# Equivalence of the Symbol Grounding and Quantum System Identification Problems

## Abstract

**:**

## 1. Introduction

How can the semantic interpretation of a formal symbol system be made intrinsic to the system, rather than just parasitic on the meanings in our heads? How can the meanings of the meaningless symbol tokens, manipulated solely on the basis of their (arbitrary) shapes, be grounded in anything but other meaningless symbols?[1] (p. 335)

## 2. Preliminaries

#### 2.1. Symbols and Grounds

#### 2.2. Systems, States and Observables

#### 2.3. Quantum and Classical

#### 2.4. Summary

## 3. Solving the QSIP Requires Solving the SGP

- (1)
- The state $|\mathbf{U}\rangle $ of any isolated quantum system $\mathbf{U}$ may be represented as a unit vector in a Hilbert space ${\mathcal{H}}_{\mathbf{U}}$.
- (2)
- The time evolution of $|\mathbf{U}\rangle $ is unitary, and may be represented by a propagator ${e}^{-(i/\hslash ){H}_{\mathbf{U}}\mathit{t}}$ where ${H}_{\mathbf{U}}$ is the Hamiltonian operator characterizing $\mathbf{U}$.
- (3)
- Measurements of $|\mathbf{U}\rangle $ may be represented as actions by a positive operator-valued measure (POVM), a collection $\left\{{E}_{i}^{\mathbf{U}}\right\}$ of positive semi-definite Hilbert-space automorphisms that sum to the Identity, on ${\mathcal{H}}_{\mathbf{U}}$.
- (4)
- The components of an isolated composite system $\mathbf{U}$ may be represented by a tensor-product structure (TPS) of ${\mathcal{H}}_{\mathbf{U}}$.

## 4. Solving the SGP Requires Solving the QSIP

^{rd}and higher-order terms are neglected, as a sum ${H}_{\mathbf{U}}={\sum}_{\mathit{ij}}{\mathit{H}}_{\mathit{ij}}$, where ${H}_{ij}$ describes the pairwise interaction between two physical degrees of freedom i and j of $\mathbf{U}$, and that allows alternative TPSs ${\mathcal{H}}_{\mathbf{S}}\otimes {\mathcal{H}}_{\mathbf{E}}$ and ${\mathcal{H}}_{{\mathbf{S}}^{\prime}}\otimes {\mathcal{H}}_{{\mathbf{E}}^{\prime}}$ to describe the same universe [18,19]. Any assumption that particular systems, and hence particular TPSs, are “preferred” by physical dynamics violates decompositional equivalence. It cannot, therefore, be assumed that the “environment” only encodes information about the states of particular systems; if the environment is assumed to encode information about the states of systems embedded in it, it must be assumed to encode information about the states of all such systems. In this case, however, observers must be regarded as choosing which encoded information to extract from the environment, which is precisely the assumption of relevance that the environment as witness formulation was designed to avoid [52].

## 5. The Unsolvability of the QSIP Renders the SGP Unsolvable

## 6. So What?

## 7. Conclusions

## Acknowledgments

## Conflict of Interest

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Equivalence of the Symbol Grounding and Quantum System Identification Problems. *Information* **2014**, *5*, 172-189.
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Equivalence of the Symbol Grounding and Quantum System Identification Problems. *Information*. 2014; 5(1):172-189.
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2014. "Equivalence of the Symbol Grounding and Quantum System Identification Problems" *Information* 5, no. 1: 172-189.
https://doi.org/10.3390/info5010172