# Some Consequences of the Thermodynamic Cost of System Identification

## Abstract

**:**

## 1. Introduction

## 2. Formalizing System Identification as a Search Process

Finite-resource restriction: No observer can employ more that a finite number of finite-resolution observational outcomes to identify a system of interest.

## 3. Distinguishing Reference from Pointer Degrees of Freedom

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

## 4. System Identification Cannot Be Arbitrarily Refined

**Theorem**

**1.**

**Proof.**

## 5. System Identification at Multiple Times

**Theorem**

**2.**

**Proof.**

## 6. Joint System Identification by Multiple Observers

**Theorem**

**3.**

**Proof.**

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ADC | Analog-to-Digital Converter |

CHSH | Clauser-Horne-Shimony-Holt |

EPR | Einstein-Podolsky-Rosen |

LOCC | Local Operations, Classical Communication |

POVM | Positive Operator-Valued Measure |

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**Figure 1.**(

**a**) A classical observer interacts with a system of interest; both are embedded in a surrounding environment. (

**b**) Interactions between observer (O), system of interest (S) and environment (E) enabling environmental decoherence. The Hamiltonian ${H}_{OS}$ transfers outcome information from S to O; ${H}_{SE}$, and ${H}_{OE}$ decohere S and O respectively. Adapted from Figure 1 in ref. [29].

**Figure 3.**The first three components of ${\Pi}^{(i,m)}(t)$ of Equation (2) in the first and ${n}^{th}$ cycles of deploying the ${M}_{i}$.

Step t | Measure ${\mathit{M}}_{\mathit{i}},\mathit{i}\le \mathit{n}$ | Outcome ${\mathit{x}}_{\mathit{i}}\in \{0,1\}$ |
---|---|---|

1 | 1 | 1 |

2 | 2 | 0 |

3 | 2 | 1 |

$...$ | $...$ | $...$ |

N | 4 | 0 |

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Fields, C. Some Consequences of the Thermodynamic Cost of System Identification. *Entropy* **2018**, *20*, 797.
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Fields C. Some Consequences of the Thermodynamic Cost of System Identification. *Entropy*. 2018; 20(10):797.
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Fields, Chris. 2018. "Some Consequences of the Thermodynamic Cost of System Identification" *Entropy* 20, no. 10: 797.
https://doi.org/10.3390/e20100797