Isotropy and control of dissipative quantum dynamics

We investigate the problem of what evolutions an open quantum system described by a time-local master equation can undergo with universal coherent controls. A series of conditions is given which exclude channels from being reachable by any unitary controls, assuming that the coupling to the environment is not being modified. These conditions primarily arise by defining decay rates for the generator of the dynamics of the open system, and then showing that controlling the system can only make these rates more isotropic. This forms a series of constraints on the shape and nonunitality of allowed evolutions, as well as an expression for the time required to reach a given goal. We give numerical examples of the usefulness of these criteria and explore some similarities they have with quantum thermodynamics.

Figure 1

We illustrate the principal ideas of this paper by showing a cross section of state space at two different times, with the small arrows indicating the direction of flow induced by a Lindbladian (the generator of memoryless dissipative dynamics). The large arrows correspond to the available controls. We see that different parts of the space are contracting at different rates; by rotating the system in time with Hamiltonian controls, some of these decay rates can be averaged together.

Figure 2

The graphs characterize the strength of Eqs. (5, 6, 7) for generalized amplitude damping Lindbladians as a function of the purity of the drift steady state—itself a function of the bath temperature. The top graph shows what fraction of the randomly generated channels satisfied all Eqs. (5)–(7) and, out of those, how many could be reached with a numerical optimization package [46] to within a distinguishability (given by the diamond norm) of at least $0.1\%$. The bottom graph shows what fraction of the same channels is ruled out by each of the conditions individually.

Figure 3

Sketch of a qutrit in a $\mathrm{\Lambda }$ configuration, where the Lindbladian causes the excited level to decay to the two ground states at different rates. The skew, ${\gamma }_1/{\gamma }_2$, characterizes the asymmetry in the system.

Figure 4

The graph shows how the dynamical map anisotropy conditions relate to the controllability of the nonsymmetric $\mathrm{\Lambda }$ system outlined in Fig. 3. The system drift had a skew of 10 and we attempted to reach maps generated by the same Lindbladian but with skews between 1 and 20. Plotted is the minimal distance (given by the diamond norm) to targets with different skews that could be reached with a numerical optimization package [46]. The thick red line at $10.3$ is the nonreachable boundary given by the majorization criteria—everything to the right of it is excluded—in excellent agreement with the numerical results. The slight bump just below 10 is due to the difficulty of numerically finding solutions which require rapidly oscillating control Hamiltonians.