P14: Quantum-information approach to geometric pumping
Within the RTG we studied [1,2] the geometric phase in the response of an observable measured externally to a driven open quantum system. We found that it can be obtained most straightforwardly when the mixed state is considered as a vector evolving in a linear space of operators (Liouville space). Physically, this geometric phase is accumulated by the external detector  and the related gauge transformations correspond to recalibrations of the mixed state of this detector. In the Liouville-space approach, linear physical restrictions (Hermiticity and trace preservation) on the dynamics are easily translated into separate restrictions on the gauge freedom of each individual state-evolution mode. However, their inseparable sum, the mixed-state Liouville-vector, is also subject to a non-trivial global constraint: The density operator that it represents must be positive (non-negative eigenvalues) which is obeyed automatically only for closed system dynamics. This fundamental constraint has so far been completely ignored in all treatments of geometric phases of open systems. For example, it has not been shown – or even addressed – whether, after the adiabatic approximation in Liouville space, the approximate evolution is still (completely) positive and strictly-speaking still makes any physical sense (no negative probabilities).
This constraint of positivity also leads to striking issues when deriving the full counting statistics (FCS) of transport from an open system including an ideal detector. One finds that after a possible gauge transformation originally positive quantities in the transformed “picture” may turn negative . Importantly, we have shown that this issue can only be noticed in the higher moments / cumulants of the statistics, i.e., in the noise (or higher ones). Although the loss of positivity does not mean the final results computed in this gauge are unphysical (they are gauge invariant), this puzzling situation reflects that Liouville space is “too big”: It contains more operators than just the mixed states and gauge transformations are ignorant of the boundary between them.
In this project, we aim to set up a new approach to geometric pumping based on quantum-measurement and information theory in which all gauge transformations make physical sense, just as they do in the familiar setting of closed systems. The project involves three steps. We will first express the exact dynamics of the open system (including a detector) in terms of measurement-outcome conditioned evolutions. Here, we can rely on the established framework of so-called “quantum instruments” and Kraus superoperators. In our other RTG project with V. Reimer, we have made progress in applying this formalism to open systems with continuous environments and have explicitly computed such Kraus operators for exactly solvable systems. One can then explicitly see that they already contain the time-evolution scales of the dissipative system that would result when tracing out the effective environment (discarding measurement outcomes) before actually performing this trace. This indicates that it may be possible to take a second step and perform the adiabatic and long-time limit relative to the dissipative time-evolution scales before tracing over the effective environment. This has the crucial advantage of automatically enforcing positivity of the density operator, before and after any gauge transformations. One should note that, for continuous environments, the trace over the environment and the long-time limit are non-trivial operations which cannot be interchanged without careful consideration. In the computation of the Kraus operators this can be carefully done, providing a new route to clarify, or even solve, the outstanding issue of “unphysical” gauge transformations in Liouville-space pumping formalism. Thirdly, we will explore practically fea-sible techniques for performing the trace over the environment and derive an adiabatic theo-rem and geometric phase in which all geometric quantities comply with all the fundamental requirements of quantum statistics (normalization, Hermicity as well as positivity).
 T. Pluecker, M. R. Wegewijs, and J. Splettstoesser, Phys. Rev. B 95, 155431 (2017)
 T. Pluecker, M. R. Wegewijs, and J. Splettstoesser, submitted to Phys. Rev. B, arXiv:1711.10431