The meaning of redundancy and consensus in quantum objectivity

Diana A. Chisholm1,2, Luca Innocenti1, and G. Massimo Palma1,3

1Università degli Studi di Palermo, Dipartimento di Fisica e Chimica – Emilio Segrè, via Archirafi 36, I-90123 Palermo, Italy
2Centre for Quantum Materials and Technologies, School of Mathematics and Physics, Queen's University Belfast, BT7 1NN, United Kingdom
3NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, 56127 Pisa, Italy

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While the terms "redundancy" and "consensus" are often used as synonyms in the context of quantum objectivity, we show here that these should be understood as two related but distinct notions, that quantify different features of the quantum-to-classical transition. We show that the two main frameworks used to measure quantum objectivity, namely spectrum broadcast structure and quantum Darwinism, are best suited to quantify redundancy and consensus, respectively. Furthermore, by analyzing explicit examples of states with nonlocally encoded information, we highlight the potentially stark difference between the degrees of redundancy and consensus. In particular, this causes a break in the hierarchical relations between spectrum broadcast structure and quantum Darwinism. Our framework provides a new perspective to interpret known and future results in the context of quantum objectivity, paving the way for a deeper understanding of the emergence of classicality from the quantum realm.

A quantum state is said to be objective if multiple observers are able to recover information about the state and agree among themselves. This is in turn possible only if said information was encoded multiple times into the surrounding environment. In this paper, we show how it is not always possible for the observers to extract all of the relevant information that was initially encoded into the environment. We do this by introducing two quantities with a rigorous definition and a clear operative interpretation: "redundancy", which quantifies how many times the information was written into the environment, and "consensus", which is the maximum number of observers able to extract said information.

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Cited by

[1] Xiao-Kan Guo and Zhiqiang Huang, "On the relation between quantum Darwinism and approximate quantum Markovianity", Physics Letters A 491, 129204 (2023).

[2] Tae-Hun Lee and Jarosław K. Korbicz, "Encoding position by spins: Objectivity in the boson-spin model", arXiv:2401.07690, (2024).

[3] Tiago Debarba, Marcus Huber, and Nicolai Friis, "Broadcasting Quantum Information using Finite Resources", arXiv:2403.07660, (2024).

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