# The ABC of Deutsch–Hayden Descriptors

## Abstract

**:**

## 1. Motivation

## 2. A Question of Picture

## 3. Tracking Observables

#### 3.1. The Descriptor of a 1-Qubit Network

**Example**

**1.**

#### 3.2. Descriptors of an n-Qubit Network

#### 3.3. The Main Simplification

**Example**

**2.**

#### 3.4. The Algebra of Descriptors

**Remark**

**1.**

**Remark**

**2.**

#### 3.5. One More Simplification

## 4. Evolution from the Future?

#### 4.1. The Functional Representation of a Gate

**Example**

**3.**

#### 4.2. Back in Order!

## 5. The Action on Descriptors

**Example**

**4.**

#### 5.1. Locality and Completeness

**Example**

**5.**

#### 5.2. The Cnot

## 6. Superdense Coding, Revisited

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bell, J.S. On the Einstein Podolsky Rosen paradox. Physics
**1964**, 1, 195–200. [Google Scholar] [CrossRef][Green Version] - Aspect, A.; Grangier, P.; Roger, G. Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bell’s inequalities. Phys. Rev. Lett.
**1982**, 49, 91. [Google Scholar] [CrossRef][Green Version] - Bennett, C.H.; Brassard, G.; Crépeau, C.; Jozsa, R.; Peres, A.; Wootters, W.K. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett.
**1993**, 70, 1895. [Google Scholar] [CrossRef] [PubMed][Green Version] - Deutsch, D.; Hayden, P. Information flow in entangled quantum systems. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2000**, 456, 1759–1774. [Google Scholar] [CrossRef][Green Version] - Brassard, G.; Raymond-Robichaud, P. Parallel lives: A local-realistic interpretation of “nonlocal” boxes. Entropy
**2019**, 21, 87. [Google Scholar] [CrossRef] [PubMed][Green Version] - Schilpp, P.A. Albert Einstein: Philosopher-Scientist, 3rd ed.; The Open Court Publishing Co.: Chicago, IL, USA, 1970; Volume 7. [Google Scholar]
- Gottesman, D. The Heisenberg representation of quantum computers. In Group 22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics; Preprint quant-ph/9807006; International Press: Cambridge, MA, USA, 1999; pp. 32–43. [Google Scholar]
- Deutsch, D. Quantum computational networks. Proc. R. Soc. A Math. Phys. Eng. Sci.
**1989**, 425, 73–90. [Google Scholar] - Bennett, C.H.; Wiesner, S.J. Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett.
**1992**, 69, 2881. [Google Scholar] [CrossRef] [PubMed][Green Version]

Bits $\mathit{i},\mathit{j}$ | State at Time 4 | State at Time 6 |
---|---|---|

$0,0$ | $|{\mathrm{\Phi}}^{+}\rangle =\frac{|00\rangle +|11\rangle}{\sqrt{2}}$ | $|00\rangle =|ij\rangle $ |

$0,1$ | $|{\mathrm{\Psi}}^{+}\rangle =\frac{|01\rangle +|10\rangle}{\sqrt{2}}$ | $|01\rangle =|ij\rangle $ |

$1,0$ | $|{\mathrm{\Phi}}^{-}\rangle =\frac{|00\rangle -|11\rangle}{\sqrt{2}}$ | $|10\rangle =|ij\rangle $ |

$1,1$ | $|{\mathrm{\Psi}}^{-}\rangle =\frac{|01\rangle -|10\rangle}{\sqrt{2}}$ | $|11\rangle =|ij\rangle $ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bédard, C.A. The ABC of Deutsch–Hayden Descriptors. *Quantum Rep.* **2021**, *3*, 272-285.
https://doi.org/10.3390/quantum3020017

**AMA Style**

Bédard CA. The ABC of Deutsch–Hayden Descriptors. *Quantum Reports*. 2021; 3(2):272-285.
https://doi.org/10.3390/quantum3020017

**Chicago/Turabian Style**

Bédard, Charles Alexandre. 2021. "The ABC of Deutsch–Hayden Descriptors" *Quantum Reports* 3, no. 2: 272-285.
https://doi.org/10.3390/quantum3020017