Theory: many worlds, tree and convergences

The many-worlds simulation compares three hypotheses: the "Tree", "Classical convergence" and "Quantum convergence" regimes. This document describes them with the notation of js/model.js and, for each one, separates published theory from the simulator's working hypotheses. References in square brackets point to the bibliography at the end. Usage and parameters: many-worlds-simulation-guide.en.md; Italian version: many-worlds-convergence-theory.it.md.

1. The starting point: quantum measurement

A quantum state evolves deterministically and linearly (Schrödinger equation). When an observable with N possible outcomes is measured, the Born rule gives outcome c the probability |a_c|², where a_c is the amplitude of component c [Born 1926]. In the standard formulation measurement is a second, discontinuous kind of evolution: the state "collapses" onto the observed component [von Neumann 1932]. The three simulated hypotheses give different answers to the question "what happens to the other N−1 components?".

The simulator keeps things minimal. At every step t it repeats the same measurement with N equiprobable outcomes (c = 0, …, N−1), and a world carries only its record k = Σ c, the sum of the outcomes. The graph shows the record centred, x = 2k − (N−1)·t, so the distribution starts at x = 0 and stays symmetric.

2. "Tree" regime: the many-worlds interpretation

2.1 The theory

Everett [1957] proposed to drop collapse. The universal wave function never reduces, and after a measurement the observer is also in superposition, with one copy per outcome. Each copy has a memory ("relative state") consistent with its own outcome and cannot notice the others. The name "many worlds" and the spread of the idea are due to DeWitt [1970; DeWitt and Graham 1973].

Branches become independent through decoherence. Interaction with the environment (photons, gas, apparatus) ties each outcome to nearly orthogonal environment states, and within extremely short times the interference terms between branches become unobservable [Zeh 1970; Zurek 1991, 2003; Joos et al. 2003]. In the modern version of the interpretation the "worlds" are emergent structures in the wave function, identified by decoherence, and need not be postulated [Wallace 2012; Saunders et al. 2010; Vaidman 2021].

2.2 The model in the simulator

The "Tree" regime only counts. Every measurement multiplies the worlds by N, so after t measurements there are

treeWorlds = N^t

distinct worlds, each with weight (squared modulus of the amplitude) N^(−t) and identified by its outcome sequence (the base-N digits of the node index). Node colour and the histogram at the bottom show how the worlds are distributed over the record k: the fraction with record k is the multinomial coefficient divided by N^t, i.e. the classical probability of the next regime. Hovering a world highlights those with the same record, which stay distinct in the tree even though their "content" is identical.

The limit of 4096 nodes (T ≤ 12 for N = 2) is purely graphical.

3. "Classical convergence" regime: worlds with the same record are the same world

3.1 The hypothesis

The simulator explores the hypothesis that the structure is not a tree. Two branches that reach the same state (here: same record k at the same step t) are a single world. The graph of worlds becomes a lattice with

distinct = (N−1)·t + 1

nodes at step t instead of N^t: linear growth, not exponential.

The idea is not part of the standard many-worlds interpretation, but it touches the literature at several points:

• Everett identified branches with the observer's memory sequences [Everett 1957, §5]. If the observer records only the total, two histories with the same total have the same memory.
• In the consistent histories formalism [Griffiths 1984; Omnès 1992; Gell-Mann and Hartle 1990] a history is a sequence of projectors. With a coarse graining that distinguishes only the record, histories with the same record become the same coarse-grained history. The condition that lets such histories be given classical probabilities (decoherence of histories) is exactly what separates this regime from the next.
• In the modern many-worlds interpretation, branches with the same macroscopic record and a different environment stay distinct, because what separates them is the state of the environment and not the observer's memory [Wallace 2012, ch. 3]. The classical regime therefore amounts to assuming that the record is the only information that survives.

3.2 The model in the simulator

If the probabilities of converging branches add without interference, the record performs a classical random walk [Feller 1968, ch. III]: each step adds a c uniform in {0, …, N−1}. The number of outcome sequences reaching node k (paths) follows the recursion

paths_{t+1}(k + c) += paths_t(k)        for c = 0, …, N−1

(for N = 2 this is Pascal's triangle and the lattice is the Galton board), and

pClassical(k, t) = paths_t(k) / N^t
flowClassical[c] = pClassical / N

c has mean (N−1)/2 and variance (N²−1)/12, so x = 2k − (N−1)·t has mean 0 and standard deviation

sigmaClassical = sqrt( t · (N²−1) / 3 )

that is, σ = √t for N = 2. The distribution widens as the square root of time (diffusion) and by the central limit theorem tends to a Gaussian. The effective number of worlds 1/Σp² (effectiveClassical) also grows as √t.

4. "Quantum convergence" regime: the amplitudes of converging branches add

4.1 The theory

If two branches really merge into the same state, quantum mechanics does not add their probabilities. The amplitudes add, and the probability is the squared modulus of the sum. This is the superposition principle in the form of Feynman's sum over paths [Feynman 1948; Feynman and Hibbs 1965]: the amplitude of reaching a point is the sum of the amplitudes of all histories that reach it, and different histories interfere. Decoherence removes the interference and brings back the classical regime; without decoherence, branches can cancel or reinforce each other.

The many-worlds literature has explicitly discussed bringing "already separated" branches of a measurement back into interference. Deutsch [1985] proposed a thought experiment: an observer (a quantum computer) performs a measurement, records the outcome and is then made to interfere with itself, which shows an observable difference between collapse and no collapse. Hanson [2003] instead studied what happens to the Born rule if small-amplitude worlds are "mangled" by interference with larger worlds. In both cases branch convergence is a theoretical possibility and does not describe the ordinary dynamics of macroscopic objects.

4.2 The model in the simulator: coined quantum walk

The usual way to propagate an amplitude on a lattice unitarily is the coined quantum walk [Aharonov, Davidovich and Zagury 1993; Meyer 1996; Ambainis et al. 2001; Nayak and Vishwanath 2000; Kempe 2003; Venegas-Andraca 2012]. Adding the amplitudes of converging branches directly is not enough, because the resulting map does not preserve the norm. So each node gets an internal degree of freedom with N components, the "coin", which here coincides with the outcome of the last measurement.

The state at step t is the set of complex amplitudes ψ_t(k, c). One step works like this:

1. coin: at every node the unitary matrix U is applied, here the discrete Fourier transform (function dftCoin)

   U[c][d] = exp(2πi · c · d / N) / sqrt(N)

   which for N = 2 is the Hadamard matrix; the DFT coin is the usual choice for coins with N > 2 faces [Mackay et al. 2002; Tregenna et al. 2003];

2. shift: component c moves to node k + c:

   ψ_{t+1}(k + c, c) = Σ_d U[c][d] · ψ_t(k, d)

The initial state is ψ_0(0, 0) = 1. The readings are

pQuantum(k, t)  = Σ_c |ψ_t(k, c)|²
flowQuantum[c]  = |(U ψ_t(k, ·))_c|²       (probability leaving along outcome c)

Branches converging on a node arrive in different coin components and meet in the coin of the next step, where the amplitudes add and interfere. The sum of the probabilities stays 1 at every step. In the graph, nodes with pQuantum < 10⁻⁹ are empty and edges with flowQuantum < 10⁻⁹ dashed: these are the branches cancelled by interference (suppressedNodes, suppressedEdges).

For N = 2 the model is the Hadamard walk, whose exact values are known: at step 3, on the records x = −3, −1, 1, 3, the probabilities are 1/8, 5/8, 1/8, 1/8 [Kempe 2003, §3]. The distribution is asymmetric because the initial coin state |0⟩ is not symmetric, a known property of the walk. The Hadamard walk with a localised initial state also coincides with the "Feynman checkerboard", the discretisation of the Dirac equation in 1+1 dimensions [Feynman and Hibbs 1965, problem 2-6; Meyer 1996].

4.3 The observable difference: ballistic spreading

The main result of quantum walk theory is that the standard deviation grows linearly with t (ballistic propagation), while in the classical walk it grows as √t [Nayak and Vishwanath 2000; Ambainis et al. 2001]. Konno's limit theorem [2002] gives the shape of the distribution: two peaks at the front, non-Gaussian. In the simulator sigmaQuantum grows as ≈ 0.46·t for N = 2, and effectiveQuantum (effective number of worlds) grows linearly instead of as √t. Values computed by buildLattice and statistics:

(seed 1; for N = 2 and D = 0 the quantum effective number is close to the classical one because the Hadamard distribution puts much of the probability on the two peaks).

5. Phase disorder: from quantum to classical

The parameter D ∈ [0, 1] multiplies the amplitudes of every node, before the coin, by a random phase e^{iφ} with |φ| ≤ π·D, drawn at every step from a seeded generator (seededRandom). The evolution stays unitary, but converging branches arrive with random relative phases. This is a minimal model of decoherence: the environment "marks" each branch with a phase the other branch does not know.

The quantum walk literature distinguishes two cases:

• static disorder (phases fixed in time, different for each node): produces Anderson localisation [Anderson 1958] and the distribution stays confined near the origin;
• dynamic disorder (new phases at every step, as here): destroys interference and brings propagation back to the diffusive regime σ ∝ √t of the classical walk [Brun, Carteret and Ambainis 2003; Kendon and Tregenna 2003]. Schreiber et al. [2011] observed both behaviours with photons.

The simulator uses dynamic disorder. For D → 1 sigmaQuantum again grows as √t (≈ 0.93·√t for N = 2, versus √t in the classical regime), the empty nodes and dashed edges change with the seed and the histogram approaches the classical outline. D therefore interpolates between regime 4 (D = 0) and a "noisy" version of regime 3. It is the same transition by which, in the many-worlds interpretation, decoherence makes branches effectively independent [Zurek 2003].

6. What the model does not say

• The measurements are all identical and equiprobable and the only state is the record k: the model does not describe any particular physical system.
• "Same record = same world" is the hypothesis being explored. In the standard interpretation branches stay distinct because of the environment (§3.1).
• The coin is a modelling choice, needed for unitarity. Different coins (for example the Grover coin for N ≥ 3) give different distributions [Tregenna et al. 2003].
• The tree view counts worlds and simulates nothing: each node weighs N^(−t), with no amplitudes.

References

Many-worlds interpretation and decoherence

• Everett, H. (1957). "Relative State" Formulation of Quantum Mechanics. Reviews of Modern Physics 29(3), 454–462.
• DeWitt, B. S. (1970). Quantum mechanics and reality. Physics Today 23(9), 30–35.
• DeWitt, B. S., Graham, N. (eds.) (1973). The Many-Worlds Interpretation of Quantum Mechanics. Princeton University Press.
• Deutsch, D. (1985). Quantum theory as a universal physical theory. International Journal of Theoretical Physics 24(1), 1–41.
• Hanson, R. (2003). When worlds collide: quantum probability from observer selection? Foundations of Physics 33(7), 1129–1150.
• Saunders, S., Barrett, J., Kent, A., Wallace, D. (eds.) (2010). Many Worlds? Everett, Quantum Theory, and Reality. Oxford University Press.
• Wallace, D. (2012). The Emergent Multiverse: Quantum Theory according to the Everett Interpretation. Oxford University Press.
• Vaidman, L. (2021). Many-Worlds Interpretation of Quantum Mechanics. In The Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/qm-manyworlds/
• Zeh, H. D. (1970). On the interpretation of measurement in quantum theory. Foundations of Physics 1(1), 69–76.
• Zurek, W. H. (1991). Decoherence and the transition from quantum to classical. Physics Today 44(10), 36–44.
• Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics 75(3), 715–775.
• Joos, E., Zeh, H. D., Kiefer, C., Giulini, D., Kupsch, J., Stamatescu, I.-O. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed. Springer.

Consistent histories and coarse graining

• Griffiths, R. B. (1984). Consistent histories and the interpretation of quantum mechanics. Journal of Statistical Physics 36(1–2), 219–272.
• Omnès, R. (1992). Consistent interpretations of quantum mechanics. Reviews of Modern Physics 64(2), 339–382.
• Gell-Mann, M., Hartle, J. B. (1990). Quantum mechanics in the light of quantum cosmology. In W. H. Zurek (ed.), Complexity, Entropy and the Physics of Information, Addison-Wesley, 425–458.

Measurement, Born rule, sum over paths

• Born, M. (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37, 863–867.
• von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Springer.
• Feynman, R. P. (1948). Space-time approach to non-relativistic quantum mechanics. Reviews of Modern Physics 20(2), 367–387.
• Feynman, R. P., Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals. McGraw-Hill.

Classical random walk

• Feller, W. (1968). An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed. Wiley.

Quantum walks

• Aharonov, Y., Davidovich, L., Zagury, N. (1993). Quantum random walks. Physical Review A 48(2), 1687–1690.
• Meyer, D. A. (1996). From quantum cellular automata to quantum lattice gases. Journal of Statistical Physics 85(5–6), 551–574.
• Nayak, A., Vishwanath, A. (2000). Quantum walk on the line. arXiv:quant-ph/0010117.
• Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J. (2001). One-dimensional quantum walks. Proceedings of the 33rd ACM Symposium on Theory of Computing (STOC), 37–49.
• Konno, N. (2002). Quantum random walks in one dimension. Quantum Information Processing 1(5), 345–354.
• Mackay, T. D., Bartlett, S. D., Stephenson, L. T., Sanders, B. C. (2002). Quantum walks in higher dimensions. Journal of Physics A 35(12), 2745–2753.
• Kempe, J. (2003). Quantum random walks: an introductory overview. Contemporary Physics 44(4), 307–327. arXiv:quant-ph/0303081.
• Tregenna, B., Flanagan, W., Maile, R., Kendon, V. (2003). Controlling discrete quantum walks: coins and initial states. New Journal of Physics 5, 83.
• Venegas-Andraca, S. E. (2012). Quantum walks: a comprehensive review. Quantum Information Processing 11(5), 1015–1106. arXiv:1201.4780.

Disorder, decoherence and localisation in quantum walks

• Anderson, P. W. (1958). Absence of diffusion in certain random lattices. Physical Review 109(5), 1492–1505.
• Brun, T. A., Carteret, H. A., Ambainis, A. (2003). Quantum random walks with decoherent coins. Physical Review A 67, 032304.
• Kendon, V., Tregenna, B. (2003). Decoherence can be useful in quantum walks. Physical Review A 67, 042315.
• Schreiber, A., Cassemiro, K. N., Potoček, V., Gábris, A., Jex, I., Silberhorn, C. (2011). Decoherence and disorder in quantum walks: from ballistic spread to localization. Physical Review Letters 106, 180403.