Admissible Causal Structures and Correlations

TL;DR

This work examines which causal structures among local quantum experiments can be realized without deviating from local quantum theory, using the process-matrix framework. It establishes a necessary graph-theoretic criterion, the siblings-on-cycles property, and conjectures its sufficiency, supported by a constructive quantum causal-model realization and numerical checks up to six-node graphs. The paper shows that graphs with cycles lacking siblings are inadmissible, and provides explicit model-parameter constructions that realize chordless siblings-on-cycles graphs, while analyzing how these structures constrain causal versus non-causal correlations through causal games and inequalities. The results clarify the limits of indefinite causal order in quantum theory, offer a graph-theoretic lens to assess admissibility, and point to future avenues in theory-independence, spacetime embedding, and potential information-processing protocols.

Abstract

It is well-known that if one assumes quantum theory to hold locally, then processes with indefinite causal order and cyclic causal structures become feasible. Here, we study qualitative limitations on causal structures and correlations imposed by local quantum theory. For one, we find a necessary graph theoretic criterion--the "siblings-on-cycles" property--for a causal structure to be admissible: Only such causal structures admit a realization consistent with local quantum theory. We conjecture that this property is moreover sufficient. This conjecture is motivated by an explicit construction of quantum causal models, and supported by numerical calculations. We show that these causal models, in a restricted setting, are indeed consistent. For another, we identify two sets of causal structures that, in the classical-deterministic case, give rise to causal and non-causal correlations respectively.

Figures (6)

• Figure 1: (a) A quantum circuit and its acyclic causal structure. (b) The quantum switch---an instance of the process-matrix framework---has a cyclic causal structure: Depending on the prepared state at $P$, a quantum system is sent through the H-shaped region from $A$ to $B$ or from $B$ to $A$. (c) If $A$ is traversed by a closed time-like curve, then $A$'s output influences the input, and a departure from quantum theory becomes necessary.
• Figure 2: Schematic of three parties and a process. The gray area represents the process: It takes the systems on the future boundaries of the parties and maps it to the past boundaries of the parties. The red connections indicate an example where the parties are causally ordered increasingly. A priori, however, the process is not assumed to respect any ordering of the parties.
• Figure 3: Example of a four-node causal model. The state on the input space of node $A$ is obtained by evolving the state on the output space of nodes $C,D$ through the channel $\rho_{A\mid C,D}$, and similarly for the other nodes.
• Figure 4: Characterization of all pairwise non-isomorphic causal structures for three parties. Graphs 7-14 are inadmissible (Theorem \ref{['thm:inadmissible']}). In the classical-deterministic case, graph 16 leads to non-causal correlations (Theorem \ref{['thm:noncausal']}), and the others to causal correlations only (Theorem \ref{['thm:causal']}). Graph 16 is also the causal structure of AF/BW process afbaumeler2016space. Graph 15 is the causal structure of the quantum switch without a region in the global future ( cf. Figure \ref{['fig:introb']}).
• Figure 5: Cases in which the directed cycle $C$ appears in the graph. Solid lines represent edges, dashed ones paths. (a) If the common parent $p$ of $c_i,c_j$ is an element in $C$, then $C$ is not induced. (b) If there exists a directed path from a node in $C$ to $p$, then the graph contains a non-induced cycle. (c) The only possibility is that there exists a set $V'$ of nodes without paths from $C$ to $V'$.

Theorems & Definitions (27)

• Definition 1: Causal correlations oreshkov2012oreshkov2016alastaircorr2016
• Definition 2: Process oreshkov2012
• Theorem 1: Classical-determintic process baumeler2016
• Theorem 2: Equivalence of antinomies baumeler2020equivalence
• Lemma 3: Component-wise non-signaling and reduced process baumeler2019baumeler2020equivalence
• Definition 3: Causal model, consistency, and faithfulness barrett2021
• Definition 4: Siblings-on-cycles graph
• Lemma 4: Quantum signaling path
• proof
• Theorem 5: Inadmissible causal structures