qit.utils.superop_fp

qit.utils.superop_fp(L, tol=None)

Fixed point states of a Liouvillian superoperator.

Finds the intersection of the kernel of the Liouvillian L with the set of valid state operators, thus giving the set of fixed point states for the quantum channel represented by the master equation

\[\dot{\rho} = \text{inv_vec}(L \text{vec}(\rho)).\]

Let L.shape == (D, D) (and d = sqrt(D) be the dimension of the Hilbert space).

Returns the D*n array A, which contains as its columns a set of n vectorized orthogonal Hermitian matrices (with respect to the Hilbert-Schmidt inner product) which “span” the set of FP states in the following sense:

\[vec(\rho) = A c, \quad \text{where} \quad c \in \text{R}^n \quad \text{and} \quad c_1 = 1.\]

A[:,0] is the shortest vector in the Hermitian kernel of L that has trace 1, the other columns of A are traceless and normalized. Hence, A defines an (n-1)-dimensional hyperplane.

A valid state operator also has to fulfill \(\rho \ge 0\). These operators form a convex set in the Hermitian trace-1 hyperplane defined by A. Currently this function does nothing to enforce positivity, it is up to the user to choose the coefficients \(a_k\) such that this condition is satisfied.

Parameters

• L (array) – Liouvillian superoperator

• tol (float) – tolerance, singular values of L that are <= tol are treated as zero

Returns

array containing n vectorized Hermitian matrices as its columns, shape == (D, n)

Return type

array,