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

Carla Ferradini, Giulia Mazzola, V. Vilasini

Abstract

Can the direction of time and the causal structure of space-time be inferred from operational principles? Causal models and tensor networks offer complementary perspectives: the former encodes cause-effect relations via directed graphs, with intrinsic ordering; the latter describes multipartite systems on undirected graphs, without presupposing directionality. We construct two-way mappings between these two frameworks, linking direction agnostic correlation functions and operational notions of signalling. This clarifies the operational meaning of causal influence in tensor networks and introduces discrete "space-time rotations'' of causal models which preserve signalling relations. Applying our framework to holographic tensor networks, we use tools from causal inference, like graph-separation, to analyse emergent causal structures. By permitting cyclic and indefinite causal structures, our results enable transfer of techniques across tensor networks and a range of causality frameworks.

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

Abstract

Can the direction of time and the causal structure of space-time be inferred from operational principles? Causal models and tensor networks offer complementary perspectives: the former encodes cause-effect relations via directed graphs, with intrinsic ordering; the latter describes multipartite systems on undirected graphs, without presupposing directionality. We construct two-way mappings between these two frameworks, linking direction agnostic correlation functions and operational notions of signalling. This clarifies the operational meaning of causal influence in tensor networks and introduces discrete "space-time rotations'' of causal models which preserve signalling relations. Applying our framework to holographic tensor networks, we use tools from causal inference, like graph-separation, to analyse emergent causal structures. By permitting cyclic and indefinite causal structures, our results enable transfer of techniques across tensor networks and a range of causality frameworks.
Paper Structure (46 sections, 10 theorems, 111 equations, 1 figure)

This paper contains 46 sections, 10 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of contributions and structure of the paper
  3. Notation
  4. Graph notation
  5. Choi-Jamiołkowski isomorphism
  6. Tensor networks and causal influence
  7. Definition of tensor network
  8. Example
  9. Causal influence as defined in Cotler_2019
  10. Example
  11. Remarks and open questions regarding this notion causal influence
  12. On which Hilbert spaces are the operators $U_{\region{A}}$ and $O_{\region{B}}$ acting?
  13. Why are unitaries and Hermitian operators introduced?
  14. What is the role of the denominator?
  15. Open questions
  16. ...and 31 more sections

Key Result

lemma 1

A superoperator $M_{B|A}\in\linops\left(\linops\left(\hilmaparg{A}\right),\linops\left(\hilmaparg{B}\right)\right)$ is completely positive if and only if $\CJ_{B|A}(M_{B|A})$ is positive and it is trace preserving if and only if $\Tr_{B}\left[\CJ_{B|A}(M_{A|B})\right] = \frac{\id_A}{d_{A}}$.

Figures (1)

  • Figure 1: Visual intuition behind main results Our work defines operationally motivated mappings between (possibly cyclic) quantum causal models and tensor networks, going in both directions. Through this, a causal model on the directed graph $\graphnamedir$ can be uniquely mapped to a tensor network on the undirected graph $\graphnameud$. From the tensor network on $\graphnameud$, we can obtain four different causal models, one for each choice of directions of edges. The choice associated with the directions of $\graphnamedir$ will lead again to the original causal model. While, other choices, which lead to causal models on the directed graphs on the right, might require self-loops to give a valid model. Such possibility can be easily checked through evaluating a partial trace. We show that the mappings preserve certain useful properties, allow to interpret alternative causal models obtained through the mapping as "rotations" of the original model, and to apply graph-separation theorems from causal models to analyse tensor networks.

Theorems & Definitions (46)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • definition 1: Tensor network
  • definition 2: Link state and total state
  • definition 3: Tensor contraction
  • remark 1
  • definition 4: Quantum causal influence --- Cotler_2019
  • definition 5: Causal graph
  • ...and 36 more