Emergent causal order and time direction: bridging causal models and tensor networks

1. Problem Statement

The fundamental question addressed is whether the direction of time and the causal structure of space-time can be inferred from operational principles rather than assumed a priori.

• The Conflict: Standard Causal Models (classical and quantum) rely on directed graphs where arrows encode cause-effect relations and input-output asymmetry. This assumes a predefined temporal direction. Conversely, Tensor Networks (used in condensed matter, quantum gravity, and holography) are defined on undirected graphs. They treat all tensor legs symmetrically, encoding correlations without presupposing a global or local time direction.
• The Gap: While tensor networks offer a framework for emergent space-time, they lack a clear operational definition of "causal influence" that distinguishes genuine causation from mere correlation. Existing proposals (e.g., [CHQY19]) define causal influence via correlation functions but lack a rigorous connection to established operational notions of causality (like interventions and signalling). Conversely, causal models struggle to describe scenarios where causal order is indefinite or dynamical.

2. Methodology

The authors construct a rigorous mathematical framework to bridge these two paradigms by defining explicit, two-way mappings between Quantum Causal Models (specifically allowing for cyclic structures) and Tensor Networks.

Key Technical Tools:

• Choi-Jamiołkowski Isomorphism: Used to map linear superoperators (causal mechanisms) to density operators (tensors) and vice versa.
• Cyclic Causal Modeling: The authors utilize the framework from [FGV25b], which allows for causal models on cyclic graphs using a specific probability rule based on self-cycle composition (loop composition of quantum channels).
• Graph Separation Theorems: They leverage $d$-separation (for acyclic graphs) and $p$-separation (for cyclic graphs) to infer conditional independencies and no-signalling relations directly from graph topology.
• Interventions: The framework defines "time-reversible interventions" (unitaries and Hermitian instruments) to probe causal influence without imposing artificial time directions.

3. Key Contributions

A. Bidirectional Mappings

The paper establishes two primary mappings:

1. Causal Model $\to$ Tensor Network ($\phi$):

   • Maps a causal model on a directed graph $\vec{G}$ to a unique tensor network on an undirected graph $\bar{G}$ (same vertices/edges, directions removed).
   • Causal mechanisms (CPTP maps) are converted into density operators (tensors) via the Choi-Jamiołkowski isomorphism.
   • Result: A unique tensor network is generated for any given causal model.

2. Tensor Network $\to$ Causal Model ($\psi$):

   • Restricted Mapping: For a specific choice of edge directions, a tensor network maps to a causal model only if the tensors satisfy a partial trace condition (ensuring the resulting maps are CPTP).
   • Generalized Mapping: To handle any choice of edge directions (even if the trace condition fails), the authors introduce self-loops and ancillary systems. This allows the construction of a valid (possibly cyclic) causal model for any orientation of the undirected graph.
   • Result: A single tensor network can generate a family of distinct causal models, each corresponding to a different assignment of edge directions.

B. Operational Causal Influence

The authors critique the existing definition of causal influence in tensor networks ([CHQY19]) and propose a modified, Operational Causal Influence:

• Critique: The original definition uses a correlation function ratio that lacks a direct operational interpretation in terms of conditional probabilities and can be ill-defined if the denominator is zero.
• Proposal: They define causal influence based on conditional probabilities of outcomes under freely chosen interventions.
  $$M(U_A : O_B | \mathcal{O}_B) = P(y | L, \mathcal{O}_B)$$
  where $L$ represents a "virtual" measurement outcome linked to the tensor network contraction.
• Significance: This definition ensures that causal influence captures genuine causation (dependence on interventions) rather than just statistical correlations.

C. Equivalence of Signalling and Causal Influence

Theorem 8 establishes a central result:

A signals to B in a causal model (via time-reversible interventions) if and only if there is non-zero operational causal influence from A to B in the corresponding tensor network.

This bridges the operational notion of signalling (intervention-based) with the correlation-based notion of influence in tensor networks, proving that the latter is a valid measure of causality.

D. Discrete "Space-Time Rotations"

A profound consequence of the mapping is the discovery of equivalence classes of causal models:

• A single tensor network can be mapped to multiple causal models with different causal graphs (different edge directions).
• Despite having different causal structures (e.g., different light cones or time flows), these models share identical signalling relations (no-signalling constraints).
• The authors term these distinct models "discrete space-time rotations" of each other, analogous to dual unitaries. This suggests that the "direction of time" in a tensor network is not unique but depends on the chosen causal interpretation, yet the physical constraints (signalling) remain invariant.

4. Results and Applications

Application to Holographic Tensor Networks

The authors apply their framework to Holographic Tensor Networks (models of AdS/CFT using perfect tensors):

• Geodesic Criterion vs. Graph Separation: Previous work relied on "minimal geodesics" to determine if an operator in the bulk can be "pushed" to the boundary to determine causal influence.
• New Insight: By mapping the holographic network to a causal model, the authors use $d$-separation (for acyclic choices) and $p$-separation (for cyclic choices) to rigorously determine no-signalling.
  • If the regions are $d$-separated in the mapped causal graph, there is no causal influence.
  • If the graph requires self-loops (cyclic) and the regions are $p$-connected, causal influence is possible unless fine-tuning is present.
• This provides a purely graph-theoretic method to analyze causal structure in holography without relying solely on geometric intuition.

5. Significance and Outlook

• Emergent Causality: The work demonstrates that causal structure and time direction can be treated as emergent properties derived from correlation structures (tensor networks) rather than fundamental inputs.
• Unified Language: It provides a unified language to translate techniques between causal inference (e.g., using separation theorems) and quantum many-body physics/holography.
• Indefinite Causal Order: The framework naturally accommodates Indefinite Causal Order (ICO) and higher-order quantum operations, as the generalized mapping handles cyclic causal structures. This opens pathways to study quantum gravity regimes where space-time geometry is in superposition.
• Fine-Tuning vs. Structure: The results highlight a connection between "fine-tuned" no-signalling relations in one causal model and "sound" graph-separation properties in another model within the same equivalence class (rotation).

In summary, the paper successfully constructs a bridge between the direction-agnostic world of tensor networks and the direction-dependent world of causal models, providing operational definitions for emergent time and causal influence, and offering new tools to analyze complex quantum systems like holographic codes.