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arXiv:2307.13218 · Synthese

Decoherence, branching, and the Born rule in a mixed-state Everettian multiverse

Eugene Y. S. Chua, Eddy Keming Chen

Published Apr 4, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract In Everettian quantum mechanics, justifications for the Born rule appeal to self-locating uncertainty or decision theory. Such justifications have focused exclusively on a pure-state Everettian multiverse, represented by a wavefunction. Recent works in quantum foundations suggest that it is viable to consider a mixed-state Everettian multiverse, represented by a (mixed-state) density matrix. Here, we discuss the conceptual foundations for decoherence and branching in a mixed-state multiverse, and extend arguments for the Born rule to this setting. This extended framework provides a unification of ‘classical’ and ‘quantum’ probabilities, and additional theoretical benefits, for the Everettian picture.

DOI
10.1007/s11229-025-04949-0

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Abstract In Everettian quantum mechanics, justifications for the Born rule appeal to self-locating uncertainty or decision theory. Such justifications have focused exclusively on a pure-state Everettian multiverse, represented by a wavefunction. Recent works in quantum foundations suggest that it is viable to consider a mixed-state Everettian multiverse, represented by a (mixed-state) density matrix. Here, we discuss the conceptual foundations for decoherence and branching in a mixed-state multiverse, and extend arguments for the Born rule to this setting. This extended framework provides a unification of ‘classical’ and ‘quantum’ probabilities, and additional theoretical benefits, for the Everettian picture.
A longstanding issue in attempts to understand the Everett (many-worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period, it is tempting to regard each branch as equiprobable, but we argue that the temptation should be resisted. Applying lessons from this analysis, we demonstrate (using methods similar to those of Zurek’s envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In doing so, we rely on a single key principle: changes to the environment alone do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers. 1. Introduction2. Preliminaries: Many-Worlds, Self-locating Uncertainty, and Branch-counting 2.1. The many-worlds interpretation2.2. Self-locating uncertainty and the Everettian multiverse2.3. Indifference and the quantitative probability problem2.4. Against branch-counting3. The Epistemic Irrelevance of the Environment 3.1. The epistemic separability principle3.2. Deriving the Born rule4. Varieties of Uncertainty 4.1. Epistemic separability principle and indifference4.2. Mixed uncertainties4.3. Large universe cosmology and the quantum multiverse5. Probability in Practice 5.1. Betting and branching5.2. Theory confirmation6. Comparison to Other Approaches 6.1. Zurek’s envariance-based derivation of the Born rule6.2. The decision-theoretic programme7. ConclusionAppendix
Vaidman's self-locating uncertainty (SLU) represents an innovative and influential approach to deriving Born rule probabilities in the Many-Worlds Interpretation (MWI) of quantum mechanics, offering a promising epistemic foundation for understanding probabilities in a deterministic multiverse. This paper examines a potential inconsistency in the SLU framework: while the concept relies on observer branching to generate multiple copies and thus enable uncertainty about branch location, such branching results in pure, deterministic local states for each copy, making the mixed reduced density matrix (RDM)—a key element in deriving amplitude-squared probabilities—unavailable. Conversely, if the mixed RDM is retained, the observer remains unentangled and spans all branches, eliminating the multiplicity essential to SLU. The derivations by McQueen and Vaidman appear to draw on both SLU (which presupposes branching) and the mixed RDM (which presupposes no branching), without fully addressing this tension. Our analysis suggests that the inconsistency poses a significant challenge to grounding the Born rule purely in SLU, independent of specific branching models. The results highlight unresolved questions in the foundations of Everettian probability and point toward the need for further refinement or supplementation of the SLU approach.
Quantum tunneling is analyzed within the Everettian interpretation, where decoherence generates distinct reflected and transmitted branches. The tunneling probability is obtained from relative branch weights via the Born rule without invoking collapse. Tunneling time τTτ_T τT​ is reformulated as the branching duration τT∼NBτBτ_T \sim N_B τ_BτT​∼NB​τB​, where NB\ N_B  NB​ is the number of environmental degrees of freedom and τBτ_BτB​ is the decoherence time required to distinguish branches. Macroscopic tunneling in Josephson junctions is also analyzed.
The Everett interpretation of quantum mechanics faces a well‑known challenge: how can probabilities emerge in a deterministic theory where all measurement outcomes occur? A prominent response appeals to self‑locating uncertainty (SLU): after branching, copies of an observer are uncertain about which branch they inhabit during the post-measurement, pre-observation period, and this uncertainty is then quantified to yield the Born rule. This paper examines whether the notion of SLU can be coherently instantiated in the Everettian framework. It argues that the two conditions required for SLU—having a branch‑relative identity (Recording Condition) and being uncertain about that identity (Uncertainty Condition)—are difficult to satisfy simultaneously in any physical system. Any attempt to realize SLU appears to force a decomposition of the observer into a memory module that records the outcome (and thus has branch‑relative identity) and a decision module that is uncertain (and thus lacks branch‑relative identity). Neither module individually possesses SLU, and it is not obvious that they can be combined into a unified epistemic agent that does. The analysis complements and deepens existing critiques and suggests that SLU‑based derivations of the Born rule may rest on a problematic foundation.
The measurement problem remains one of the most profound puzzles in the foundation of quantum mechanics. It contains two distinct aspects: the collapse of the wavefunction and the origin of probabilities. While Everettian Quantum Mechanics (EQM) and the Many Worlds Interpretation (MWI) work around the collapse aspect, by positing that the universe branches through pure unitary evolution, it leaves unresolved the question of why measurement outcomes occur with frequencies given by the Born Rule. This paper explores a recent derivation of the Born Rule by Carroll and Sebens based on principles of Self-Locating Uncertainty (SLU) and the Epistemic Separability Principle (ESP). By examining this framework in detail, beginning with an equal amplitude case and extending to an unequal amplitude case, we show how these principles account for the probabilities given by the Born Rule, hence showing that SLU and the ESP, within the scope of EQM, offer a solution to the measurement problem. We further show the significance and implications of deriving the Born Rule and highlight the relevance of the measurement problem to more recent developments like quantum gravity.
Abstract This introductory chapter summarizes the principal positive theses of the book (Section 1), namely, that the unitary evolution of the quantum state, for sufficiently large systems of particles in thermal environments, yields a multiplicity of branching structures (worlds) obeying approximately classical equations, a result that follows from decoherence theory with no special assumptions or additions to the quantum formalism; and that further, the branching of worlds has a natural interpretation in terms of probability, as quantified by the Born rule in terms of ratios of squared norms of branch amplitudes. In particular, an agent who knows the branching structures of the quantum state that will result from his actions is rationally compelled to order his preferences among actions by the expected utilities of those branching structures as defined by the Born rule. He is, moreover, entitled to view a branching process as involving uncertainty as to which branch he will find himself. But independent of the truth of these claims, the theory is still testable: an agent who does not know the branching structure of the state, and who is indeed undecided between a many-worlds and a one-world theory, is rationally bound to update her credences in such theories by conditionalisation on observed statistical evidence, just as she is bound to do in the case of conventional quantum mechanics. Section 2 summarizes some of the arguments and counter-arguments for these claims as developed in subsequent chapters. Section 3 provides some more mathematical and historical background to the Everett interpretation, including a brief introduction to the decoherent histories formalism.