# Quantum Darwinism in a Composite System: Objectivity versus Classicality

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Composite System Model

## 3. Quantum Darwinism and Objectivity

## 4. Strong Quantum Darwinism

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivations of Equations (6) and (7)

#### Appendix A.1. Direct Approach

#### Appendix A.2. Koashi–Winter Relation

## Appendix B. Testing the Strong Quantum Darwinism Criteria

## References

- Ollivier, H.; Poulin, D.; Zurek, W.H. Objective Properties from Subjective Quantum States: Environment as a Witness. Phys. Rev. Lett.
**2004**, 93, 220401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Blume-Kohout, R.; Zurek, W.H. Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information. Phys. Rev. A
**2006**, 73, 062310. [Google Scholar] [CrossRef] [Green Version] - Zurek, W.H. Quantum Darwinism. Nat. Phys.
**2009**, 5, 181. [Google Scholar] [CrossRef] - Horodecki, R.; Korbicz, J.K.; Horodecki, P. Quantum origins of objectivity. Phys. Rev. A
**2015**, 91, 032122. [Google Scholar] [CrossRef] [Green Version] - Korbicz, J.K.; Horodecki, P.; Horodecki, R. Objectivity in a Noisy Photonic Environment through Quantum State Information Broadcasting. Phys. Rev. Lett.
**2014**, 112, 120402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Le, T.P.; Olaya-Castro, A. Strong Quantum Darwinism and Strong Independence are Equivalent to Spectrum Broadcast Structure. Phys. Rev. Lett.
**2019**, 122, 010403. [Google Scholar] [CrossRef] [Green Version] - Balaneskovica, N.; Mendler, M. Dissipation, dephasing and quantum Darwinism in qubit systems with random unitary interactions. Eur. Phys. J. D
**2016**, 70, 177. [Google Scholar] [CrossRef] - Le, T.P.; Olaya-Castro, A. Objectivity (or lack thereof): Comparison between predictions of quantum Darwinism and spectrum broadcast structure. Phys. Rev. A
**2018**, 98, 032103. [Google Scholar] [CrossRef] [Green Version] - Le, T.P.; Olaya-Castro, A. Witnessing non-objectivity in the framework of strong quantum Darwinism. Quantum Sci. Technol.
**2020**, 5, 045012. [Google Scholar] [CrossRef] - Korbicz, J.K. Roads to objectivity: Quantum Darwinism, Spectrum Broadcast Structures, and Strong quantum Darwinism. arXiv
**2020**, arXiv:2007.04276. [Google Scholar] - Li, S.W.; Cai, C.Y.; Liu, X.F.; Sun, C.P. Objectivity in Quantum Measurement. Found. Phys.
**2018**, 48, 654. [Google Scholar] [CrossRef] [Green Version] - Pleasance, G.; Garraway, B.M. Application of quantum Darwinism to a structured environment. Phys. Rev. A
**2017**, 96, 062105. [Google Scholar] [CrossRef] [Green Version] - Brandão, F.G.S.L.; Piani, M.; Horodecki, P. Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information. Nat. Commun.
**2015**, 6, 7908. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ryan, E.; Paternostro, M.; Campbell, S. Quantum Darwinism in a structured spin environment. arXiv
**2020**, arXiv:2011.13385. [Google Scholar] - Campbell, S.; Çakmak, B.; Müstecaplıoğlu, O.E.; Paternostro, M.; Vacchini, B. Collisional unfolding of quantum Darwinism. Phys. Rev. A
**2019**, 99, 042103. [Google Scholar] [CrossRef] [Green Version] - Oliveira, S.M.; de Paula, A.L.; Drumond, R.C. Quantum Darwinism and non-Markovianity in a model of quantum harmonic oscillators. Phys. Rev. A
**2019**, 100, 052110. [Google Scholar] [CrossRef] [Green Version] - Giorgi, G.L.; Galve, F.; Zambrini, R. Quantum Darwinism and non-Markovian dissipative dynamics from quantum phases of the spin-1/2 XX model. Phys. Rev. A
**2015**, 92, 022105. [Google Scholar] [CrossRef] [Green Version] - Galve, F.; Zambrini, R.; Maniscalco, S. Non-Markovianity hinders Quantum Darwinism. Sci. Rep.
**2016**, 6, 19607. [Google Scholar] [CrossRef] [PubMed] - Milazzo, N.; Lorenzo, S.; Paternostro, M.; Palma, G.M. Role of information backflow in the emergence of quantum Darwinism. Phys. Rev. A
**2019**, 100, 012101. [Google Scholar] [CrossRef] [Green Version] - Lorenzo, S.; Paternostro, M.; Palma, G.M. Reading a Qubit Quantum State with a Quantum Meter: Time Unfolding of Quantum Darwinism and Quantum Information Flux. Open Syst. Inf. Dyn.
**2019**, 26, 1950023. [Google Scholar] [CrossRef] [Green Version] - Lampo, A.; Tuziemski, J.; Lewenstein, M.; Korbicz, J.K. Objectivity in the non-Markovian spin-boson model. Phys. Rev. A
**2017**, 96, 012120. [Google Scholar] [CrossRef] [Green Version] - Mironowicz, P.; Należyty, P.; Horodecki, P.; Korbicz, J.K. System information propagation for composite structures. Phys. Rev. A
**2018**, 98, 022124. [Google Scholar] [CrossRef] [Green Version] - Tuziemski, J.; Lampo, A.; Lewenstein, M.; Korbicz, J.K. Reexamination of the decoherence of spin registers. Phys. Rev. A
**2019**, 99, 022122. [Google Scholar] [CrossRef] [Green Version] - García-Pérez, G.; Chisholm, D.A.; Rossi, M.A.C.; Palma, G.M.; Maniscalco, S. Decoherence without entanglement and quantum Darwinism. Phys. Rev. Res.
**2020**, 2, 012061. [Google Scholar] [CrossRef] [Green Version] - Ciampini, M.A.; Pinna, G.; Mataloni, P.; Paternostro, M. Experimental signature of quantum Darwinism in photonic cluster states. Phys. Rev. A
**2018**, 98, 020101. [Google Scholar] [CrossRef] [Green Version] - Ciampini, M.A.; Pinna, G.; Paternostro, M.; Mataloni, P. Experimental Quantum Darwinism simulator using photonic cluster states. In Quantum Information and Measurement (QIM) V: Quantum Technologies; Optical Society of America: Washington, DC, USA, 2019; p. S2D.6. [Google Scholar] [CrossRef]
- Chen, M.C.; Zhong, H.S.; Li, Y.; Wu, D.; Wang, X.L.; Li, L.; Liu, N.L.; Lu, C.Y.; Pan, J.W. Emergence of classical objectivity of quantum Darwinism in a photonic quantum simulator. Sci. Bull.
**2019**, 64, 580–585. [Google Scholar] [CrossRef] [Green Version] - Unden, T.K.; Louzon, D.; Zwolak, M.; Zurek, W.H.; Jelezko, F. Revealing the Emergence of Classicality Using Nitrogen-Vacancy Centers. Phys. Rev. Lett.
**2019**, 123, 140402. [Google Scholar] [CrossRef] - Modi, K.; Brodutch, A.; Cable, H.; Paterek, T.; Vedral, V. The classical-quantum boundary for correlations: Discord and related measures. Rev. Mod. Phys.
**2012**, 84, 1655–1707. [Google Scholar] [CrossRef] [Green Version] - Campbell, S.; Vacchini, B. Collision models in open system dynamics: A versatile tool for deeper insights? EPL
**2021**, 133, 60001. [Google Scholar] [CrossRef] - Ciccarello, F.; Lorenzo, S.; Giovannetti, V.; Palma, G.M. Quantum collision models: Open system dynamics from repeated Interactions. arXiv
**2021**, arXiv:2106.11974. [Google Scholar] - Ciccarello, F. Collision models in quantum optics. Quant. Meas. Quant. Metrol.
**2017**, 4, 53. [Google Scholar] [CrossRef] [Green Version] - Ollivier, H.; Zurek, W.H. Quantum Discord: A Measure of the Quantumness of Correlations. Phys. Rev. Lett.
**2001**, 88, 017901. [Google Scholar] [CrossRef] - Henderson, L.; Vedral, V. Classical, quantum and total correlations. J. Phys. A Math. Gen.
**2001**, 34, 6899–6905. [Google Scholar] [CrossRef] - Nielsen, M.; Chuang, I. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Koashi, M.; Winter, A. Monogamy of quantum entanglement and other correlations. Phys. Rev. A
**2004**, 69, 022309. [Google Scholar] [CrossRef] [Green Version] - Touil, A.; Yan, B.; Girolami, D.; Deffner, S.; Zurek, W.H. Eavesdropping on the Decohering Environment: Quantum Darwinism, Amplification, and the Origin of Objective Classical Reality. arXiv
**2021**, arXiv:2107.00035. [Google Scholar] - Huang, Y. Computing quantum discord is NP-complete. New J. Phys.
**2014**, 16, 033027. [Google Scholar] [CrossRef] - Ferraro, A.; Aolita, L.; Cavalcanti, D.; Cucchietti, F.M.; Acín, A. Almost all quantum states have nonclassical correlations. Phys. Rev. A
**2010**, 81, 052318. [Google Scholar] [CrossRef] [Green Version] - Huang, J.H.; Wang, L.; Zhu, S.Y. A new criterion for zero quantum discord. New J. Phys.
**2011**, 13, 063045. [Google Scholar] [CrossRef] - Bylicka, B.; Chruściński, D. Circulant States with Vanishing Quantum Discord. Open Syst. Inf. Dyn.
**2012**, 19, 1250006. [Google Scholar] [CrossRef]

**Figure 1.**Schematics of the considered model. A composite system that is made up of two interacting qubits, which are also coupled to a fragmented environment. For a excitation preserving interaction between the system qubits, i.e., ${J}_{x}={J}_{y}$ in Equation (2) and pure dephasing interaction between the system qubits and the environment, the interaction Hamiltonians commute, leading to Equation (4).

**Figure 2.**Both system qubits are prepared in a symmetric initial state with ${\theta}_{{S}_{1}}={\theta}_{{S}_{2}}=\pi /6$ with the interaction parameters ${J}_{SE}=1$, $J=10$, and we consider environments of size $N=6$ (solid curves) and $N=250$ (dashed curves). (

**a**) Dynamics of mutual information between the composite system and a single environmental qubit, $\mathcal{I}({\rho}_{{S}_{1}{S}_{2}:{E}_{k}})$, (darker, black) and the entropy of the composite system, $S({\rho}_{{S}_{1}{S}_{2}})$ (lighter, red). (

**b**) Coherence present in the composite system state, $|{\rho}_{{S}_{1}{S}_{2}}^{1,2}|$ (lighter, blue) and coherences in the state of a single environmental qubit $|{\rho}_{{E}_{k}}^{1,2}|$ (darker, black). (

**c**) Quantum discord shared between the two system qubits, ${D}^{\to}({\rho}_{{S}_{1}{S}_{2}})$ (lighter, orange) and quantum discord between one of the system qubits and a single environmental constituent ${D}^{\to}({\rho}_{{S}_{1}{E}_{k}})$ (darker, green). In panels (

**a**–

**c**) the faint vertical line at $t=\pi /4$ denotes the time at which we have $\mathcal{I}({\rho}_{{S}_{1}{S}_{2}:{E}_{k}})=S({\rho}_{{S}_{1}{S}_{2}})$, i.e., the emergence of Darwinism. (

**d**) $\mathcal{I}({\rho}_{{S}_{1}{S}_{2}:{\mathcal{E}}_{f}})/S({\rho}_{{S}_{1}{S}_{2}})$ vs. the size of the environment fraction f (upper, solid) and $\mathcal{I}({\rho}_{{S}_{1}:{\mathcal{E}}_{f}})//S({\rho}_{{S}_{1}})$ (lower, dashed) at two instants of time, $t=\pi /4$ where perfect redundant encoding is observed (lighter, red) and $t=\pi /4-0.1$ (darker, blue).

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

© 2021 by the authors. 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**

Çakmak, B.; Müstecaplıoğlu, Ö.E.; Paternostro, M.; Vacchini, B.; Campbell, S.
Quantum Darwinism in a Composite System: Objectivity versus Classicality. *Entropy* **2021**, *23*, 995.
https://doi.org/10.3390/e23080995

**AMA Style**

Çakmak B, Müstecaplıoğlu ÖE, Paternostro M, Vacchini B, Campbell S.
Quantum Darwinism in a Composite System: Objectivity versus Classicality. *Entropy*. 2021; 23(8):995.
https://doi.org/10.3390/e23080995

**Chicago/Turabian Style**

Çakmak, Barış, Özgür E. Müstecaplıoğlu, Mauro Paternostro, Bassano Vacchini, and Steve Campbell.
2021. "Quantum Darwinism in a Composite System: Objectivity versus Classicality" *Entropy* 23, no. 8: 995.
https://doi.org/10.3390/e23080995