Structure of Quantum Theory

The recently introduced frameworks of quantum circuits with classical or quantum control (QC-CCs or QC-QCs) of causal order provide models of higher-order generalised quantum circuits in which a classical or quantum control system determines the order in which different agents apply their operations. Classical control endows QC-CCs with a well-defined causal order, whereas quantum control endows QC-QCs with a so-called indefinite causal order. Importantly, in both frameworks the control state need not be fixed in advance but may instead be established during the computation, giving rise to so-called dynamical causal order.
While the information-processing advantages of processes with indefinite causal order have been extensively investigated in various information processing tasks of quantum metrology, quantum communication, or quantum channel discrimination, the computational power of dynamical causal order—either classically or coherently controlled—has remained largely unexplored.
In this work, we investigate the information-processing advantages of quantum circuits with dynamical causal order over their non-dynamical counterparts through a new multipartite quantum channel discrimination task. In the tripartite setting, we establish a separation showing that both QC-CCs and QC-QCs with dynamical causal order outperform the corresponding quantum circuits with non-dynamical causal order. We then extend our analysis to the fourpartite setting, where we prove a separation at the corresponding quantum channel discrimination task between three distinct notions of dynamicality: non-dynamical causal order, dynamical but non-influenceable causal order, and dynamical and influenceable causal order. Our results therefore allow us to better understand the extent to which dynamical causal order—whether classically or coherently controlled—can be a computational resource.