# 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 |

## References

- Boltzmann, L. Lectures on Gas Theory; Dover Press: New York, NY, USA, 1995. First published 1896. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Dev.
**1961**, 5, 183–195. [Google Scholar] [CrossRef] - Landauer, R. Information is a physical entity. Phys. A
**1999**, 263, 63–67. [Google Scholar] [CrossRef] [Green Version] - Spekkens, R.W. Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A
**2007**, 75, 032110. [Google Scholar] [CrossRef] - Bartlett, S.D.; Rudolph, T.; Spekkens, R.W. Reconstruction of gaussian quantum mechanics from Liouville mechanics with an epistemic restriction. Phys. Rev. A
**2012**, 86, 012103. [Google Scholar] [CrossRef] - Jennings, D.; Leifer, M. No return to classical reality. Contempl. Phys.
**2016**, 57, 60–82. [Google Scholar] [CrossRef] - Krechmer, K. Relative measurement theory: The unification of experimental and theoretical measurements. Measurement
**2018**, 116, 77–82. [Google Scholar] [CrossRef] - Landsman, N.P. Between classical and quantum. In Handbook of the Philosophy of Science: Philosophy of Physics; Butterfield, J., Earman, J., Eds.; Elsevier: Amsterdam, The Netherlands, 2007; pp. 417–553. [Google Scholar]
- Schlosshauer, M. Decoherence and the Quantum to Classical Transition; Springer: Berlin, Germany, 2007. [Google Scholar]
- Moore, E.F. Gedankenexperiments on sequential machines. In Autonoma Studies; Shannon, C.W., McCarthy, J., Eds.; Princeton University Press: Princeton, NJ, USA, 1956; pp. 129–155. [Google Scholar]
- Fields, C. Bell’s theorem from Moore’s theorem. Int. J. Gen. Syst.
**2013**, 42, 376–385. [Google Scholar] [CrossRef] - Fields, C. Building the observer into the system: Toward a realistic description of human interaction with the world. Systems
**2016**, 4, 32. [Google Scholar] [CrossRef] - Hopcroft, J.E.; Ullman, J.D. Introduction to Automata Theory, Languages and Computation; Addison-Wesley: Reading, MA, USA, 1979. [Google Scholar]
- Grinbaum, A. How device-independent approaches change the meaning of physical theory. Stud. Hist. Philos. Mod. Phys.
**2017**, 58, 22–30. [Google Scholar] [CrossRef] [Green Version] - Emary, C.; Lambert, N.; Nori, F. Leggett–Garg inequalities. Rep. Prog. Phys.
**2014**, 77, 039501. [Google Scholar] [CrossRef] - Mermin, D. Hidden variables and the two theorems of John Bell. Rev. Mod. Phys.
**1993**, 65, 803–815. [Google Scholar] [CrossRef] [Green Version] - Fields, C. If physics is an information science, what is an observer? Information
**2012**, 3, 92–123. [Google Scholar] [CrossRef] - Fields, C. A model-theoretic interpretation of environment-induced superselection. Int. J. Gen. Syst.
**2012**, 41, 847–859. [Google Scholar] [CrossRef] [Green Version] - Kupervasser, O. Application of New Cybernetics in Physics; Elsevier: Amsterdam, The Netherlands, 2017. [Google Scholar]
- Von Neumann, J. The Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1955. [Google Scholar]
- Zeh, D. On the interpretation of measurement in quantum theory. Found. Phys.
**1970**, 1, 69–76. [Google Scholar] [CrossRef] - Zeh, D. Toward a quantum theory of observation. Found. Phys.
**1973**, 3, 109–116. [Google Scholar] [CrossRef] [Green Version] - Zurek, W.H. Pointer basis of the quantum apparatus: Into what mixture does the wave packet collapse? Phys. Rev. D
**1981**, 24, 1516–1525. [Google Scholar] [CrossRef] - Zurek, W.H. Environment-induced superselection rules. Phys. Rev. D
**1982**, 26, 1862–1880. [Google Scholar] [CrossRef] - Joos, E.; Zeh, D. The emergence of classical properties through interaction with the environment. Z. Phys. B Condens. Matter
**1985**, 59, 223–243. [Google Scholar] [CrossRef] - Zurek, W.H. Decoherence, einselection and the existential interpretation (the rough guide). Philos. Trans. R. Soc. A
**1998**, 356, 1793–1821. [Google Scholar] [Green Version] - Zurek, W.H. Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys.
**2003**, 75, 715–775. [Google Scholar] [CrossRef] [Green Version] - Tegmark, M. How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem. Phys. Rev. D
**2012**, 85, 123517. [Google Scholar] [CrossRef] - Ollivier, H.; Poulin, D.; Zurek, W.H. Objective properties from subjective quantum states: Environment as a witness. Phys. Rev. Lett.
**2004**, 93, 220401. [Google Scholar] [CrossRef] [PubMed] - Ollivier, H.; Poulin, D.; Zurek, W.H. Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe. Phys. Rev. A
**2005**, 72, 042113. [Google Scholar] [CrossRef] - Blume-Kohout, R.; Zurek, W.H. Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information. Phys. Rev. A
**2006**, 73, 062310. [Google Scholar] [CrossRef] - Zurek, W.H. Quantum Darwinism. Nat. Phys.
**2009**, 5, 181–188. [Google Scholar] [CrossRef] - Fuchs, C. QBism, the perimeter of Quantum Bayesianism. arXiv, 2010; arXiv:1003.5201v1. [Google Scholar]
- Rovelli, C. Relational quantum mechanics. Int. J. Theor. Phys.
**1996**, 35, 1637–1678. [Google Scholar] [CrossRef] [Green Version] - Peres, A. Unperformed experiments have no results. Am. J. Phys.
**1978**, 46, 745–747. [Google Scholar] [CrossRef] - Cabello, A. A simple explanation of Born’s rule. arXiv, 2018; arXiv:1801.06347. [Google Scholar]
- Bell, J.S. Against measurement. Phys. World
**1990**, 3, 33–41. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.M. Quantum information becomes classical when distributed to many users. Phys. Rev. Lett.
**2006**, 97, 250503. [Google Scholar] [CrossRef] [PubMed] - Korbicz, J.K.; Horodecki, P.; Horodecki, R. Objectivity in a noisy photonic environment through quantum state information broadcasting. Phys. Rev. Lett.
**2014**, 112, 120402. [Google Scholar] [CrossRef] [PubMed] - Fields, C. Quantum Darwinism requires an extra-theoretical assumption of encoding redundancy. Int. J. Theor. Phys.
**2010**, 49, 2523–2527. [Google Scholar] [CrossRef] - Pusey, M.F.; Barrett, J.; Rudolph, T. On the reality of the quantum state. Nat. Phys.
**2012**, 8, 475–478. [Google Scholar] [CrossRef] [Green Version] - Ekert, A.K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett.
**1991**, 67, 661–663. [Google Scholar] [CrossRef] [PubMed] - Gisin, N.; Thew, R. Quantum communication. Nat. Photonics
**2007**, 1, 165–171. [Google Scholar] [CrossRef] [Green Version] - Fine, A. Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett.
**1982**, 48, 291–295. [Google Scholar] [CrossRef] - Mermin, N.D. Quantum mysteries revisited. Am. J. Phys.
**1990**, 58, 731–734. [Google Scholar] [CrossRef] - Hensen, B.; Bernien, H.; Dreau, A.E.; Reiserer, A.; Kalb, N.; Blok, J.; Ruitenberg, M.S.; Vermeulen, R.F.L.; Schouten, R.N.; Abellán, C.; et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature
**2015**, 526, 682–686. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Giustina, M.; Versteegh, M.A.M.; Wengerowsky, S.; Handsteiner, J.; Hochrainer, A.; Phelan, K.; Steinlechner, F.; Kofler, J.; Larsson, J.-A.; Abellán, C.; et al. A significant-loophole-free test of Bell’s theorem with entangled photons. Phys. Rev. Lett.
**2015**, 115, 250401. [Google Scholar] [CrossRef] [PubMed] - Shalm, L.K.; Meyer-Scott, E.; Christensen, B.G.; Bierhorst, P.; Wayne, M.A.; Stevens, M.J.; Gerrits, T.; Glancy, S.; Hamel, D.R.; Allman, M.S.; et al. A strong loophole-free test of local realism. Phys. Rev. Lett.
**2015**, 115, 250402. [Google Scholar] [CrossRef] [PubMed] - Hofer-Szabó, G. How human and nature shake hands: The role of no-conspiracy in physical theories. Stud. Hist. Philos. Mod. Phys.
**2017**, 57, 89–97. [Google Scholar] [CrossRef] [Green Version] - Bartlett, S.D.; Rudolph, T.; Spekkens, R.W. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys.
**2007**, 79, 555–609. [Google Scholar] [CrossRef] [Green Version] - Coecke, B. Quantum picturalism. Contempl. Phys.
**2010**, 51, 59–83. [Google Scholar] [CrossRef] [Green Version] - Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Informational derivation of quantum theory. Phys. Rev. A
**2011**, 84, 012311. [Google Scholar] [CrossRef]

**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.
https://doi.org/10.3390/e20100797

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Fields C.
Some Consequences of the Thermodynamic Cost of System Identification. *Entropy*. 2018; 20(10):797.
https://doi.org/10.3390/e20100797

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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