Reconstructing non-Markovian quantum dynamics with limited control

The dynamics of an open quantum system can be fully described and tomographically reconstructed if the experimenter has complete control over the system of interest. Most real-world experiments do not fulfill this assumption, and the amount of control is restricted by the respective experimental setup. That is, the set of performable manipulations of the system is limited. In this paper, we provide a general reconstruction scheme that yields an operationally well-defined description of non-Markovian quantum dynamics for a large class of realistic experimental situations. The resultant “restricted” descriptor for the process, surprisingly, does not have the property of complete positivity. Based on these restricted dynamics, we construct witnesses for correlations and the presence of memory effects. We demonstrate the applicability of our framework for the two important cases where the set of performable operations comprises only unitary operations or projective measurements, respectively, and show that it provides a powerful tool for the description of quantum control experiments.

Figure 1

Depiction of a multistep process. The local operations ${\mathcal{A}}_k$ act at times $t_k$. In between, the dynamics is governed by the system-environment unitaries ${\mathcal{U}}_{k+1:k}$. The final state ${\rho }^{\prime }={\rho }^{\prime }\left({\mathbf{A}}_{N-1:0}\right)$ is determined by QST at the time $t_N$. The corresponding process tensor ${\mathcal{T}}^{N:0}$ is depicted by the dotted line.

Figure 2

Preparation of a product state. Two copies of the initial state ${\rho }_{\mathcal{SE}}$ are created and their system degrees of freedom are swapped by the operation $S_{12}^{\left(\mathcal{S}\right)}$. Subsequently, the degrees of freedom of the second copy are disregarded.