional CP.

liouville_is_CP(superoperator: ndarray, basis: Basis, return_eig: bool | None = False, atol: float | None = None) → bool | Tuple[bool, Tuple[ndarray, ndarray]]

Test if a Liouville superoperator is completely positive (CP).

Parameters:

superoperator: ndarray, shape (…, d**2, d**2)

The superoperator in Liouville representation to be checked for CPness.

basis: Basis, shape (d**2, d, d)

The operator basis defining the Liouville representation.

return_eig: bool, optional

Return the tuple of eigenvalues and eigenvectors of the Choi matrix. The default is False.

atol: float, optional

Absolute tolerance for the complete positivity.

Returns:

CP: bool, (shape (…,))

The (array, if broadcasted) of bools indicating if superoperator is CP.

(D, V): Tuple[ndarray, ndarray]

The eigenvalues and eigenvectors of the Choi matrix (only if return_eig is True).

See also

liouville_representation

Calculate Liouville representation of a unitary.

Liouville_to_choi

Convert from Liouville to Choi matrix representation.

liouville_is_cCP

Test if a superoperator is conditional CP.

Notes

A superoperator \(\mathcal{S}\) is completely positive (CP) if and only if its Choi matrix representation is positive semidefinite:

\[\mathcal{S}\text{ is CP }\Leftrightarrow \mathrm{choi}(\mathcal{S})\geq 0.\]

liouville_is_cCP(superoperator: ndarray, basis: Basis, return_eig: bool | None = False, atol: float | None = None) → bool | Tuple[bool, Tuple[ndarray, ndarray]]

Test if a Liouville superoperator is conditional completely positive.

Parameters:

superoperator: ndarray, shape (…, d**2, d**2)

The superoperator in Liouville representation to be checked for cCPness

basis: Basis, shape (d**2, d, d)

The operator basis defining the Liouville representation.

return_eig: bool, optional

Return the tuple of eigenvalues and eigenvectors of the Choi matrix projected on the complement of the maximally entangled state. The default is False.

atol: float, optional

Absolute tolerance for the complete positivity.

Returns:

cCP: bool, (shape (…,))

The (array, if broadcasted) of bools indicating if superoperator is cCP

(D, V): Tuple[ndarray, ndarray]

The eigenvalues and eigenvectors of the projected Choi matrix (only if return_eig is True).

See also

liouville_representation

Calculate Liouville representation of a unitary.

Liouville_to_choi

Convert from Liouville to Choi matrix representation.

liouville_is_CP

Test if a superoperator is CP.

Notes

A superoperator \(\mathcal{S}\) is conditional completely positive (cCP) if and only if its Choi matrix projected on the complement of the maximally entangled state is positive semidefinite:

\[\mathcal{S}\text{ is cCP }\Leftrightarrow Q\mathrm{choi}(\mathcal{S})Q\geq 0\]

with \(Q = \mathbb{I} - |\Omega\rangle\langle\Omega|\).

liouville_representation(U: ndarray, basis: Basis) → ndarray

Get the Liouville representaion of the unitary U with respect to the basis.

Parameters:

U: ndarray, shape (…, d, d)

The unitary.

basis: Basis, shape (d**2, d, d)

The basis used for the representation, e.g. a Pauli basis.

Returns:

R: ndarray, shape (…, d**2, d**2)

The Liouville representation of U.

Notes

The Liouville representation of a unitary quantum operation \(\mathcal{U}:\rho\rightarrow U\rho U^\dagger\) is given by

\[\mathcal{U}_{ij} = \mathrm{tr}(C_i U C_j U^\dagger)\]

with \(C_i\) elements of the basis spanning \(\mathbb{C}^{d\times d}\) with \(d\) the dimension of the Hilbert space.

liouville_to_choi(superoperator: ndarray, basis: Basis) → ndarray

Convert from Liouville to Choi matrix representation.

Parameters:

superoperator: ndarray, shape (…, d**2, d**2)

The Liouville representation of a superoperator.

basis: Basis, shape (d**2, d, d)

The operator basis defining the Liouville representation.

Returns:

choi: ndarray, shape (…, d**2, d**2)

The Choi matrix representation of the superoperator.

See also

liouville_representation

Calculate Liouville representation of a unitary.

liouville_is_CP

Test if a superoperator is completely positive (CP).

liouville_is_cCP

Test if a superoperator is conditional CP.

Notes

The Choi matrix is given by

\[\begin{split}\mathrm{choi}(\mathcal{S}) &= (\mathbb{I}\otimes\mathcal{S}) (|\Omega\rangle\langle\Omega|) \\ &= \sum_{ij} E_{ij}\otimes\mathcal{S}(E_{ij}) \\ &= \sum_{ij}\mathcal{S}_{ij} C_j^T\otimes C_i\end{split}\]

where \(|\Omega\rangle\) is a maximally entangled state, \(E_{ij} = |i\rangle\langle j|\), and \(C_i\) are the basis elements that define the Liouville representation \(\mathcal{S}_{ij}\) [Mer13].

References

[Mer13]

Merkel, S. T. et al. Self-consistent quantum process tomography. Physical Review A - Atomic, Molecular, and Optical Physics, 87, 062119 (2013). https://doi.org/10.1103/PhysRevA.87.062119