[LW Meta: This post is (a) about how wavefunction branching explains measurements and why chaotic dynamics don't cause the quantum state to become non-classical-looking (b) the mathematical problem of formally defining wavefunction branches. The former will be familiar to people who've read Eliezer's quantum sequence, but the latter is new to LW to my knowledge!]

2.5.4.1 In 2.5.2 I left open two questions about quantum mechanics and seemingly-classical reality. Namely, (b) how does the world stay in a relatively-classical state, and (c) what happens when we take a measurement? It turns out the answer to both questions is the same, and it involves the information-proliferating dynamics discussed in section 2.4.

2.5.4.2 Let's examine the Stern-Gerlach experiment in more detail. In this experiment a stream of neutral silver atoms is propelled through a non-uniform magnetic field and strikes a screen. The interaction of the magnetic field with the spin is described by the Hamiltonian

where is the magnetic moment, is the spin angular momentum, is the signed gyromagnetic ratio, and is the magnetic field at the particle's position. The field gradient exerts opposite forces on the spin-up and spin-down components of their wavefunction, producing two "lines" in the surface where the atoms strike.

2.5.4.3 So this is a "measurement event", or more precisely many measurement events. Let's consider a variant in which we only let one atom through the magnetic field. And further, let's imagine that its spin is in the state (this could be arranged by allowing a very sparse stream of atoms through another Stern-Gerlach magnet measuring a different axis for example). In this case we won't see two lines, but rather(using a microscope perhaps) the atom being in one location or the other(with probability in each case) -- a "measurement event".

2.5.4.4 What exactly is happening to the quantum state in this case? Let's forget about measurement events and just track the unitary dynamics.

2.5.4.4.1 Initially we have a silver atom, with a relatively spatially localized wavepacket, with amplitude spread evenly over the discrete states and .

2.5.4.4.2 As it moves forward through the field, the and components of the wavepacket are differentially accelerated by the magnetic field, so the atom's state becomes more like

with the two components becoming spatially separated.

2.5.4.4.3 Now the components of the atom's wavefunction strike the surface. They begin interacting with the wavefunctions of the atoms in the surface and with the light reflecting off it. Now, since interactions happen in the full Hilbert space -- aka complex functions over the whole configuration space -- the overall state evolves to something like

(a phonon is a disturbance in the atomic lattice of the surface).

2.5.4.4.4 As time evolution continues, more photons and phonons will scatter off the particle, and those that have scattered will interact with other things. A human experimenter may examine the position of the atom(with a microscope maybe) and their brain will become entangled with the atom's state, producing a state

2.5.4.4.5 As discussed in the last section, this ultimately will lead to entanglements forming with photons radiating into space, making the entanglement effectively irreversible.

2.5.4.5 So the upshot of the above is that the state of the particle wavefunction(spin up or spin down) gets copied into many places, causing the overall wavefunction to evolve into 2 distinct "blobs" in high-dimensional space,

2.5.4.5.1 The significance of this comes from the way that unitary evolution works. The unitarities that govern our universe have the property of locality -- i.e. for a given time interval, the projection of the evolved state vector to a given location depends only on the state in a bounded neighbourhood of the location earlier in time. In our universe, influences cannot travel faster than light speed.

2.5.4.5.2 This means that for the purposes of time evolution, each "blob" can be considered separately from the other "blob". That is, say we are evolving some system initially in the state , and evolve only the first component, getting a state . The point is, the states remain "separate" no matter what evolution is applied to the first state -- all of them must be aligned, which becomes harder and harder to arrange as more copies are made and become scrambled in innumerable environmental degrees of freedom.

2.5.4.5.2.1*(Technical) The exact meaning of this is somewhat technical. If we assume and are initially orthogonal, then any unitary evolution applied to both will have them remain orthogonal. Thus we need to consider somewhat finer-grained notions of similarity.

2.5.4.5.2.2* If we examine a subsystem, it could be the case that there is interference between the two states, even if they remain orthogonal overall. For instance, if we have two orthogonal components of an electron wavefunction it's possible for us to interfere them by projecting them onto the same screen, producing an "oscillating wave" of amplitude(as opposed to the Gaussian we would get if there was no interference).

2.5.4.5.2.3* The two components remain orthogonal, but when projected onto particular positions, they can destructively interfere. An abstraction of this is local observables -- is there any local transformation and projection we can make such that the states have a non-zero inner product? Mathematically, this could be the case if there is some operator supported only on the region in question such that . And this is just what is prevented by the records in the other locations [1] .

2.5.4.5.3 A literal example of information-copying destroying local interference can be found in modified double slit experiments. The classic double-slit experiment illustrates the wave-like nature of particles by propelling a particle(e.g. an electron or photon) through one of two slits in a wall(or equivalent, e.g. a specially tuned electromagnetic field), forward to an area which measures them(e.g. another wall which they impact), producing a characteristic interference pattern. If a system which reacts to the presence of the particles is placed on one or both slits, the interference pattern is destroyed, as the information about which path the particle took is copied(to e.g. environmental photons), preventing local interference at the detector.

2.5.4.6 The components that a wavefunction splits into when quantum-scale information is copied are sometimes referred to as branches . This phenomenon of quantum coherence being lost due to the environment copying information is also known as decoherence.

2.5.4.7 So this is what is occurring when a "measurement" takes place. In fact, the very same process explains the other mystery alluded to above -- how the world stays in a relatively classical-looking corner of Hilbert space despite this corner being relatively small. The reason this occurs is that when superpositions become too spread out, their information is recorded by other degrees of freedom. Thus classically chaotic systems such as turbulent fluids generate wavefunction branches in much the same way that a measuring device does.

2.5.5.1 So wavefunction branching explains much of how classical reality and measurement emerge from quantum mechanics. However, actually formally defining wavefunction branches in full generality turns out to be a tricky problem! I'll review various proposals before presenting my preferred account, due to Adrian Kent.

2.5.5.2 One indication of the difficulties here comes from the preferred basis problem from the decoherence literature.

2.5.5.2.1 Say we are analyzing an isolated quantum system , initially in a superposition .

2.5.5.2.2 Now it interacts with an "environment" system , which records the state of . The overall state is now . These types of states, with perfect correlations between two systems, are known as "Bell states"(due to their role in the violation of Bell's inequality).

2.5.5.2.3 Seemingly, the system has been measured in the , basis. However! We can rewrite the state of the system in the basis

and the state of the environment in the basis

In this basis, the system can be written in the form . It appears that the environment has a "record" of the system's state in this basis too!

2.5.5.2.4 So what state was the system measured in? The correlation with the environment, on its own, is not enough to tell. This is the preferred basis problem .

2.5.5.2.5 Therefore, additional principles need to be introduced. In the decoherence literature, where the preferred basis problem was first raised, one solution is called "einselection": the idea is that while many bases may be valid at an instant , most of them will not remain diagonal under further interaction between the system and environment. The basis that does remain diagonal is called the pointer basis of the interaction.

2.5.5.3 Einselection works well enough for analyzing a single measurement of a particular system. However, our goal is to analyze the branching of the universe as a whole, not just a single system. Ideally we would proceed without relying on a privileged system/environment split.

2.5.5.4 One possible way forward is to note that while a Bell state across two systems may maintain its diagonal shape in many different bases, this is not the case for Bell-like states across three or more systems, e.g. . For such systems, the "diagonalizing" basis is unique . Measurement events in the real world spread information to many places, so multipartite information copying is to be expected. The different subsystems storing information about a branch(i.e. such that the branches are orthogonal when projected onto the subsystem) are referred to as "records" of the branch.

2.5.5.5 However, this too has problems. In particular, it relies on a privileged partitioning of the environment into a set of distinct systems: different partitions may give different branch decompositions.

2.5.5.5.2 The same paper (by Jess Riedel) analyzes the conditions in which different sets of records can be combined into a joint branch decomposition. A criterion is given in terms of a preferred length scale : sets of records are compatible if each is recorded on at least 3 regions of radius , each separated from the others by a distance of . This will typically hold for records in reality due to information-amplifying dynamics, but the arbitrariness of the length scale is somewhat unsatisfying(as the paper acknowledges). [2]

2.5.5.6 Another class of proposals involves quantum circuit complexity . Quantum circuit complexity is defined analogously to classical circuit complexity: we have an input set of qubits which we transform using a standard set of quantum gates. The quantum complexity of a state is defined by the number of gates needed to produce it starting from a standard starting state(such as ). The complexity of a given unitary transformation is defined likewise.

2.5.5.6.1 The idea here is that complexity provides a good way of tracking the ways that branches chaotically become more distinct from each other -- and hence ever more difficult to interfere -- over time.

2.5.5.6.2 Weingarten defines a complexity measure on a decomposition of a wavefunction into orthogonal components. This complexity measure is defined as the sum of the Shannon entropy of the probability distribution over branches plus (a constant multiplied by) the expected squared complexity of each branch. The idea is that once a "branching event" has occurred, the complexity of preparing the superposition over the branches will increase rapidly such that it becomes favorable under the complexity measure to split into sub-branches.

2.5.5.6.3 Another complexity-based branching definition is due to Taylor and McCulloch . They instead consider the complexity of distinguishing and interfering pairs of branches. A branch decomposition is considered good if the complexity of interfering each pair of branches is larger(by some large amount) than the complexity of distinguishing between them.

2.5.5.6.4 These definitions definitely have an appeal in terms of producing, insofar as it's possible, an "objective" measure of the difficulty of re-combining the branches. There are some drawbacks, however. Both proposals need an arbitrary constant defining when the threshold for branching has been crossed. [3] Quantum circuit complexity is a theoretically intractable quantity so it seems it will be very difficult to prove anything rigorously about branching in these models. And their complexity measures don't take into account "practical" difficulties with re-combination for actual observers, in particular the difficulty of collecting information that has been carried away by photons.