# Making the Thermodynamic Cost of Active Inference Explicit

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## Abstract

**:**

## 1. Introduction

## 2. “Just Physics” versus Implemented Computation

_{B}T).

## 3. Coupling Information and Energy Flows

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

## 4. Measuring and Controlling Energy Usage

**Theorem**

**2.**

**Proof.**

## 5. Resource Usage in the Matrix Representation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AIA | Active Inference Agent |

CCCD | Cone-CoCone Diagram |

FEP | Free Energy Principle |

QRF | Quantum Refence Frame |

TFE | Thermodynamic Free Energy |

TQFT | Topological Quantum Field Theory |

VFE | Variational Free Energy |

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**Figure 1.**Generic structure of semantic interpretations of physical processes. Function f interprets, via the semantic map $\psi $, the action of the physical time-propagator $\mathcal{P}\left(t\right)$ between time points ${t}_{i}$ and ${t}_{j}$. The interpretation is semantically coherent provided the diagram commutes, i.e., provided ${f|}_{{t}_{i}\to {t}_{j}}\left(\psi {|}_{{t}_{i}}\left(S\right)\right){=\psi |}_{{t}_{j}}\left(\mathcal{P}{|}_{{t}_{i}\to {t}_{j}}\left(S\right)\right)$.

**Figure 2.**“Attaching” a CCCD to an intersystem boundary $\mathcal{B}$ depicted as an ancillary array of qubits. Operators ${M}_{i}^{k}$, $k=S$ or E, are single-bit components of the interaction Hamiltonian ${H}_{SE}$. The node C is both the limit and the colimit of the nodes ${\mathcal{A}}_{i}$; only leftward-going (cocone implementing) arrows are shown for simplicity. See [12,13,24,30] for details. Adapted from [12], CC-BY license.

**Figure 3.**Factoring a QRF Q into components ${Q}_{\chi}$ and ${Q}_{\Theta}$ allows information exchange through $\mathcal{B}$ to be separated from thermal exchange through $\mathcal{B}$. This breaks the previous qubit-exchange symmetry on $\mathcal{B}$ as discussed in Ref. [12].

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**MDPI and ACS Style**

Fields, C.; Goldstein, A.; Sandved-Smith, L.
Making the Thermodynamic Cost of Active Inference Explicit. *Entropy* **2024**, *26*, 622.
https://doi.org/10.3390/e26080622

**AMA Style**

Fields C, Goldstein A, Sandved-Smith L.
Making the Thermodynamic Cost of Active Inference Explicit. *Entropy*. 2024; 26(8):622.
https://doi.org/10.3390/e26080622

**Chicago/Turabian Style**

Fields, Chris, Adam Goldstein, and Lars Sandved-Smith.
2024. "Making the Thermodynamic Cost of Active Inference Explicit" *Entropy* 26, no. 8: 622.
https://doi.org/10.3390/e26080622