Decoherence, Branching, and the Born Rule in a Mixed-State Everettian Multiverse

Table of Links

Abstract & Introduction

Decoherence and Branching

The Born Rule

Discussion

Conclusion and References

Abstract

In Everettian quantum mechanics, justifications for the Born rule appeal to selflocating uncertainty or decision theory. Such justifications have focused exclusively on a pure-state Evere‹ian multiverse, represented by a wave function. Recent works in quantum foundations suggest that it is viable to consider a mixedstate Everettian multiverse, represented by a (mixed-state) density matrix. Here, we develop the conceptual foundations for decoherence and branching in a mixedstate multiverse, and extend the standard Everettian justifications 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.

1 Introduction

Everettian quantum mechanics (EQM) is a minimalist interpretation of quantum mechanics with some counter-intuitive features (Barrett 2023; Vaidman 2021). Instead of attempting to collapse the quantum state or adding extra variables to obtain a definite outcome for each experiment, it proposes to take unitary quantum mechanics as fundamental and replace our single-world ontology with a multiverse, where every possible outcome of an experiment is realized in some branch (a parallel world). Hence it is also sometimes called the ‘many-worlds’ interpretation.

There are two main issues with EQM, one metaphysical and the other epistemological. The metaphysical issue concerns the ontology of EQM. How do we obtain the appearance of a classical world, with definite records and observers, from the quantum state? A much discussed solution appeals to decoherence, with its ability to suppress interference and give rise to an “emergent multiverse” (Wallace 2012). The universal quantum state evolves into one with many branches, each representing an emergent (quasi-)classical world.

The epistemological issue concerns the understanding of probability in EQM. A key postulate of quantum mechanics, and a crucial element of its empirical confirmation, is the Born rule: the probability of observing a certain outcome is given by the squared amplitude of the quantum state. How should we make sense of this probability when every measurement outcome occurs on some branch of the Everettian multiverse, and what justifies the interpretation of the squared amplitutdes as probabilities? There are several responses to the probability issue. The Deutsch-Wallace program understands probability in terms of the betting preferences of agents within the multiverse, which uses a decision-theoretic representation theorem to prove that the agent’s credences must satisfy the Born rule, on pain of irrationality (e.g. Deutsch 1999, Wallace 2012). The Sebens-Carroll (2018) and McQueen-Vaidman (2018) programs understand probability in terms of self-locating uncertainty of a localized agent on some branch, employing certain epistemic principles – such as “separability” or “symmetry” – to prove that the agent’s self-locating uncertainty must satisfy the Born rule.

Promising as they may be, these defenses and justifications of EQM have an apparent limitation. They focus exclusively on the case of a universal pure state, where the quantum state of the multiverse is represented by a wave function. Defenders of EQM, like many other realist interpreters, regard the universal pure state as representing something objective and mind-independent. However, recent works in quantum foundations (Allori et al. 2013; Chen 2021; Durr et al. 2005; Maroney 2005; Robertson ¨ 2022; Wallace 2012) suggest that the above approach to realism, based on the wave function, is not the only possibility for realism about the quantum state. It’s also viable – and in some circumstances even more theoretically attractive – to take a realist stance based on the density matrix (Chen 2021). On this view, we can associate (possibly mixed-state) density matrices, rather than (necessarily pure-state) wave functions, to isolated systems and even to the entire universe. While density matrices are conventionally used to represent ignorance about some underlying wave function or the external environment, it’s also possible to regard density matrices as fundamental. On the new picture, the universe as a whole can be aptly represented by a fundamental density matrix evolving unitarily according to the von Neumann equation. In contrast, on the standard picture, it is represented as a wave function evolving unitarily according to the Schrodinger equation. If the fundamental density matrix in this new realist picture is mathematically the same as that of the “ignorance” density matrix in the standard picture, the two theories will be empirically equivalent, since they make the same statistical predictions for all experiments.

All wave functions correspond to some pure-state density matrices, but not all density matrices have corresponding wave functions. ‘us, realism based on the density matrix allows for more quantum states than realism based on the wave function. The former is also compatible with a theoretically attractive package – the Wentaculus – which provides a unified explanation for quantum phenomena and the thermodynamic arrow of time (Chen 2020, Chen 2021, Chen 2022a, Chen 2022b). Following Chen (2021, 2019), we call this new picture Density Matrix Realism (DMR) and the old one Wave Function Realism (WFR). We denote the Everettian versions of DMR and WFR as DMRE and WFRE respectively. (Note that this is a wider conception of quantum state realism than that of Albert (1996) and Ney (2021).)

This project has several conceptual payoffs. First, it requires us to clarify the ontological structure of the multiverse and the requirements of decoherence. As it turns out, branching requires decoherence but decoherence does not require a universal pure state. The story of decoherence applies both to pure and mixed states, which has been underappreciated in the literature.

Second, with the access to a larger state space, Everettians can explore new theoretical possibilities that are naturally suggested by DMR. For example, DMRE provides the basis for a unified account of probability that may be absent on WFRE. On WFRE, without knowing what the universal wave function is, we may assign a density matrix ρ to represent our epistemic state. The probabilities we extract from ρ range over various possible candidate multiverses. As such, it is not interpreted as self-locating uncertainty or betting preferences of agents within a multiverse, and must be treated as a distinct source of probability (e.g. statistical mechanical / classical probability of possible initial conditions). In contrast, DMRE allows us to regard ρ as representing the actual fundamental quantum state of the multiverse. We have the option to posit just one source of probability, corresponding to the weights associated with branches of the actual mixed-state multiverse