Quantum switch instabilities with an open control

The superposition of causal orders shows promise in various quantum technologies. However, the fragility of quantum systems arising from environmental interactions, leading to dissipative behavior and irreversibility, demands a deeper understanding of the possible instabilities in the coherent control of causal orders. In this work, we employ a collisional model to investigate the impact of an open control system on the generation of interference between two causal orders. We present the environmental instabilities for the switch of two arbitrary quantum operations and examine the influence of environmental temperature on each potential outcome of control post-selection. Additionally, we explore how environmental instabilities affect protocol performance, including switching between mutually unbiased measurement observables and refrigeration powered by causal order superposition, providing insights into broader implications. Using the quantum switch—a process involving a controlled operation followed by post-selection of the control—a system of interest can evolve under an effective dynamics governed by a superposition of operation orders. This work investigates how environmental influences on the control impact the desired superposition of orders in realistic scenarios.