Wavefunction branching: when you can't tell pure states from mixed states

Jordan K. Taylor; Ian P. McCulloch

doi:10.22331/q-2025-03-25-1670

Wavefunction branching: when you can’t tell pure states from mixed states

Jordan K. Taylor¹ and Ian P. McCulloch^1,2,3

¹School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia
²Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
³Frontier Center for Theory and Computation, National Tsing Hua University, Hsinchu 30013, Taiwan

Published:	2025-03-25, volume 9, page 1670
Editor:	Eric Cavalcanti
Eprint:	arXiv:2308.04494v4
Doi:	https://doi.org/10.22331/q-2025-03-25-1670
Citation:	Quantum 9, 1670 (2025).

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Abstract

We propose a definition of wavefunction "branchings": quantum superpositions which can't be feasibly distinguished from the corresponding mixed state, even under time evolution. Our definition is largely independent of interpretations, requiring only that it takes many more local gates to swap branches than to distinguish them. We give several examples of states admitting such branch decompositions. Under our definition, we argue that attempts to get relative-phase information between branches will fail without frequent active error correction, that branches are effectively the opposite of good error-correcting codes, that branches effectively only grow further apart in time under natural evolution, that branches tend to absorb spatial entanglement, that branching is stronger in the presence of conserved quantities, and that branching implies effective irreversibility. Identifying these branch decompositions in many-body quantum states could shed light on the emergence of classicality, provide a metric for experimental tests at the quantum/ classical boundary, and allow for longer numerical time evolution simulations. We see this work as a generalization of the basic ideas of environmentally-induced decoherence to situations with no clear system/ environment split.

Featured image: If interference complexity is high but distinguishability complexity is low between two terms in a superposition, then relative-phase information (required to tell quantum from classical) will be extremely hard to access without being accidentally disrupted.

Popular summary

TLDR: How does the classical world we observe emerge from the strange, counterintuitive rules of quantum mechanics? We propose a new criterion for when quantum states can be treated as effectively classical "branches," based on the computational difficulty of swapping states compared to distinguishing them. Unlike traditional approaches, our method doesn’t require dividing the system into "subsystem" and "environment." Instead, it provides a general framework that connects to quantum error correction, symmetries, and the growth of quantum complexity, offering new insights into the quantum-to-classical transition.

———

Quantum mechanics is incredibly successful at describing the microscopic world, but it often clashes with our everyday experience of a classical, deterministic reality. For example, Schrödinger’s famous thought experiment suggests that a cat could exist in a superposition of being both alive and dead — a scenario that seems absurd in our macroscopic world. So why don’t we observe such quantum weirdness in our daily lives?

The standard explanation is decoherence: when you focus on a small part of a quantum system and ignore the rest, the system appears to behave classically. For instance, a cat in a superposition of alive and dead states becomes indistinguishable from a classical probability distribution of the cat being either alive or dead, unless you meticulously track the relative-phase information that has spread into the environment. However, decoherence relies on splitting the quantum state into a "subsystem" (e.g., the cat) and an "environment" (e.g., the surrounding air molecules). In many cases, especially in closed quantum systems, it’s unclear how to make this split — or if it even exists.

Our work takes a different approach. Instead of assuming a subsystem/environment split, we directly identify quantum superpositions that can be treated as classical probability distributions. We introduce the concept of "good branches": if a quantum state can be written as a superposition of terms (e.g., "alive cat" + "dead cat"), and it takes far more computational effort to swap these terms than to distinguish them, then the superposition can be replaced with a classical probability distribution. For example, it’s much harder to bring a dead cat back to life (swapping states) than to determine whether a cat is alive or dead (distinguishing states). This makes "alive cat" and "dead cat" good branches. The relative-phase information between these branches — which is essential for detecting quantum superpositions — is extremely hard to access without being accidentally disrupted.

We argue that such branches are not only effectively classical but also irreversible over exponentially long times, meaning they don’t recombine easily. Additionally, branches tend to absorb entanglement over time, are strengthened by symmetries (like energy conservation), and are fundamentally different from quantum error-correcting codes, which protect quantum information rather than allowing it to decohere.

We conjecture that this process of branch formation is ubiquitous in nature, playing a key role in explaining phenomena like quantum thermalization. Our framework also opens up new avenues for testing the foundations of quantum mechanics, including the many-worlds interpretation, by providing a concrete way to quantify when and how quantum superpositions become effectively classical. This could lead to new theoretical insights, numerical simulations, and even experimental tests probing the boundary between quantum and classical behavior.

► BibTeX data

@article{Taylor2025wavefunction,
  doi = {10.22331/q-2025-03-25-1670},
  url = {https://doi.org/10.22331/q-2025-03-25-1670},
  title = {Wavefunction branching: when you can't tell pure states from mixed states},
  author = {Taylor, Jordan K. and McCulloch, Ian P.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1670},
  month = mar,
  year = {2025}
}

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[3] Robert Kudelić, "Quantum Monte Carlo randomized algorithm for the Feedback Arc Set problem", Array 31, 101011 (2026).

[4] Jay Lawrence, "The original Wigner's friend scenarios", American Journal of Physics 94 4, 298 (2026).

[5] William J. Zeng, Farrokh Labib, and Vincent Russo, "Towards violations of Local Friendliness with quantum computers", Quantum 9, 1851 (2025).

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[7] C. Jess Riedel, "Wavefunction branches demand a definition!", arXiv:2506.15663, (2025).

[8] Don Weingarten, "Vacuum Branching, Dark Energy, Dark Matter", Foundations of Physics 55 4, 55 (2025).

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