[This post describes ideas generated in discussion with Markus Hauru, Curt von Keyserlingk, and Daniel Ranard.]
An original dream of defining branches based on redundant records (aka redundant classical information, aka GHZ-like correlations) was that it would be possible to decompose the wavefunction of an evolving non-integrable quantum system at each point in time into macroscopically distinguishable branches that individually had bounded amounts of long-range entanglement (i.e., could be efficiently expressed as a matrix product state) even though the amount of long-range entanglement for the overall state diverges in time. If one could numerically perform such a decomposition, and if the branches only “fine-grain in time”, then one could classically sample from the branches to accurately estimate local observables even if the number of branches increases exponentially in time (which we expect them to do).
However, we now think that only a fairly small fraction of all long range entanglement can be attributed to redundantly recorded branches. Thus, even if we found and efficiently handled all such classical information using a decomposition into a number of branches that was increasing exponentially in time (polynomial branch entropy), most branches would nevertheless still have an entanglement entropy across any spatial partition that grew ~linearly in time (i.e., exponentially increasing bond dimension in the MPS representation) until saturating.
In this post I’ll first write down a simple model that suggests the need to generalize the idea of branches in order to account for most long-range entanglement. Then I will give some related reasons to think that this generalized structure will take the form not of a preferred basis, but rather preferred subspaces and subsystems, and together these will combine into a preferred “branch algebra”.… [continue reading]
[This post describes ideas generated in discussion with Markus Hauru, Curt von Keyserlingk, and Daniel Ranard.]
Taylor & McCulloch have a tantalizing paper about which I’ll have much to say in the future. However, for now I want to discuss the idea of the “compatibility” of branch decompositions, which is raised in their appendix. In particular, the differences between their approach and mine prompted me to think more about how we could narrow down on what sorts of logicalThis is “logic” in the same sense of identifying sets of propositions in consistent histories that comport with the axioms of a classical probability space, before discussing any questions of physics.a axioms for branches could be identified even before we pin down a physical definition. Indeed, as I will discuss below, the desire for compatibility raises the hope that some natural axioms for branches might enable the construction of a preferred decomposition of the Hilbert space into branching subspaces, and that this might be done independently of the particular overall wavefunction. However, the axioms that I write down prove to be insufficient for this task.
Logical branch axioms
Suppose we have a binary relation “” on the vectors in a (finite-dimensional) Hilbert space that indicates that two vectors (states), when superposed, should be considered to live on distinct branches. I will adopt the convention that “” is interpreted to assert that and that the branch relation holds.This doesn’t constrain us because if we just want to assert the binary relation without asserting equality of the sum to a third vector, we write without setting it equal to anything, and if we just want addition without asserting the relation, we write .… [continue reading]
This post explains the relationship between (objective) collapse theories and theories of wavefunction branching. The formalizations are mathematically very simple, but it’s surprisingly easily to get confused about observational consequences unless laid out explicitly.
(Below I work in the non-relativistic case.)
Branching: An augmentation
In its most general form, a branching theory is some time-dependent orthogonalSome people have discussed non-orthogonal branches, but this breaks the straightforward probabilistic interpretation of the branches using the Born rule. This can be repaired, but generally only by introducing additional structure or principles that, in my experience, usually turns the theory into something more like a collapse theory, which is what I’m trying to constrast with here.a decomposition of the wavefunction: where is some time-dependent set of orthogonal vectors. I’ve expressed this in the Heisenberg picture, but the Schrödinger picture wavefunction and branches are obtained in the usual (non-branch-dependent) way by evolution with the overall unitary: and .
We generally expect the branches to fine-grain in time. That is, for any two times and , it must be possible to partition the branches at the later time into subsets of child branches, each labeled by a parent branch at the earlier time, so that each subset of children sums up to its corresponding earlier-time parent: for all where and for . By the orthogonality, a child will be a member of the subset corresponding to a parent if and only if the overlap of the child and the parent is non-zero. In other words, a branching theory fine-grains in time if the elements of and are formed by taking partitions and of the same set of orthogonal vectors, where is a refinement of , and vector-summing each subset of the respective partition.… [continue reading]
We examine the emergence of objectivity for quantum many-body systems in a setting without an environment to decohere the system’s state, but where observers can only access small fragments of the whole system. We extend the result of Reidel (2017) to the case where the system is in a mixed state, measurements are performed through POVMs, and imprints of the outcomes are imperfect. We introduce a new condition on states and measurements to recover full classicality for any number of observers. We further show that evolutions of quantum many-body systems can be expected to yield states that satisfy this condition whenever the corresponding measurement outcomes are redundant.
Ollivier does a good job of summarizing why there is an urgent need to find a way to identify objectively classical variables in a many-body system without leaning on a preferred system-environment tensor decomposition. He also concisely describes the main results of my paper in somewhat different language, so some of you may find his version nicer to read.A minor quibble: Although this is of course a matter of taste, I disagree that the Shor code example was the “core of the main result” of my paper. In my opinion, the key idea was that there was a sensible way of defining redundancy at all in a way that allowed for proving statements about compatibility without recourse to a preferred non-microscopic tensor structure. The Shor-code example is more important for showing the limits of what redundancy can tell you (which is saturated in a weak sense).… [continue reading]
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.
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.
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 seven papers: Carroll & Singh [arXiv:2005.12938]; Riedel, Zurek, & Zwolak [arXiv:1312.0331]; Weingarten [arXiv:2105.04545]; Kent [arXiv:1311.0249]; Zampeli, Pavlou, & Wallden [arXiv:2205.15893]; Ollivier [arXiv:2202.06832]; and Strasberg, Reinhard, & Schindler [arXiv:2304.10258].
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.
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.
[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 for some tensor decomposition , where is an (approximately) stable orthonormal basis independent of initial state, and where for times and , where is the initial state of and 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 ( or ) and recover the classical equations of motion; Yaffe has a pleasingly general approach to that task , 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: