# Emergent classicality in general multipartite states and channels

Xiao-Liang Qi and Daniel Ranard

Department of Physics, Stanford University, Stanford, CA 94305-4060, USA

### Abstract

In a quantum measurement process, classical information about the measured system spreads throughout the environment. Meanwhile, quantum information about the system becomes inaccessible to local observers. Here we prove a result about quantum channels indicating that an aspect of this phenomenon is completely general. We show that for any evolution of the system and environment, for everywhere in the environment excluding an $O(1)$-sized region we call the "quantum Markov blanket," any locally accessible information about the system must be approximately classical, i.e. obtainable from some fixed measurement. The result strengthens the earlier result of Brandão et al. ($\textit{Nat. comm. 6:7908}$) in which the excluded region was allowed to grow with total environment size. It may also be seen as a new consequence of the principles of no-cloning or monogamy of entanglement. Our proof offers a constructive optimization procedure for determining the "quantum Markov blanket" region, as well as the effective measurement induced by the evolution. Alternatively, under channel-state duality, our result characterizes the marginals of multipartite states.

Why do quantum-mechanical systems often appear to behave classically at large scales? Part of the answer lies in decoherence: when a quantum system interacts with an environment, often the environment effectively “measures” the system. Afterward, local observers in the environment can only learn classical information about the system. This phenomenon might seem to depend on the exact form of the system-environment interaction. However, we prove that an aspect of this phenomenon is general to all quantum evolutions. The structure of quantum mechanics itself ensures that most observers can only learn compatible, classical information about any system.

We prove a general result about quantum states and channels that guarantees the above classicality. More precisely, we show that whenever a subsystem interacts with an environment, almost all local observers in the environment can only learn classical information about the system. This result also provides a new quantitative form of two well-known principles: no-cloning, i.e. the inability to clone quantum information, and monogamy of entanglement, i.e. the inability of one system to be highly entangled with multiple others.

Our proof also offers an optimization procedure for determining the effective measurement made by the environment. This procedure could be used to analyze the emergence of classicality in numerically tractable systems like spin chains. Taking advantage of our general results, future work might also identify the more specific conditions under which the quantum dynamics allow an effective classical description.

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### Cited by

[1] Thao P. Le and Alexandra Olaya-Castro, "Witnessing non-objectivity in the framework of strong quantum Darwinism", Quantum Science and Technology 5 4, 045012 (2020).

[2] Akram Touil, Bin Yan, Davide Girolami, Sebastian Deffner, and Wojciech H. Zurek, "Eavesdropping on the Decohering Environment: Quantum Darwinism, Amplification, and the Origin of Objective Classical Reality", arXiv:2107.00035.

[3] Thao P. Le, Piotr Mironowicz, and Paweł Horodecki, "Blurred quantum Darwinism across quantum reference frames", Physical Review A 102 6, 062420 (2020).

[4] Wojciech Hubert Zurek, "Emergence of the Classical from within the Quantum Universe", arXiv:2107.03378.

[5] Mark Girard, Martin Plávala, and Jamie Sikora, "Jordan products of quantum channels and their compatibility", Nature Communications 12, 2129 (2021).

[6] Eugenia Colafranceschi, Ludovico Lami, Gerardo Adesso, and Tommaso Tufarelli, "Refined diamond norm bounds on the emergence of objectivity of observables", Journal of Physics A Mathematical General 53 39, 395305 (2020).

[7] Rene Allerstorfer, Harry Buhrman, Florian Speelman, and Philip Verduyn Lunel, "New Protocols and Ideas for Practical Quantum Position Verification", arXiv:2106.12911.

[8] Alexandre Feller, Guillaume CÅ`uret Cauquil, and Benjamin Roussel, "Einselection from incompatible decoherence channels", Physical Review A 101 6, 062107 (2020).

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