Tag: branching

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Comments on Weingarten’s preferred branch

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[Added 2022-March-13: Weingarten has a new paper, discussed by me here, that mostly supercedes the content of this post. In the new approach, the preferred branch decomposition is to be generated using a modification on Nielsen’s measure of quantum circuit complexity.]

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]

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Toward relativistic branches of the wavefunction

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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]

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Branches and matrix-product states

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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\}).
… [continue reading]
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How fast do macroscopic wavefunctions branch?

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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.

Punchline

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.

Elaboration

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