Interplay between the Hilbert-space dimension of a control system and the memory induced by a quantum switch

Several recent studies have demonstrated the utility of the quantum switch as an important resource for enhancing the performance of various information processing tasks. In a quantum switch, the advantages appear significantly due to the coherent superposition of alternative configurations of the quantum components which are controlled by an additional control system. Here we explore the impact of increasing the Hilbert-space dimension of the control system on the performance of the quantum switch. In particular, we focus on a quantifier of the quantum switch through the emergence of non-Markovianity and explicitly study its behavior when we increase the Hilbert-space dimension of the control system. We observe that increasing the Hilbert-space dimension of the control system leads to the corresponding enhancement of the non-Markovian memory induced by it. Our study demonstrates how the dimension of the control system can be harnessed to improve the quantum-switch-based information processing or communication tasks.