# Reference Frame Induced Symmetry Breaking on Holographic Screens

^{1}

^{2}

^{3}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

GHP:If but only if a pair of finite quantum systems A and B have a separable joint state $|AB\rangle =|A\rangle |B\rangle $, there is a finite spacelike surface $\mathcal{B}$, with area $A\left(\mathcal{B}\right)\ge A{\left(\mathcal{B}\right)}_{min}=4\mathrm{ln}2N{l}_{P}^{2}$, N the dimension of ${H}_{AB}$ and ${l}_{P}$ the Planck length, that implements ${H}_{AB}$ as a classical channel.

## 2. Instantaneous Interactions across $\mathcal{B}$

**Lemma**

**1.**

**Proof.**

#### 2.1. Example: Scattering

#### 2.2. Example: Hawking Radiation

#### 2.3. Symmetry across $\mathcal{B}$ Corresponds to “Free Choice” of QRFs

## 3. Reference Frame Induced Decoherence

#### 3.1. QRFs for System Identification

**Lemma**

**2.**

**Proof.**

#### 3.2. Reference and Pointer Measurements

#### 3.3. Sequential Pointer Measurements Induce Decoherence

**Lemma**

**3.**

**Proof.**

[T]he formulation of the measurement problem and its resolution through the appeal to decoherence require a universe split into systems. Yet, it is far from clear how one can define systems given an overall Hilbert space ‘of everything’ and the total Hamiltonian.

#### 3.4. Example: Mass and Hawking Radiation QRFs for a BH

#### 3.5. Computation and Memory Costs Induce Coarse-Graining

## 4. Reference Frame Induced Entanglement

**Theorem**

**1.**

**Proof.**

## 5. Reference Frame Induced Contextuality

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

## 6. Writing and Reading Classical Memories

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BH | Black Hole |

CCD | Cocone Diagram |

CD | Cone Diagram |

EPR | Einstein-Podolsky-Rosen |

ER | Einstein-Rosen |

GHP | Generalized Holographic Principle |

LOCC | Local Operations, Classical Communication |

QECC | Quantum Error-Correcting Code |

QRF | Quantum Reference Frame |

## Appendix A. The Basics of Channel Theory Information Flow and Context Dependency

**Definition**

**A1.**

**Definition**

**A2.**

#### Appendix A.1. Example: Observables in Context

- (i)
- A a set of “events” (in the general sense of the term, e.g., as observed value combinations or atomic), as related to
- (ii)
- a set B of conditions specifying “objects/contents” or “influences,” and
- (iii)
- a set R of contexts (or, in certain instances, a set of “detectors”, “measurements” or “methods”).

**K**$=\{-1,1\}$. Notably, in [33], ‘${\models}_{\mathcal{A}}$’ can be realized for an inferential process by the conditional probability $p\left(a\right|x)=p(a\left|\right\{b,c\left\}\right)$, whenever defined, for $a\in A,b\in B$ and $c\in R$, and which for suitable indexing, leads to an information flow of hierarchical Bayesian inference within a CCD [33]. The background to the results in Section 5 here can be found in ([33], Section 7). In particular, ([33], Th. 7.1) states the criteria for intrinsic contextuality (non-co-deployable observables) in terms of noncommutativity of a CCD. Note that the above classifier (Chu space) formulism of contextuality is very general. Special cases of the set $X=B\times R$ are the sets of binary random variables labelled by a measurement (contents-context) system as basic to the theory of Contextuality-by-Default [78,79]. Much amounts to the question of determining the nature of an empirical model e relative to how a probability distribution can be obtained as the marginals of a global probability distribution on the outcomes to all measurements. For example, e is said to be contextual in [80] if the corresponding probability distribution cannot be obtained by such global means. This has a compatible interpretation in terms of the non-existence of a global section of a sheaf defined relative to a “measurement cover” in [81]. These methods of studying contextuality are also very general, and as for those of [33], can extend beyond quantum theory to such disciplines as linguistics and psychology. To see the explicit connections between these various approaches would indeed be a worthwhile undertaking.

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**Figure 1.**(

**a**) A gauge boson transfers asymptotically-classical momentum information across a holographic screen $\mathcal{B}$. (

**b**) The scattering process transfers no information about the entanglement entropy $\mathcal{S}\left(B\right)$.

**Figure 2.**(

**a**) Hawking pair annihilation-production near a BH is asymptotically indistinguishable from (

**b**) symmetric scattering from the stretched horizon.

**Figure 3.**A cocone diagram (CCD) is a commuting diagram depicting maps (infomorphisms) ${f}_{ij}$ between (eigen-)classifiers ${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{j}$, maps ${g}_{kl}$ from the ${\mathcal{A}}_{k}$ to one or more channels ${\mathcal{C}}_{l}$ over subsets of the ${\mathcal{A}}_{i}$, and maps ${h}_{l}$ from channels ${\mathcal{C}}_{l}$ to the colimit $\mathbf{C}$ (cf. Equation (6.7) of [32]). Such a CCD can be associated (double-headed arrows) with any subset of binary operators ${M}_{k}^{A}\dots {M}_{n}^{A}$ provided that these operators all mutually commute. The CCD specifies, in this case, a classical algorithm implemented by ${H}_{A}$. The complete set of operators ${M}_{i}^{A}$ and ${M}_{i}^{B}$ in (2) together with the array of mutually noninteracting qubits ${q}_{1}\dots {q}_{N}$ (i.e., the screen $\mathcal{B}$) implement the classical channel between A and B. Free choice of QRFs by A and B corresponds to independent, free choice of z axis by A and B at each qubit. Note that should the CCD fail to commute (in which case the colimit becomes undefined), then the ${\mathcal{A}}_{i}$ are considered as “non-co-deployable” (observables), and their corresponding distributed system exhibits intrinsic contextuality ([33], Section 7).

**Figure 4.**A cocone diagram (CCD) computing an effective (or virtual) “system state” ${\rho}^{S}$ comprises classifier channels computing an effective pointer state ${\rho}^{{P}_{i}}$ and an effective reference state ${\rho}^{R}$ (cf. [6]). These channels define the effective “subsystems” R and ${P}_{i}$ comprising S. The CCD acts on the pure physical state ${|B\rangle}^{A}$ encoded by ${H}_{AB}$ on the holographic screen $\mathcal{B}$ (blue) separating A from B. The computation represented by the CCD is implemented by the internal dynamics ${H}_{A}$.

**Figure 5.**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 macroscopic times ${\tau}_{i}$. The operators ${M}_{i}^{P}$ can equally well be generalized to subsets ${\{{M}^{P}\}}_{i}$ of mutually-commuting pointer-state observables.

**Figure 6.**Identifying a local quantum of radiation as a Hawking quantum ${r}_{H}$ from a distant BH requires a local Hawking QRF ${R}_{H}$. Lemma 3 rules this out.

**Figure 7.**A typical Bell protocol described in the lab frame. Sharing of measurements results via a classical channel is required to observe a Bell-inequality violation. If Alice’s interaction with Bob’s message is viewed as an ordinary quantum measurement, the entanglement disappears as in Section 3.4 above.

**Figure 8.**(

**a**) A Bell protocol in the frame of the entangled state (yellow circle). Alice and Bob collide at ${t}_{meas}$, at which time they share, and together measure, the entangled state. (

**b**) This is equivalent to Alice and Bob sharing an entangled QRF that reports consistent pointer outcomes to each observer.

**Figure 9.**A CD ${W}_{jj}$ (green triangle) specifies a memory-write operation of the time-stamped state $({\rho}^{R{P}_{j}},{\tau}_{j})$ to $\mathcal{B}$. The timestamp ${\tau}_{j}$ is generated by the groupoid action ${\mathcal{G}}_{ij}$.

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Fields, C.; Glazebrook, J.F.; Marcianò, A. Reference Frame Induced Symmetry Breaking on Holographic Screens. *Symmetry* **2021**, *13*, 408.
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Fields C, Glazebrook JF, Marcianò A. Reference Frame Induced Symmetry Breaking on Holographic Screens. *Symmetry*. 2021; 13(3):408.
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Fields, Chris, James F. Glazebrook, and Antonino Marcianò. 2021. "Reference Frame Induced Symmetry Breaking on Holographic Screens" *Symmetry* 13, no. 3: 408.
https://doi.org/10.3390/sym13030408