No quantum advantage for violating fixed-order inequalities?

Abstract

In standard quantum theory, the causal relations between operations are fixed. One can relax this notion by allowing for dynamical arrangements, where operations may influence the causal relations of future operations, as certified by violation of fixed-order inequalities, e.g., the k-cycle inequalities. Another, non-causal, departure further relaxes these limitations, and is certified by violations of causal inequalities. In this paper, we explore the interplay between dynamic and indefinite causality. We study the k-cycle inequalities and show that the quantum switch violates these inequalities without exploiting its indefinite nature. We further introduce non-adaptive strategies, which effectively remove the dynamical aspect of any process, and show that the k-cycle inequalities become novel causal inequalities; violating k-cycle inequalities under the restriction of non-adaptive strategies requires non-causal setups. The quantum switch is known to be incapable of violating causal inequalities, and it is believed that a device-independent certification of its causal indefiniteness requires extended setups incorporating spacelike separation. This work reopens the possibility for a device-independent certification of the quantum switch in isolation via fixed-order inequalities instead of causal inequalities. The inequalities we study here, however, turn out to be unsuitable for such a device-independent certification. In this work, we initiate the question posed by the title. This question, however, remains unanswered.