Memory in quantum processes with indefinite time direction and causal order

We examine the emergence of dynamical memory effects in quantum processes having indefinite time direction and causal order. In particular, we focus on the class of phase-covariant qubit channels, which encompasses some of the most significant paradigmatic open quantum system models. In order to assess the memory in the time evolution of the system, we utilize the trace distance and the entanglement based measures of non-Markovianity. While the indefinite time direction is obtained through the quantum time flip operation that realizes a coherent superposition of forward and backward processes, the indefinite causal order is achieved via the quantum switch map, which implements two quantum processes in a coherent superposition of their two possible orders. Considering various different families of phase-covariant qubit channels, we demonstrate that, when implemented on memoryless quantum processes, both the quantum time flip and the quantum switch operations can generate memory effects in the dynamics according to the trace distance based measure under certain conditions. On the other hand, with respect to the entanglement based measure, we show that neither the quantum time flip nor the quantum switch could induce dynamical memory for any of the considered phase-covariant channels.