The solution of the inverse problem for the time-dependent density matrix means finding the time-dependent Hamiltonian operator $\hat{H}$ corresponding to a given time-dependent density matrix $\hat{\rho }$. We point out that the standard requirements for the time-dependent density operator imply a number of compatibility relations. Accordingly, a given $N\times N$ density matrix might result from a whole family of Hamiltonians, characterized by $N$ free, real parameters. The cases of a 2×2 and 3×3 density matrix and of the free-fermion chain are worked out explicitly.

• Received 6 July 1981

DOI:https://doi.org/10.1103/PhysRevA.25.1815