# Gravity, Quantum Fields and Quantum Information: Problems with Classical Channel and Stochastic Theories

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## Abstract

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## 1. Preamble

## 2. Problem 1: Quantum Fields Misconstrued as Information Channels

#### 2.1. Quantum Channels

- In a direct interaction channel, the emitter and receiver interact through a unitary map $\widehat{U}$ on ${\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}$. Then$$\begin{array}{c}\hfill C\left[{\widehat{\rho}}_{e}\right]=T{r}_{{\mathcal{H}}_{A}}\left(\right)open="("\; close=")">\widehat{U}({\widehat{\rho}}_{e}\otimes {\widehat{\rho}}_{0}){\widehat{U}}^{\u2020},\end{array}$$
- In a mediated interaction channel, there exists a mediating system described by the Hilbert space ${\mathcal{H}}_{M}$. Both emitter and receiver interact through the mediator, but not directly with each other. Then, the Hamiltonian is $\widehat{H}={\widehat{H}}_{A}+{\widehat{H}}_{B}+{\widehat{H}}_{M}+{\widehat{H}}_{AM}+{\widehat{H}}_{BM}$, where ${\widehat{H}}_{AM}$ describes the interaction of the emitter with the mediator and ${\widehat{H}}_{BM}$ describes the interaction of the receiver with the mediator. These interactions are usually viewed as permanent, i.e., that they cannot be switched on or off.

#### 2.2. Problems with Treating QFT as Quantum Channel

- A consistent theory of interactions exists only if receiver and emitter are also treated by QFT. However, then, the Hamiltonian of the system cannot be defined on a tensor product Hilbert space ${\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}\otimes {\mathcal{H}}_{med}$, as in a mediated interaction channel. This statement follows from classic theorems by Haag [1] and Hall and Wightman [2]. This means that the degrees of freedom of the field are intertwined with those of the emitter and the receiver, in a way that it is impossible to disentangle. The root of this problem is phenomena such as vacuum polarization—it is impossible to describe the field vacuum as a factorized state or even an entangled state in a factorized single state.

- The notion of a quantum channel originates from Quantum Information Theory (QIT), which has mainly been developed in the context of non-relativistic quantum mechanics, a small corner of full QFT. Current QIT is problematic when basic relativistic principles—both special and general—such as causality and covariance, need be accounted for. A relativistic QIT that expresses all informational notions in terms of quantum fields is currently missing, largely because of difficulties in formulating a comprehensive QFT theory of measurement [3].

- The naive idea of a field as an object that mediates interaction is insufficient to describe the actual theories of mediating interactions, namely, gauge field theories. The reason is that it does not take into account the presence of constraints, which are a consequence of the gauge symmetry. The same issue appears in the treatment of gravity.

#### 2.3. Implications for Gravity

#### 2.4. Event Formalism and Closed Timelike Curves

## 3. Problem 2: Quantum Processes/Fluctuations Cannot Be Replaced by Classical Stochastic Processes/Noises

#### 3.1. Fluctuations as Sources of Gravitational Decoherence—What Is Missing or Misleading

#### 3.2. Classical Stochastic Processes or Noises Miss Out Important Information in Quantum Theories

## 4. Problem 3: How Are Semiclassical and Stochastic Theories Related to Their Quantum Origins? How Does Noise Enter?

#### 4.1. Semiclassical Theory as Large N Limit of Quantum Theory

#### 4.2. Stochastic Semiclassical Theory: Noise Can Be Defined for Quantum Fluctuations in Gaussian Systems

## 5. Conclusions

- (i)
- Staying in the confines of nonrelativistic quantum mechanics to describe quantum field processes and quantum information;
- (ii)
- Using information channels to describe quantum and gravitational effects in substitution of QFT and GR;
- (iii)
- Introducing classical stochastic sources or processes to mock up quantum field fluctuations or replace quantum field processes, leading to theories with inconsistencies or pathologies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Anastopoulos, C.; Hu, B.-L.
Gravity, Quantum Fields and Quantum Information: Problems with Classical Channel and Stochastic Theories. *Entropy* **2022**, *24*, 490.
https://doi.org/10.3390/e24040490

**AMA Style**

Anastopoulos C, Hu B-L.
Gravity, Quantum Fields and Quantum Information: Problems with Classical Channel and Stochastic Theories. *Entropy*. 2022; 24(4):490.
https://doi.org/10.3390/e24040490

**Chicago/Turabian Style**

Anastopoulos, Charis, and Bei-Lok Hu.
2022. "Gravity, Quantum Fields and Quantum Information: Problems with Classical Channel and Stochastic Theories" *Entropy* 24, no. 4: 490.
https://doi.org/10.3390/e24040490