# Holographic Screens Are Classical Information Channels

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## Abstract

**:**

## 1. Introduction

## 2. Implementation of Classical Communication by ${\mathit{H}}_{\mathit{AB}}$

## 3. Timelike and Lightlike Communication Channels

## 4. Holographic Screens Encode Interaction Eigenvalues

**Theorem**

**1**.

**A:**- The state $|U\rangle =|AB\rangle $ is separable: $|AB\rangle =|A\rangle |B\rangle $.
**B:**- The systems A and B communicate via an ideal, ancillary classical information channel with finite capacity.
**C:**- The eigenvalues of the interaction ${H}_{AB}$ are written on a finite, ancillary holographic screen at the $A-B$ boundary.

**Proof.**

**A**→**B:****([11] Theorem 1)**If A and B are separable, the interaction ${H}_{AB}={H}_{U}-({H}_{A}+{H}_{B})$ can be written in the form (1), with the N Hermitian operators ${M}_{i}^{k}$, $k=A$ or B, having binary eigenvalues. The $A-B$ interaction at any time t is, in this case, completely specified by an N-bit string. Hence, nothing is lost by replacing the interaction with an exchange of N-bit strings, i.e., with finite-bandwidth classical communication. There are no intervening systems to introduce noise and energy is perfectly conserved; hence the channel is ideal and can be considered to be ancillary.**B**→**C:**- A classical information channel can be timelike or lightlike. The information encoded into the channel by A (B) must be within the past lightcone of A’s (B’s) end of the channel, while the information that is received from the channel by A (B) can only flow into the future lightcone of A’s (B’s) end of the channel. These past and future light cones are light-sheets of the two ends of the channel, and define equal areas $A\left({B}_{k}\right)$ of the boundaries ${B}_{A}$ and ${B}_{B}$ of the two channel ends by (3). These boundaries are by definition holographic screens for A and B, respectively. As the only information exchanged through the channel consists of encodings of eigenvalues of ${H}_{AB}$, this is the only information on the relevant light-sheets and the only information encoded on the boundaries. The boundaries have no degrees of freedom on which ${H}_{AB}$ depends; hence, they are ancillary.
**C**→**B:**- The eigenvalues of ${H}_{AB}$ have a finite binary encoding, hence the intervening screen has finite area. As it encodes classical information that is accessible to both A and B, it is a classical channel between A and B.
**B**→**A:**- In order for A and B to exchange finite classical information specifying their states, their states must be well-defined. As $AB=U$, $|U\rangle $ must be separable as $|U\rangle =|A\rangle |B\rangle $.

## 5. Discussion

#### 5.1. Serialization Induces Decoherence

#### 5.2. Net Mass-Energy Transfer Alters Channel Width

#### 5.3. Spacelike Separation Decreases Communication Bandwidth

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AdS/CFT | Anti de Sitter/Conformal Field Theory |

BH | Black hole |

HP | Holographic Principle |

LOCC | Local operations, classical communication |

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**Figure 1.**An N-qubit array serving as a classical channel C between A and B. The two systems alternate preparing and measuring the state of the array.

**Figure 2.**The holographic principle relates the quantity of information writable on or readable from a channel to its spacelike area. (

**a**) a purely-timelike channel, e.g., an ideal memory. Upward arrows indicate unidirectional B to A information flow; downward arrows indicate A to B information flow. With both sets of arrows, the channel is reversible. Both of the systems experience past-to-future causality in both unidirectional and bidirectional scenarios. (

**b**) a ideal lightlike channel, equivalent to an ideal memory with its input and output surfaces displaced in space.

**Figure 3.**Net mass-energy transfers from B to A. (

**a**) transferring a system X (red triangle) with which B interacts but A does not increases the horizon width. (

**b**) transferring a system X with which A interacts, but B does not decrease the horizon width.

**Figure 4.**The area that can be both encoded by future directed light-sheets of B and sampled by future-directed light-sheets of A decreases by $1/{r}^{2}$, where r is the spacelike separation of A and B.

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Fields, C.; Marcianò, A.
Holographic Screens Are Classical Information Channels. *Quantum Rep.* **2020**, *2*, 326-336.
https://doi.org/10.3390/quantum2020022

**AMA Style**

Fields C, Marcianò A.
Holographic Screens Are Classical Information Channels. *Quantum Reports*. 2020; 2(2):326-336.
https://doi.org/10.3390/quantum2020022

**Chicago/Turabian Style**

Fields, Chris, and Antonino Marcianò.
2020. "Holographic Screens Are Classical Information Channels" *Quantum Reports* 2, no. 2: 326-336.
https://doi.org/10.3390/quantum2020022