Knot invariants and indefinite causal order

TL;DR

This work investigates indefinite causal order in superpositions of finitely many quasiclassical spacetimes by building a knot-theoretic representation of how events order across branches. It introduces explicit causal-order quantifiers (pairwise, longitudinal, transverse, and total) and shows that maximal indefiniteness corresponds to torus knots, while definite order corresponds to the unknot; for certain subsequences, the quadratic term of the Alexander-Conway polynomial directly encodes indefinitiveness, implying topological protection under topology-preserving maps. The authors develop both fine- and coarse-grained knot constructions via Dowker–Thistlethwaite codes, provide an operational encoding of order for arbitrary $N$ and $M$ with a unitary transformation, and introduce quantum coherence-weighted measures of causal order. They connect quantum diffeomorphism invariance with knot invariants, relate their framework to the process-matrix formalism, and outline numerous future directions, including extensions to infinite event/spacetime limits, connections to braid groups, and potential experimental realizations of higher-order quantum switches. Overall, the paper offers a novel, topologically grounded framework for classifying and manipulating indefinite causal order in quantum gravitational contexts, with potential implications for quantum technologies exploiting causal superpositions.

Abstract

We explore indefinite causal order between events in the context of quasiclassical spacetimes in superposition. We introduce several new quantifiers to measure the degree of indefiniteness of the causal order for an arbitrary finite number of events and spacetime configurations in superposition. By constructing diagrammatic and knot-theoretic representations of the causal order between events, we find that the definiteness or maximal indefiniteness of the causal order is topologically invariant. This reveals an intriguing connection between the field of quantum causality and knot theory. Furthermore, we provide an operational encoding of indefinite causal order and discuss how to incorporate a measure of quantum coherence into our classification.

Figures (15)

• Figure 1: A strategy to identify points $p\in\mathcal{M_A}$, $q \in \mathcal{M_B}$, and $r\in\mathcal{M_C}$ across a superposition of three spacetimes. Following Kabel2024, we identify those points at which a chosen set of reference fields $\chi$ takes on the same values, that is, points for which $\chi^\mathcal{A}(p) = \chi^\mathcal{B}(q) = \chi^\mathcal{C}(r) = \chi \in \mathbb{R}^d$.
• Figure 2: Measures of indefinite causal order for $N=4$ events across a superposition of $M=3$ spacetimes. The dashed black line describes pairwise causal order$\mathfrak{s}_{12}^{\mathcal{AB}}=-1$; the blue rectangle captures the longitudinal causal order$\mathfrak{l}^{\mathcal{A}\mathcal{B}}=2$, as well the causal indefiniteness $\delta(\mathcal{A},\mathcal{B})=2$; the orange shape represents the transverse causal order$\mathfrak{t}_{34}=-1$. The total causal order is $\mathfrak{s}_{\mathrm{tot}} = 2$ and the total causal indefiniteness is $\Delta = 8$.
• Figure 3: Indefinite causal order of three events in a superposition of $M=3$ spacetimes : $ABC-BCA-CAB$.
• Figure 4: A representation of indefinite causal order for two events $\mathcal{E}_1 = A$ and $\mathcal{E}_2 = B$ in a superposition of two spacetimes $(\mathcal{M}_\mathcal{X}, g_\mathcal{X})$, where $\mathcal{X} = \mathcal{A}, \mathcal{B}$. In each spacetime $(\mathcal{M}_{\mathcal{X}}, g_{\mathcal{X}})$, $\tau_\mathcal{X}$ denotes the proper time of a specified timelike worldline. The events are ordered according to the pairwise causal order $s_{12}^{\mathcal{X}}$ between them such that $A \prec B$ iff $s_{12}^\mathcal{X}=\text{sign}(\Delta \tau_\mathcal{X}) = 1$ (that is, iff $A$ occurs at an earlier proper time $\tau_\mathcal{X}$ than $B$.)
• Figure 5: Braid diagram for three spacetimes in superposition and two events. A disadvantage of the braid diagram is that this construction depends explicitly on the choice of reference spacetime -- $(\mathcal{M}_\mathcal{B}, g_\mathcal{B})$ in the left and $(\mathcal{M}_\mathcal{A}, g_\mathcal{A})$ in the right diagram.

Theorems & Definitions (72)

• Definition 1: Event
• Definition 2: Comparison map
• Definition 3: Causal order between two events
• Proposition 1
• proof
• Definition 4: Ordered collection of events
• Definition 5: Pairwise causal order
• Proposition 2
• proof
• Proposition 3