# Quantum Darwinism and the spreading of classical information in non-classical theories

Roberto D. Baldijao1,2, Marius Krumm2,3, Andrew J. P. Garner2, and Markus P. Mueller2,4,5

1Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Campinas, SP 13083-859, Brazil
2Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
3Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
4Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Vienna, Austria
5Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada

### Abstract

Quantum Darwinism posits that the emergence of a classical reality relies on the spreading of classical information from a quantum system to many parts of its environment. But what are the essential physical principles of quantum theory that make this mechanism possible? We address this question by formulating the simplest instance of Darwinism – CNOT-like $fan$-$out$ interactions – in a class of probabilistic theories that contain classical and quantum theory as special cases. We determine necessary and sufficient conditions for any theory to admit such interactions. We find that every theory with non-classical features that admits this idealized spreading of classical information must have both entangled states and entangled measurements. Furthermore, we show that Spekkens' toy theory admits this form of Darwinism, and so do all probabilistic theories that satisfy principles like strong symmetry, or contain a certain type of decoherence processes. Our result suggests the counter-intuitive general principle that in the presence of local non-classicality, a classical world can only emerge if this non-classicality can be "amplified" to a form of entanglement.

Quantum mechanics predicts that physical objects can have several contradictory properties at once – a phenomenon called superposition. For example, a single electron can travel along several distinct paths at the same time. Why, then, do we never see a single car take both lanes on a highway at once, and moreover always agree with other observers about the single lane that the car is in? To explain objective reality in a quantum world, physicists have proposed "Quantum Darwinism" – essentially, a mechanism that carries information from one system to many observers. In our work, we investigate the physical principles behind this, using a more general formalism that drops many details specific to quantum mechanics. We discuss the sufficient structure needed to enable such Darwinism in physical theories, and also present a non-quantum example. Furthermore, we show that a physical theory only allows for Darwinism if it also admits entanglement. That is, quite surprisingly, to tame superpositions and present an ordinary classical world, physics must necessarily invoke an even more extraordinary phenomenon.

### ► References

[1] W. H. Zurek. Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys., 75 (3): 715–775, 2003. 10.1103/​revmodphys.75.715.
https:/​/​doi.org/​10.1103/​revmodphys.75.715

[2] W. H. Zurek. Relative States and the Environment: Einselection, Envariance, Quantum Darwinism, and the Existential Interpretation. Pre-print, arXiv:0707.2832, 2007. URL https:/​/​arxiv.org/​abs/​0707.2832.
arXiv:0707.2832

[3] W. H. Zurek. Quantum Darwinism. Nature Physics, 5 (3): 181–188, 2009. 10.1038/​nphys1202.
https:/​/​doi.org/​10.1038/​nphys1202

[4] F. G. S. L. Brandão, M. Piani, and P. Horodecki. Generic emergence of classical features in quantum darwinism. Nature Communications, 6 (1), 2015. 10.1038/​ncomms8908.
https:/​/​doi.org/​10.1038/​ncomms8908

[5] P. A. Knott, T. Tufarelli, M. Piani, and G. Adesso. Generic emergence of objectivity of observables in infinite dimensions. Phys. Rev. Lett., 121: 160401, 2018. 10.1103/​PhysRevLett.121.160401.
https:/​/​doi.org/​10.1103/​PhysRevLett.121.160401

[6] R. Horodecki, J. K. Korbicz, and P. Horodecki. Quantum origins of objectivity. Phys. Rev. A, 91 (3): 032122, 2015. 10.1103/​PhysRevA.91.032122.
https:/​/​doi.org/​10.1103/​PhysRevA.91.032122

[7] T. P. Le and A. Olaya-Castro. Objectivity (or lack thereof): Comparison between predictions of quantum darwinism and spectrum broadcast structure. Phys. Rev. A, 98 (3), 2018. 10.1103/​physreva.98.032103.
https:/​/​doi.org/​10.1103/​physreva.98.032103

[8] R. Blume-Kohout and W. H. Zurek. Quantum darwinism in quantum brownian motion. Phys. Rev. Lett., 101: 240405, 2008. 10.1103/​PhysRevLett.101.240405.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.240405

[9] R. Blume-Kohout and W. H. Zurek. A simple example of quantum darwinism'': Redundant information storage in many-spin environments. Foundations of Physics, 35 (11): 1857–1876, 2005. 10.1007/​s10701-005-7352-5.
https:/​/​doi.org/​10.1007/​s10701-005-7352-5

[10] C. J. Riedel and W. H. Zurek. Quantum darwinism in an everyday environment: Huge redundancy in scattered photons. Phys. Rev. Lett., 105: 020404, 2010. 10.1103/​PhysRevLett.105.020404.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.020404

[11] M. Zwolak, H. T. Quan, and W. H. Zurek. Redundant imprinting of information in nonideal environments: Objective reality via a noisy channel. Phys. Rev. A, 81: 062110, 2010. 10.1103/​PhysRevA.81.062110.
https:/​/​doi.org/​10.1103/​PhysRevA.81.062110

[12] M. Zwolak, H. T. Quan, and W. H. Zurek. Quantum darwinism in a mixed environment. Phys. Rev. Lett., 103: 110402, 2009. 10.1103/​PhysRevLett.103.110402.
https:/​/​doi.org/​10.1103/​PhysRevLett.103.110402

[13] P. A. M. Dirac. The Principles of Quantum Mechanics. Oxford University Press, Oxford, 4th edition, 1958.

[14] J. J. Sakurai and J. Napolitano. Modern Quantum Mechanics. Addison-Wesley, 2nd edition, 2011.

[15] M. Schlosshauer. Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys., 76 (4): 1267–1305, 2004. 10.1103/​RevModPhys.76.1267.
https:/​/​doi.org/​10.1103/​RevModPhys.76.1267

[16] W. K. Wootters and W. H. Zurek. A single quantum cannot be cloned. Nature, 299 (5886): 802–803, 1982. 10.1038/​299802a0.
https:/​/​doi.org/​10.1038/​299802a0

[17] L. Hardy. Quantum Theory From Five Reasonable Axioms. Pre-print, arXiv:quant-ph/​0101012, 2001. URL https:/​/​arxiv.org/​abs/​quant-ph/​0101012.
arXiv:quant-ph/0101012

[18] J. Barrett. Information processing in generalized probabilistic theories. Phys. Rev. A, 75: 032304, 2007. 10.1103/​PhysRevA.75.032304.
https:/​/​doi.org/​10.1103/​PhysRevA.75.032304

[19] H. Barnum, J. Barrett, M. Leifer, and A. Wilce. Teleportation in general probabilistic theories, 2012.

[20] H. Barnum, J. Barrett, M. Leifer, and A. Wilce. Generalized no-broadcasting theorem. Phys. Rev. Lett., 99 (24), 2007. 10.1103/​physrevlett.99.240501.
https:/​/​doi.org/​10.1103/​physrevlett.99.240501

[21] A. J. P. Garner, O. C. O. Dahlsten, Y. Nakata, M. Murao, and V. Vedral. A framework for phase and interference in generalized probabilistic theories. New J. Phys., 15 (9): 093044, 2013. 10.1088/​1367-2630/​15/​9/​093044.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​9/​093044

[22] O. C. O. Dahlsten, A. J. P. Garner, and V. Vedral. The uncertainty principle enables non-classical dynamics in an interferometer. Nature Communications, 5 (4592), 2014. 10.1038/​ncomms5592.
https:/​/​doi.org/​10.1038/​ncomms5592

[23] J. G. Richens, J. H. Selby, and S. W. Al-Safi. Entanglement is necessary for emergent classicality in all physical theories. Phys. Rev. Lett., 119: 080503, 2017. 10.1103/​PhysRevLett.119.080503.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.080503

[24] G. Chiribella, G. M. D'Ariano, and P. Perinotti. Probabilistic theories with purification. Phys. Rev. A, 81 (6), 2010. 10.1103/​physreva.81.062348.
https:/​/​doi.org/​10.1103/​physreva.81.062348

[25] P. Janotta and R. Lal. Generalized probabilistic theories without the no-restriction hypothesis. Phys. Rev. A, 87 (5), 2013. 10.1103/​physreva.87.052131.
https:/​/​doi.org/​10.1103/​physreva.87.052131

[26] H. Barnum, M. P. Müller, and C. Ududec. Higher-order interference and single-system postulates characterizing quantum theory. New J. Phys., 16 (12): 123029, 2014a. 10.1088/​1367-2630/​16/​12/​123029.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​12/​123029

[27] R. W. Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Phys. Rev. A, 75 (3): 32110, 2007. 10.1103/​PhysRevA.75.032110.
https:/​/​doi.org/​10.1103/​PhysRevA.75.032110

[28] A. S. Holevo. Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Peredachi Inf., 9 (3): 177–183, 1973.

[29] M. Zwolak, C. J. Riedel, and W. H. Zurek. Amplification, decoherence and the acquisition of information by spin environments. Scientific Reports, 6 (1): 25277, 2016. 10.1038/​srep25277.
https:/​/​doi.org/​10.1038/​srep25277

[30] T. K. Unden, D. Louzon, M. Zwolak, W. H. Zurek, and F. Jelezko. Revealing the emergence of classicality using nitrogen-vacancy centers. Phys. Rev. Lett., 123: 140402, 2019. 10.1103/​PhysRevLett.123.140402.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.140402

[31] M. A. Ciampini, G. Pinna, P. Mataloni, and M. Paternostro. Experimental signature of quantum darwinism in photonic cluster states. Phys. Rev. A, 98: 020101, 2018. 10.1103/​PhysRevA.98.020101.
https:/​/​doi.org/​10.1103/​PhysRevA.98.020101

[32] C. J. Riedel and W. H. Zurek. Redundant information from thermal illumination: quantum darwinism in scattered photons. New Journal of Physics, 13 (7): 073038, jul 2011. 10.1088/​1367-2630/​13/​7/​073038.
https:/​/​doi.org/​10.1088/​1367-2630/​13/​7/​073038

[33] M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens. Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories. PRX Quantum, 2 (2), Apr 2021. ISSN 2691-3399. 10.1103/​prxquantum.2.020302.
https:/​/​doi.org/​10.1103/​prxquantum.2.020302

[34] G. Chiribella, G. M. D'Ariano, and P. Perinotti. Informational derivation of quantum theory. Phys. Rev. A, 84 (1): 012311, 2011. 10.1103/​PhysRevA.84.012311.
https:/​/​doi.org/​10.1103/​PhysRevA.84.012311

[35] Ll. Masanes and M. P. Müller. A derivation of quantum theory from physical requirements. New J. Phys., 13 (6): 063001, 2011. 10.1088/​1367-2630/​13/​6/​063001.
https:/​/​doi.org/​10.1088/​1367-2630/​13/​6/​063001

[36] M. P. Müller and Ll. Masanes. Three-dimensionality of space and the quantum bit: an information-theoretic approach. New J. Phys., 15 (5): 053040, 2013. 10.1088/​1367-2630/​15/​5/​053040.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​5/​053040

[37] M. P. Müller. Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes). SciPost Phys. Lect. Notes, page 28, 2021. 10.21468/​SciPostPhysLectNotes.28.
https:/​/​doi.org/​10.21468/​SciPostPhysLectNotes.28

[38] R. Webster. Convexity. Oxford University Press, Oxford, 1994. ISBN 0-19-853147-8.

[39] G. Chiribella and C. M. Scandolo. Operational axioms for diagonalizing states. Electronic Proceedings in Theoretical Computer Science, 195: 96–115, 2015. 10.4204/​EPTCS.195.8.
https:/​/​doi.org/​10.4204/​EPTCS.195.8

[40] M. P. Müller and C. Ududec. Structure of reversible computation determines the self-duality of quantum theory. Phys. Rev. Lett., 108: 130401, 2012. 10.1103/​PhysRevLett.108.130401.
https:/​/​doi.org/​10.1103/​PhysRevLett.108.130401

[41] S. W. Al-Safi and J. Richens. Reversibility and the structure of the local state space. New J. Phys., 17 (12): 123001, 2015. 10.1088/​1367-2630/​17/​12/​123001.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​12/​123001

[42] S. Massar and M. K. Patra. Information and communication in polygon theories. Phys. Rev. A, 89: 052124, 2014. 10.1103/​PhysRevA.89.052124.
https:/​/​doi.org/​10.1103/​PhysRevA.89.052124

[43] P. Janotta and H. Hinrichsen. Generalized probability theories: what determines the structure of quantum theory? Journal of Physics A: Mathematical and Theoretical, 47 (32): 323001, 2014. 10.1088/​1751-8113/​47/​32/​323001.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​32/​323001

[44] B. Coecke and C. Heunen. Pictures of complete positivity in arbitrary dimension. Information and Computation, 250: 50–58, 2016. 10.1016/​j.ic.2016.02.007.
https:/​/​doi.org/​10.1016/​j.ic.2016.02.007

[45] D. Gross, M. P. Müller, R. Colbeck, and O. C. O. Dahlsten. All reversible dynamics in maximally nonlocal theories are trivial. Phys. Rev. Lett., 104: 080402, 2010. 10.1103/​PhysRevLett.104.080402.
https:/​/​doi.org/​10.1103/​PhysRevLett.104.080402

[46] H. Barnum, M. P. Müller, and C. Ududec. Higher-order interference and single-system postulates characterizing quantum theory. New J. Phys., 16 (12): 123029, 2014b. 10.1088/​1367-2630/​16/​12/​123029.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​12/​123029

[47] M. F. Pusey. Stabilizer Notation for Spekkens' Toy Theory. Foundations of Physics, 42 (5): 688–708, 2012. 10.1007/​s10701-012-9639-7.
https:/​/​doi.org/​10.1007/​s10701-012-9639-7

[48] B. Coecke, B. Edwards, and R. W. Spekkens. Phase Groups and the Origin of Non-locality for Qubits. Electronic Notes in Theoretical Computer Science, 270 (2): 15–36, 2011. 10.1016/​j.entcs.2011.01.021.
https:/​/​doi.org/​10.1016/​j.entcs.2011.01.021

[49] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner. Bell nonlocality. Reviews of Modern Physics, 86 (2): 419–478, 2014. 10.1103/​RevModPhys.86.419.
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[50] D. Gottesman. The Heisenberg Representation of Quantum Computers. In Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, pages 32–43, 1999. URL https:/​/​arxiv.org/​abs/​quant-ph/​9807006.
arXiv:quant-ph/9807006

[51] C. M. Scandolo, R. Salazar, J. K. Korbicz, and P. Horodecki. Universal structure of objective states in all fundamental causal theories. Physical Review Research, 3 (3), Aug 2021. ISSN 2643-1564. 10.1103/​physrevresearch.3.033148.
https:/​/​doi.org/​10.1103/​physrevresearch.3.033148

[52] C. M. Lee and J. Barrett. Computation in generalised probabilisitic theories. New J. Phys., 17 (17), 2015. 10.1088/​1367-2630/​17/​8/​083001.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​8/​083001

[53] A. J. P. Garner. Interferometric Computation Beyond Quantum Theory. Foundations of Physics, 2018. 10.1007/​s10701-018-0142-7.
https:/​/​doi.org/​10.1007/​s10701-018-0142-7

[54] K. H. Wan, O. C. O. Dahlsten, H. Kristjánsson, R. Gardner, and M. S. Kim. Quantum generalisation of feedforward neural networks. npj Quantum Information, 3 (1), 2017. 10.1038/​s41534-017-0032-4.
https:/​/​doi.org/​10.1038/​s41534-017-0032-4

[55] P. Janotta, C. Gogolin, J. Barrett, and N. Brunner. Limits on nonlocal correlations from the structure of the local state space. New J. Phys., 13 (6): 063024, 2011. 10.1088/​1367-2630/​13/​6/​063024.
https:/​/​doi.org/​10.1088/​1367-2630/​13/​6/​063024

[56] Ll. Masanes, M. P. Müller, R. Augusiak, and D. Pérez-García. Existence of an information unit as a postulate of quantum theory. Proceedings of the National Academy of Sciences of the United States of America, 110 (41): 16373–16377, 2013. 10.1073/​pnas.1304884110.
https:/​/​doi.org/​10.1073/​pnas.1304884110

[57] L. Hardy. Disentangling nonlocality and teleportation. Pre-print, arXiv:quant-ph/​9906123, 1999. URL https:/​/​arxiv.org/​abs/​quant-ph/​9906123.
arXiv:quant-ph/9906123

[58] A. J. P. Garner. Phase and interference phenomena in generalised probabilistic theories. PhD thesis, University of Oxford, 2015. URL https:/​/​ora.ox.ac.uk/​objects/​uuid:c0017faf-cbe0-4365-a1ff-080fa031d006.
https:/​/​ora.ox.ac.uk/​objects/​uuid:c0017faf-cbe0-4365-a1ff-080fa031d006

[59] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000. ISBN 0521635039.

### Cited by

[1] Wojciech Hubert Zurek, "Emergence of the Classical World from Within Our Quantum Universe", From Quantum to Classical; Essays in Honour of H.-Dieter Zeh 23 (2022).

[2] Carlo Maria Scandolo, Roberto Salazar, Jarosław K. Korbicz, and Paweł Horodecki, "Universal structure of objective states in all fundamental causal theories", arXiv:1805.12126.

[3] Sidiney B. Montanhano, "Characterization of Contextuality with Semi-Module Čech Cohomology and its Relation with Cohomology of Effect Algebras", arXiv:2104.11411.

The above citations are from SAO/NASA ADS (last updated successfully 2022-10-01 22:32:34). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-10-01 22:32:32).