# Representing Measurement as a Thermodynamic Symmetry Breaking

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## Abstract

**:**

## 1. Introduction

**Assumption**

**1**

**.**We assume a decomposition of $U=$ “everything” into $OW$, where O is the “observer” and W is the “world” with which the observer interacts. We require that the state $|U\rangle $ be separable as $|U\rangle =|OW\rangle =|O\rangle |W\rangle $, at least up to some recoherence time that is long with respect to any time interval of interest.

**Assumption**

**2**

**.**We assume the Hilbert space ${\mathcal{H}}_{U}={\mathcal{H}}_{O}\otimes {\mathcal{H}}_{W}$ has finite dimension ${d}_{U}={d}_{O}+{d}_{W}$.

- We formulate the distinction between system identification and pointer-state measurement as a collection of equivalence relations on cocones, and showing that: (1) transitions between cocone equivalence classes can be represented more generally as groupoid operations; and (2) these groupoid operations correspond to entanglement swaps that result in O-relative decoherence [27].
- We show that free-energy acquisition and waste-heat dissipation into the “environment” component of W can generically have non-negligible effects on observational outcomes due to entanglement swapping/contextuality.

## 2. Interaction as Mutual Measurement by $\mathit{O}$ and $\mathit{W}$

#### 2.1. Sequential Measurements

#### 2.2. Mutual Measurement Is Classical Communication

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 3. System Identification and Measurement by $\mathit{O}$

#### 3.1. Systems Require Reference Components with Invariant States

#### 3.2. Reference Components Can Be Represented as Cocones over One-Bit Classifiers

**Definition**

**1.**

**Definition**

**2.**

- The ${\mathcal{A}}_{ij}$ must be representable as a finite nonredundant set $\left\{{\mathcal{A}}_{k}\right\}$ with infomorphisms $fij:{\mathcal{A}}_{i}\to {\mathcal{A}}_{j}$.
- There exist infomorphisms ${h}_{i}:{\mathcal{C}}_{i}\to \mathbf{C}$ and ${h}_{ij}:{\mathcal{C}}_{i}\to {\mathcal{C}}_{j}$.
- All compositions of infomorphisms with codomain $\mathbf{C}$ commute.

#### 3.3. Measuring the Pointer State $|P\rangle $ of an Identified System S

#### 3.4. Sequential Measurements Induce Decoherence

#### 3.5. Entanglement Swapping Induces Contextuality

#### 3.6. CCD Commutativity Enforces Bayesian Coherence

- that only the shortest paths between objects in a diagram (such as a CCD) are labeled, and that the probability of a such a path is the product of the probabilities of its component arrows; and
- that the probabilities of all paths sum to unity.

**Definition**

**3.**

## 4. Thermodynamic Asymmetries and Their Effects

#### 4.1. Information Processing Demands Are Asymmetrical between R, P and E

#### 4.2. Thermodynamic Interactions with E Generically Disturb $|P\rangle $

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EPR | Einstein–Podolsky–Rosen |

IGUS | Information Gathering and Using System |

CCD | Cocone Diagram |

CCCD | Cone–Cocone Diagram |

CbD | Contextuality by Default |

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**Figure 1.**Observer O and world W exchange bits via an ancillary array of non-interacting qubits. Bit values are preserved if a quantum reference frame (here, a z axis) is shared a priori.

**Figure 2.**To be identifiable by observation, a system S must have a reference component R with a time-invariant state $|R\rangle $. To be of interest for measurements, S must also have a pointer component P with a time-varying state $|P\rangle $.

**Figure 3.**A cocone diagram (CCD) is a commuting diagram depicting maps (infomorphisms) ${f}_{ij}$ between classifiers ${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{j}$, maps ${g}_{k}l$ from the ${\mathcal{A}}_{k}$ to one or more channels ${\mathcal{C}}_{l}$ over a subset of the ${\mathcal{A}}_{i}$, and maps ${h}_{l}$ from channels ${\mathcal{C}}_{l}$ to the colimit $\mathbf{C}$ (cf. Equation (6.7) of [26]).

**Figure 4.**A sequence of CCDs identifying R (blue triangles) and measuring pointer components ${P}_{i},{P}_{j},{P}_{k}\dots {P}_{l}$. Transitions between CCDs are implemented by groupoid elements, e.g., ${\mathcal{G}}_{ij}$ and labeled by discrete times, e.g., ${\tau}_{i}$. The operators ${M}_{k}^{P}$ can equally well be generalized to subsets ${\left\{{M}^{P}\right\}}_{k}$ of mutually-commuting pointer-state observables.

**Figure 5.**Bayesian coherence is obtained if probabilities along all paths in a diagram sum to unity. Probabilities are not assigned within the R-identifying cocone (blue triangle).

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Fields, C.; Glazebrook, J.F.
Representing Measurement as a Thermodynamic Symmetry Breaking. *Symmetry* **2020**, *12*, 810.
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Fields C, Glazebrook JF.
Representing Measurement as a Thermodynamic Symmetry Breaking. *Symmetry*. 2020; 12(5):810.
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**Chicago/Turabian Style**

Fields, Chris, and James F. Glazebrook.
2020. "Representing Measurement as a Thermodynamic Symmetry Breaking" *Symmetry* 12, no. 5: 810.
https://doi.org/10.3390/sym12050810