dq.slindbladian

slindbladian(H: QArrayLike, jump_ops: list[QArrayLike]) -> QArray

Returns the Lindbladian superoperator (in matrix form).

The Lindbladian superoperator $\mathcal{L}$ is defined by: $$\mathcal{L} (\rho) = -i[H,\rho] + \sum_{k=1}^N \mathcal{D}[L_k] (\rho),$$

where $H$ is the system Hamiltonian, $\{L_k\}$ is a set of $N$ jump operators (arbitrary operators) and $\mathcal{D}[L]$ is the Lindblad dissipation superoperator (see dq.sdissipator()).

The vectorized form of this superoperator is: $$-i (I_n \otimes H) + i (H^\mathrm{T} \otimes I_n) + \sum_{k=1}^N \left( L_k^* \otimes L_k - \frac{1}{2} (I_n \otimes L_k^\dag L_k) - \frac{1}{2} (L_k^\mathrm{T} L_k^* \otimes I_n) \right).$$

Note

This superoperator is also sometimes called Liouvillian.

Parameters

• H (qarray-like of shape (..., n, n)) –

  Hamiltonian.

• jump_ops (list of qarray-like, each of shape (..., n, n)) –

  List of jump operators.

Returns

(qarray of shape (..., n^2, n^2)) Lindbladian superoperator.

See also

• dq.lindbladian(): applies the Lindbladian superoperator to a state using only $n\times n$ matrix multiplications.