# 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 |

## References

- ’t Hooft, G. Dimensional reduction in quantum gravity. In Salamfestschrift; Ali, A., Ellis, J., Randjbar-Daemi, S., Eds.; World Scientific: Singapore, 1993; pp. 284–296. [Google Scholar]
- Susskind, L. The world as a hologram. J. Math. Phys.
**1995**, 36, 6377–6396. [Google Scholar] [CrossRef][Green Version] - Bousso, R. The holographic principle. Rev. Mod. Phys.
**2002**, 74, 825–874. [Google Scholar] [CrossRef][Green Version] - Bekenstein, J.D. Black holes and information theory. Contemp. Phys.
**2004**, 45, 31–43. [Google Scholar] [CrossRef][Green Version] - 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]
- Zurek, W.H. Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys.
**2003**, 75, 715–775. [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] - Chitambar, E.; Leung, D.; Mančinska, L.; Ozols, M.; Winter, A. Everything you always wanted to know about LOCC (but were afraid to ask). Comms. Math. Phys.
**2014**, 328, 303–326. [Google Scholar] [CrossRef][Green Version] - Fields, C.; Marcianò, A. Sharing nonfungible information requires shared nonfungible information. Quantum Rep.
**2019**, 1, 252–259. [Google Scholar] [CrossRef][Green Version] - Fields, C.; Glazebrook, J.F. Representing measurement as a thermodynamic symmetry breaking. Symmetry
**2020**, 12, 810. [Google Scholar] [CrossRef] - Addazi, A.; Chen, P.; Fabrocini, F.; Fields, C.; Greco, E.; Lutti, M.; Marcianò, A.; Pasechnik, R. Generalized holographic principle, gauge invariance and the emergence of gravity à la Wilczek. arXiv
**2020**, arXiv:2004.13751v1. [Google Scholar] - 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][Green Version] - Wheeler, J.A. Law without law. In Quantum Theory and Measurement; Wheeler, J.A., Zurek, W.H., Eds.; Princeton University Press: Princeton, NJ, USA, 1983; pp. 182–213. [Google Scholar]
- Bohr, N. The quantum postulate and the recent development of atomic theory. Nature
**1928**, 121, 580–590. [Google Scholar] [CrossRef][Green Version] - Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Devel.
**1961**, 5, 183–195. [Google Scholar] [CrossRef] - Fields, C. Decoherence as a sequence of entanglement swaps. Results Phys.
**2019**, 12, 1888–1892. [Google Scholar] [CrossRef] - Landauer, R. Information is a physical entity. Phys. A
**1999**, 263, 63–67. [Google Scholar] [CrossRef][Green Version] - Rovelli, C. Black holes have more states than those giving the Bekenstein-Hawking entropy: A simple argument. arXiv
**2017**, arXiv:1710:00218. [Google Scholar] - Rovelli, C. The subtle unphysical hypothesis of the firewall theorem. Entropy
**2019**, 21, 839. [Google Scholar] [CrossRef][Green Version] - Hawking, S.W. Breakdown of predictability in gravitational collapse. Phys. Rev. D
**1976**, 14, 2460–2473. [Google Scholar] [CrossRef] - Susskind, L.; Thorlacius, L. Gedanken experiments involving black holes. Phys. Rev. D
**1994**, 49, 966–974. [Google Scholar] [CrossRef][Green Version] - Almheiri, A.; Marolf, D.; Polchinski, J.; Sully, J. Black Holes: Complementarity or firewalls? J. High Energy Phys.
**2013**, 2013, 62. [Google Scholar] [CrossRef][Green Version] - Susskind, L. Entanglement is not enough. arXiv
**2014**, arXiv:1411.0690. [Google Scholar] [CrossRef][Green Version] - Fields, C. Some consequences of the thermodynamic cost of system identification. Entropy
**2018**, 20, 797. [Google Scholar] [CrossRef][Green Version] - Angelo, R.M.; Brunner, N.; Popescu, S.; Short, A.J.; Skrzypczyk, P. Physics within a quantum reference frame. J. Phys. A
**2011**, 2011 44, 145304. [Google Scholar] [CrossRef][Green Version] - Verlinde, E. On the origin of gravity and the laws of Newton. J. High Energy Phys.
**2011**, 2011, 29. [Google Scholar] [CrossRef][Green Version] - Maldacena, J. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys.
**1998**, 2, 231–252. [Google Scholar] [CrossRef] - Chen, P.; Chiang, H.-W.; Fields, C.; Marcianò, A. (Center for Field Theory and Particle Physics & Department of Physics, Fudan University, Shanghai 200433, China). Work in progress.

**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