Weingarten’s branches from quantum complexity

Don Weingarten’s newI previously blogged about earlier work by Weingarten on a related topic. This new paper directly addresses my previous concerns.a   attack [2105.04545] on the problem of defining wavefunction branches is the most important paper on this topic in several years — and hence, by my strange twisted lights, one of the most important recent papers in physics. Ultimately I think there are significant barriers to the success of this approach, but these may be surmountable. Regardless, the paper makes tons of progress in understanding the advantages and drawbacks of a definition of branches based on quantum complexity.

Here’s the abstract:

Beginning with the Everett-DeWitt many-worlds interpretation of quantum mechanics, there have been a series of proposals for how the state vector of a quantum system might split at any instant into orthogonal branches, each of which exhibits approximately classical behavior. Here we propose a decomposition of a state vector into branches by finding the minimum of a measure of the mean squared quantum complexity of the branches in the branch decomposition. In a non-relativistic formulation of this proposal, branching occurs repeatedly over time, with each branch splitting successively into further sub-branches among which the branch followed by the real world is chosen randomly according to the Born rule. In a Lorentz covariant version, the real world is a single random draw from the set of branches at asymptotically late time, restored to finite time by sequentially retracing the set of branching events implied by the late time choice. The complexity measure depends on a parameter b with units of volume which sets the boundary between quantum and classical behavior.
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Living bibliography for the problem of defining wavefunction branches

[Last updated: Nov 27, 2021.]

This post is (a seed of) a bibliography covering the primordial research area that goes by some of the following names:

Although the way this problem tends to be formalized varies with context, I don’t think we have confidence in any of the formalizations. The different versions are very tightly related, so that a solution in one context is likely give, or at least strongly point toward, solutions for the others.

As a time-saving device, I will mostly just quote a few paragraphs from existing papers that review the literature, along with the relevant part of their list of references. Currently I am drawing on four papers: Carroll & Singh [arXiv:2005.12938]; Riedel, Zurek, & Zwolak [arXiv:1312.0331]; Weingarten [arXiv:2105.04545]; and Kent [arXiv:1311.0249].

I hope to update this from time to time, and perhaps turn it into a proper review article of its own one day. If you have a recommendation for this bibliography (either a single citation, or a paper I should quote), please do let me know.

Carroll & Singh

From “Quantum Mereology: Factorizing Hilbert Space into Subsystems with Quasi-Classical Dynamics” [arXiv:2005.12938]:

While this question has not frequently been addressed in the literature on quantum foundations and emergence of classicality, a few works have highlighted its importance and made attempts to understand it better.

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Branches as hidden nodes in a neural net

I had been vaguely aware that there was an important connection between tensor network representations of quantum many-body states (e.g., matrix product states) and artificial neural nets, but it didn’t really click together until I saw Roger Melko’s nice talk on Friday about his recent paper with Torlai et al.:There is a title card about “resurgence” from Francesco Di Renzo’s talk at the beginning of the talk you can ignore. This is just a mistake in KITP’s video system.a  

[Download MP4]   [Other options]

In particular, he sketched the essential equivalence between matrix product states (MPS) and restricted Boltzmann machinesThis is discussed in detail by Chen et al. See also good intuition and a helpful physicist-statistician dictionary from Lin and Tegmark.b   (RBM) before showing how he and collaborators could train an efficient RBM representations of the states of the transverse-field Ising and XXZ models with a small number of local measurements from the true state.

As you’ve heard me belabor ad nauseum, I think identifying and defining branches is the key outstanding task inhibiting progress in resolving the measurement problem. I had already been thinking of branches as a sort of “global” tensor in an MPS, i.e., there would be a single index (bond) that would label the branches and serve to efficiently encode a pure state with long-range entanglement due to the amplification that defines a physical measurement process. (More generally, you can imagine branching events with effects that haven’t propagated outside of some region, such as the light-cone or Lieb-Robinson bound, and you might even make a hand-wavy connection to entanglement renormalization.)… [continue reading]

Models of decoherence and branching

[This is akin to a living review, which will hopefully improve from time to time. Last edited 2020-4-8.]

This post will collect some models of decoherence and branching. We don’t have a rigorous definition of branches yet but I crudely define models of branching to be models of decoherenceI take decoherence to mean a model with dynamics taking the form U \approx \sum_i \vert S_i\rangle\langle S_i |\otimes U^{\mathcal{E}}_i for some tensor decomposition \mathcal{H} = \mathcal{S} \otimes \mathcal{E}, where \{\vert S_i\rangle\} is an (approximately) stable orthonormal basis independent of initial state, and where \mathrm{Tr}[ U^{\mathcal{E}}_i \rho^{\mathcal{E} \dagger}_0 U^{\mathcal{E}}_j ] \approx 0 for times t \gtrsim t_D and i \neq j, where \rho^{\mathcal{E}}_0 is the initial state of \mathcal{E} and t_D is some characteristic time scale.a   which additionally feature some combination of amplification, irreversibility, redundant records, and/or outcomes with an intuitive macroscopic interpretation.

(Note in particular that I am not just listing models for which you can mathematically take a classical limit (\hbar\to 0 or N\to\infty) and recover the classical equations of motion; Yaffe has a pleasingly general approach to that task [1], but I’ve previously sketched why that’s an incomplete explanation for classicality.)

I have the following desiderata for models, which tend to be in tension with computational tractability:

  • physically realistic
  • symmetric (e.g., translationally)
  • no ad-hoc system-environment distinction
  • Ehrenfest evolution along classical phase-space trajectories (at least on Lyapunov timescales)

Regarding that last one: we would like to recover “classical behavior” in the sense of classical Hamiltonian flow, which (presumably) means continuous degrees of freedom.… [continue reading]

Comments on Weingarten’s preferred branch

[Added 2022-March-13: Weingarten has a new paper, discussed by me here, that mostly supercedes the content of this post.]

A senior colleague asked me for thoughts on this paper describing a single-preferred-branch flavor of quantum mechanics, and I thought I’d copy them here. Tl;dr: I did not find an important new idea in it, but this paper nicely illustrates the appeal of Finkelstein’s partial-trace decoherence and the ambiguity inherent in connecting a many-worlds wavefunction to our direct observations.

We propose a method for finding an initial state vector which by ordinary Hamiltonian time evolution follows a single branch of many-worlds quantum mechanics. The resulting deterministic system appears to exhibit random behavior as a result of the successive emergence over time of information present in the initial state but not previously observed.

We start by assuming that a precise wavefunction branch structure has been specified. The idea, basically, is to randomly draw a branch at late times according to the Born probability, then to evolve it backwards in time to the beginning of the universe and take that as your initial condition. The main motivating observation is that, if we assume that all branch splittings are defined by a projective decomposition of some subsystem (‘the system’) which is recorded faithfully elsewhere (‘the environment’), then the lone preferred branch — time-evolving by itself — is an eigenstate of each of the projectors defining the splits. In a sense, Weingarten lays claim to ordered consistency [arxiv:gr-qc/9607073] by assuming partial-trace decoherenceNote on terminology: What Finkelstein called “partial-trace decoherence” is really a specialized form of consistency (i.e.,[continue reading]

Toward relativistic branches of the wavefunction

I prepared the following extended abstract for the Spacetime and Information Workshop as part of my continuing mission to corrupt physicists while they are still young and impressionable. I reproduce it here for your reading pleasure.

Finding a precise definition of branches in the wavefunction of closed many-body systems is crucial to conceptual clarity in the foundations of quantum mechanics. Toward this goal, we propose amplification, which can be quantified, as the key feature characterizing anthropocentric measurement; this immediately and naturally extends to non-anthropocentric amplification, such as the ubiquitous case of classically chaotic degrees of freedom decohering. Amplification can be formalized as the production of redundant records distributed over spatial disjoint regions, a certain form of multi-partite entanglement in the pure quantum state of a large closed system. If this definition can be made rigorous and shown to be unique, it is then possible to ask many compelling questions about how branches form and evolve.

A recent result shows that branch decompositions are highly constrained just by this requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. However, this result is grounded in a preferred tensor decomposition into independent microscopic subsystems associated with spatial locality. This structure breaks down in a relativistic setting on scales smaller than the Compton wavelength of the relevant field. Indeed, a key insight from algebraic quantum field theory is that finite-energy states are never exact eigenstates of local operators, and hence never have exact records that are spatially disjoint, although they can approximate this arbitrarily well on large scales.… [continue reading]

Branches and matrix-product states

I’m happy to use this bully pulpit to advertise that the following paper has been deemed “probably not terrible”, i.e., published.

When the wave function of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops “branches”: a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structure associated with spatial locality, I show that branch decompositions are highly constrained just by the requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. Speculatively, objective branch decompositions may speed up numerical simulations of nonstationary many-body states, illuminate the thermalization of closed systems, and demote measurement from fundamental primitive in the quantum formalism.

Here’s the figureThe editor tried to convince me that this figure appeared on the cover for purely aesthetic reasons and this does not mean my letter is the best thing in the issue…but I know better!a   and caption:

Spatially disjoint regions with the same coloring (e.g., the solid blue regions \mathcal{F}, \mathcal{F}', \ldots) denote different records for the same observable (e.g., \Omega_a = \{\Omega_a^{\mathcal{F}},\Omega_a^{\mathcal{F}'},\ldots\}).
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How fast do macroscopic wavefunctions branch?

Over at PhysicsOverflow, Daniel Ranard asked a question that’s near and dear to my heart:

How deterministic are large open quantum systems (e.g. with humans)?

Consider some large system modeled as an open quantum system — say, a person in a room, where the walls of the room interact in a boring way with some environment. Begin with a pure initial state describing some comprehensible configuration. (Maybe the person is sitting down.) Generically, the system will be in a highly mixed state after some time. Both normal human experience and the study of decoherence suggest that this state will be a mixture of orthogonal pure states that describe classical-like configurations. Call these configurations branches.

How much does a pure state of the system branch over human time scales? There will soon be many (many) orthogonal branches with distinct microscopic details. But to what extent will probabilities be spread over macroscopically (and noticeably) different branches?

I answered the question over there as best I could. Below, I’ll reproduce my answer and indulge in slightly more detail and speculation.

This question is central to my research interests, in the sense that completing that research would necessarily let me give a precise, unambiguous answer. So I can only give an imprecise, hand-wavy one. I’ll write down the punchline, then work backwards.


The instantaneous rate of branching, as measured in entropy/time (e.g., bits/s), is given by the sum of all positive Lyapunov exponents for all non-thermalized degrees of freedom.

Most of the vagueness in this claim comes from defining/identifying degree of freedom that have thermalized, and dealing with cases of partial/incomplete thermalization; these problems exists classically.


The original question postulates that the macroscopic system starts in a quantum state corresponding to some comprehensible classical configuration, i.e.,… [continue reading]