No quantum advantage for violating fixed-order inequalities?

TL;DR

This work investigates whether quantum control of causal order can yield advantages in fixed-order inequalities, focusing on the $k$-cycle game. Using the process matrix framework, it shows that cyclic and sparse quantum switches can deterministically win the $k$-cycle game under adaptive strategies, while non-adaptive strategies suppress dynamical causality. Moreover, when all parties are treated as playing with non-adaptive control, the $n$-cycle inequality becomes a facet of the causal polytope, and causal or fixed-order correlations cannot beat the bound, though non-causal processes can violate it. The results challenge the straightforward device-independent certification of isolated quantum switches via fixed-order inequalities and raise questions about graph-based generalizations and potential quantum-over-classical advantages in other setups.

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.

Figures (3)

• Figure 1: In a fixed spacetime (a), the causal relations among the parties are fixed beforehand and independent of their actions. (b) In contrast, when the parties are placed on a dynamical spacetime, such as provided by general relativity, then each party's action, e.g., of party $A$, may not only influence the information accessible within the future light cone, but also the causal relations among the parties in the future, i.e., $C, D$.
• Figure 2: We investigate the $k$-cycle game within the process formalism. In case (a), two additional (non-playing) parties, $P$ and $F$, can influence the game. In particular, $P$ and $F$ might employ adaptive strategies where their actions depend on the value of $s$, which specifies who among the playing parties $A,B,C,D, \dots$ is the sender. If they use a non-adaptive strategy, their operations are fixed, and the scenario depicted in (a) is equivalent to the situation shown in (b), where all the parties involved are playing parties. We refer to the game in this scenario as the $n$-cycle game.
• Figure 3: (a) In the cyclic quantum switch, the target system traverses in sequence all parties in a cyclic order. The control system specifies the initial party. (b) The sparse quantum switch: Depending on the value of the control system, a single link between two parties is activated. All remaining parties are causally disconnected.