Numerical Lindblad evolution and positivity

For a finite-dimensional system, take Hermitian H, dimensionless jump operators L-r, and nonnegative rates gamma-r with inverse-time units. The GKSL equation combines the Hamiltonian commutator with jump and anticommutator terms. With fixed coefficients this generates a completely positive trace-preserving semigroup. Complete positivity matters because the map must remain positive when the system is considered alongside an untouched ancillary system, not merely for a few tested density matrices.

A physical derivation may invoke weak coupling, short environmental correlations and a secular or other controlled approximation. Those assumptions are not implied by observing that a fitted generator has a convenient matrix form. Strongly correlated initial system-environment states, memory effects or an inadequate Hilbert cutoff can require a different model. Identify whether the equation is stipulated phenomenology, a derived approximation or a benchmark, and vary numerical controls separately from those physical assumptions.

Choose zero Hamiltonian and a jump operator taking excited state one to ground state zero. The excited population decays with the rate gamma, while an off-diagonal coherence decays at half that rate. The ground population follows from unit trace. A coherent initial pure state generally becomes mixed at intermediate times before reaching the ground state. Matching only its populations would miss part of the benchmark.

For a Hermitian two-by-two trace-one matrix, nonnegative diagonal entries and nonnegative determinant ensure positive semidefiniteness. Its purity is another useful diagnostic, but purity alone does not replace the eigenvalue test for general matrices. The Bloch vector of a mixed qubit lies inside the unit sphere; a pure-state-only display can reproduce its populations while having too much coherence. Such a display must not be interpreted as a dissipative evolution solver.

A generic ODE integrator can preserve trace and Hermiticity while generating a negative eigenvalue at a finite timestep. Forward Euler makes this visible immediately for a sufficiently large amplitude-damping step. Smaller steps can reduce a particular violation, but positivity for a few initial states does not establish complete positivity of the map for every system-and-ancilla state. Exponential or specially constructed channel updates offer stronger structure when their implementation is controlled.

Monitor trace, Hermiticity, the smallest eigenvalue, observable convergence and solver residuals together. Clipping negative eigenvalues or renormalizing after a step changes the method and can hide rather than quantify its error. Adaptive tolerances and Hilbert dimension must be varied until the target observable stabilizes; positivity at each finite size does not certify that the chosen bath model or truncated operators describe the intended experiment. Numerical stability and physical validity remain separate questions.