Conditions for strictly purity-decreasing quantum Markovian dynamics

For non-diagonalizable L, the spectral theorem of linear algebra asserts that it is possible to expand L in its generalized eigen-supervectors.

“For nondiagonalizable L , the spectral theorem [31] of linear algebra asserts that it is possible to expand L in its generalized eigen-supervectors.”

For finite-dimensional separable Hilbert spaces, quantum mechanics is essentially matrix analysis.

“Indeed, quantum mechanics in separable Hilbert spaces turns out to be essentially matrix analysis [30, 31].”

The differential equation dx(t)/dt = A x(t) has solution x(t) = e^{tA} c.

“We know from matrix analysis (via the theory of matrix functions) that the differential equation in (45) has as solution [31, 32, 33]”

If a given eigenvalue λ_α has g_α < p_α, with p_α ≥ 2, the spectral theorem asserts it is possible to find p_α independent eigenvectors {|ζ_α^(m)〉〉}, m=1,...,p_α, all with eigenvalue λ_α.

“If a given eigenvalue λ α has g α &lt; p α , with p α ≥ 2, the spectral theorem [24, 31, 46, 47] of linear algebra asserts that it is possible to find p α independent eigenvectors, {| ζ α ( m ) 〉 〉} , m = 1 , 2 , . . . , p α , all with eigenvalue λ α .”

In the Markovian limit with independent white noises, the dissipator depends only on the average density operator and takes diagonal Lindblad form.

“In particular, the dissipator now only depends on the average density operator hρ st ðtÞi, that we hereafter denote by ρðtÞ to simplify the notation.”