Superposition of causal order in quantum walks: non-Markovianity and causal asymmetry

Abstract

We set the criteria for a quantum walk to exhibit nontrivial dynamics when placed in an indefinite causal order and study two-period quantum walks when the evolution operator is arranged in a causally ordered sequence and in an indefinite causal order using quantum switch. When either forward or backward causal sequence is implemented, one observes a causal asymmetry in the dynamics, in the sense that the reduced dynamics of the coin state is more non-Markovian for one particular temporal order of operations than that of the other. When the dynamics is defined using evolution operators in a superposition of causal orders, the reduced dynamics of the coin space exhibit higher non-Markovian behavior than either of the definite causal orders. This effect can be interpreted as a Parrondo-like effect in non-Markovianity of the reduced state dynamics of the coin. We further generalize the qualitative description of our results pertaining to the dynamics when the walk has a higher number of periods.

Communication advantage of quantum compositions of channels from non-Markovianity

“…It was also shown in Ref. [44] that in quantum random walks where the dynamics of the reduced coin state is non-Markovian [45], by placing the walk's evolution operators in an indefinite causal order, there is an increase in non-Markovianity as compared to a fixed causal order. Even more recently, it was shown that for channels that are non-Markovian but do not offer backflow of information, known as eternally non-Markovian channels [46], such that its non-Markovianity is "hidden," placing these channels in the quantum switch setup can activate this hidden non-Markovianity to allow backflow of information [47].…”

Effects of non-Markovianity in quantum compositions of channels with indefinite causal order

“…We suggest that the quantum switch's advantages has nonnegligible contributions from non-Markovian effects. As far as we are aware, despite some works discussing non-Markovianity and indefinite causal order together [22][23][24][25], we are the first to make this connection and proposal 1 .…”

“…For example, it was shown in Ref. [23] that in quantum random walks where the dynamics of the reduced coin state are non-Markovian [24], by placing the walk's evolution operators in an indefinite causal order, there is an increase in non-Markovianity as compared to a fixed causal order. It was also shown that for channels that are non-Markovian but do not offer backflow of information, e.g., eternally non-Markovian channels [76], such that its non-Markovianity is "hidden", placing these channels in the quantum switch setup can activate this hidden non-Markovianity to allow backflow of information [25].…”