Shearing Off the Tree:
Emerging Branch Structure and Born’s Rule in an Equilibrated Multiverse
Philipp Strasberg and Joseph Schindler
F´ısica Teorica: Informaci´o i Fenomens Quantics, Departament de F´ısica,
Universitat Autonoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
(Dated: February 3, 2026)
Within the many worlds interpretation (MWI) it is believed that, as time passes on, the linearity
of the Schr¨odinger equation together with decoherence generate an exponentially growing tree of
branches where “everything happens”, provided the branches are defined for a decohering basis. By
studying an example, using exact numerical diagonalization of the Schr¨odinger equation to compute
the decoherent histories functional, we find that this picture needs revision. Our example shows
decoherence for histories defined at a few times, but a significant fraction (often the vast majority)
of branches shows strong interference effects for histories of many times. In a sense made precise
below, the histories independently sample an equilibrated quantum process, and, remarkably, we
find that only histories that sample frequencies in accordance with Born’s rule remain decoherent.
Our results suggest that there is more structure in the many worlds tree than previously anticipated,
influencing arguments of both proponents and opponents of the MWI.
Recent decades have seen intensified research interests
in Everett’s “relative state” formulation of quantum mechanics [1], which became widely known as the many
worlds interpretation (MWI) [2–9]. One reason of its
increased popularity is the development of decoherence
theory [10, 11], which can be used to investigate which
“worlds” behave classical, i.e., are decohered.
Provided one focuses on a basis that decoheres, the
many worlds “Multiverse” is conventionally pictured as
a branching tree with an exponentially growing number
of branches as time passes on (we restrict the discussion to nonrelativistic quantum mechanics with a primitive time parameter). This is easily seen for a many
worlds description of an idealized frequency experiment,
in which a system in a superposition is (re)prepared and
measured many times, see Fig. 1(a). Note, however, that
a branching could happen for any quantum fluctuation
suitably coupled to a macroscopic degree of freedom that
decoheres. This is the standard picture of the Multiverse
used by both proponents and opponents of the MWI [2–
9] and sometimes even classical toy-many-worlds-models
are used to explain aspects of the MWI.
Yet, the question how long this picture can remain true
has not been addressed, it is only clear that decoherence
can not proliferate forever in any (effectively) finite dimensional system. Here, we investigate this question in
detail by means of a clear example using decoherent histories. We find that decoherence does not stop globally
but continues unchanged on a (typically extremely small)
subset of branches. This suggests that the many worlds
tree has a non-trivial and potentially rich structure.
We continue with the general mathematical formalism
and explain how to investigate the decoherence properties of exponentially many branches. Then, we specify a model that can be related to sampling an equilibrated quantum process in an idealized frequency experiment with L independent and repeated trials (however,
even if one rejects this interpretation, the numerical results clearly demonstrate the hitherto unobserved phe0
0
0
1
1
11 1 1
1
0 0
0
0
Figure 1. (a) Tree structure of the Multiverse for a repeated
binary “0 or 1” measurement: The standard account posits
the reality of all branches (dashed and solid lines), but our
results show that only a subset decoheres (solid lines). (b)
Trajectory for the probability to obtain outcome ‘one’ conditioned on one branch (pink line in (a)). The numerical
parameters of (b) are the same as in Fig. 2 with D = 25000.
nomenon of a non-trivial (de)coherence structure among
the branches). At the end, we discuss the remarkable observation that all decohering branches sample frequencies
in accordance with Born’s rule.
Formalism. To access the decoherence of branches at
many times L ≫ 1 it is natural to use the decoherent histories formalism [12–17]. We are aware of controversies
about its use in relation to interpretations of quantum
physics, e.g., in Ref. [18]. In relation to our work, we reject this criticism as we only use the mathematical framework and do not postulate decoherent histories (instead,
arXiv:2310.06755v3 [quant-ph] 2 Feb 2026
2
we study their emergence based solely on the Schr¨odinger
equation). Seen from that perspective, little controversy
remains about the fact that decoherent histories capture
an essential aspect of classicality, even though they might
not capture all aspects. In particular, various researchers
have established a close connection between decoherent
histories and the formation of stable records or memories,
an important prerequisite to identify classical branches
that could support observers [19–26].
Mathematically, we use a complete set of orthogonal
projectors Πx associated with some abstract and for simplicity binary property x ∈ {0, 1} that decomposes the
Hilbert space of an isolated system. The initial state is
|ψ(t0)⟩ and the unitary time evolution operator from tk
to tℓ is denoted Uℓ,k. Then, the state compatible with
having properties xi at times ti for i ∈ {1, 2, . . . , L} is
|ψ(x)⟩ = ΠxL UL,L−1 · · · Πx2U2,1Πx1U1,0|ψ(t0)⟩ (1)
and x = (xL, . . . , x2, x1) is called a history. Note that x
merely labels the branches |ψ(x)⟩ of a unitarily evolving
global pure state as per
|ψ(tL)⟩ = UL,0|ψ(t0)⟩ =
X
x
|ψ(x)⟩. (2)
No actual measurement happens in the isolated system.
Now, two different histories x and y are decoherent if
ϵ(x, y) ≡
|⟨ψ(y)|ψ(x)⟩|
p
⟨ψ(x)|ψ(x)⟩
p
⟨ψ(y)|ψ(y)⟩
≪ 1. (3)
Note that ϵ(x, y) ≤ 1 by Cauchy-Schwarz’ inequality.
Unfortunately, the 2L many relative states |ψ(x)⟩ make
the evaluation of Eq. (3) for L ≫ 1 practically impossible.
Therefore, we focus our attention on a specific question
(whose motivation will become clear below) and introduce the relative state
|ψ(m)⟩ ≡ X
x
δm,n1(x)
|ψ(x)⟩, m ∈ {0, 1, . . . , L}. (4)
Here, δm,n is the Kronecker symbol and n1(x) ≡
PL
i=1 xi
the
P
net number of ‘ones’ in x. These states still satisfy
m |ψ(m)⟩ = |ψ(tL)⟩ and P
m ⟨ψ(m)|ψ(m)⟩ = 1. They
therefore label a complete set of branches in which potential observers care only about the relative frequency
m/L of ‘ones’ but forgot the information about which
particular sequence x was realized.
Toy model. Within this binary setting any Hamiltonian can be written as
H =
H00 H01
H10 H11
(5)
with Hij = ΠiHΠj . For reasons we explain below,
we take H00 (H11) to be diagonal matrices with d0
(d1) evenly spaced eigenenergies in the interval [0, δϵ]
(in the simulation we set δϵ = 0.5) such that the total Hilbert space dimension is D = d0 + d1. These
subspaces interact via H01 = H
†
10 = λR, where R is
a random matrix uniformly filled with zero-mean-unitvariance Gaussian random numbers and λ is a coupling
strength. We will restrict the discussion now to the relevant weak coupling regime, described by the condition
c ≡ 8λ
2d0d1/(Dδϵ2
) ≪ 1, but we later also study numerically the strong coupling regime.
We first note about this model that decoherence for
small L is established [27–29]. In addition, within an
open system approach, Π0 and Π1 can be seen as projectors onto pointer states of a two-level system weakly
coupled to an environment of dimension D/2. Environmentally induced decoherence has been also established
in this case (e.g., in Refs. [30–33]), but it is not necessary to refer to any system-environment tensor product
structure in the following. Moreover, equilibration and
thermalization of this model is also well established (e.g.,
in Refs. [34–38]) and the characteristic relaxation time
scale is well described by τ = δϵ/(2πλ2D) [37].
This toy model can therefore be regarded as an
archetype model behaving in agreement with decoherence
theory and statistical mechanics. Moreover, the success
of random matrix theory to describe generic properties
of complex systems [39–45] motivates the conjecture that
it also captures relevant aspects of realistic systems, in
particular owing to the close connection between random
matrix theory and realistic quantum many-body systems
(eigenstate thermalization hypothesis [44–49]).
Finally, we choose a Haar random initial state |ψ(t0)⟩
and times tj+1 − tj ≫ τ (numerically, we randomly sample tj+1 − tj ∈ [19.5τ, 20.5τ ]). The latter choice ensures that the projectors Πx reach their equilibrium value
⟨Πx⟩ ≈ dx/D (with exponentially suppressed fluctuations around its mean [50–54]) before each tj , as shown
in Fig. 1(b). Thus, from a macroscopic point of view the
systems looks identically prepared at each tj , although
subtle, hidden microscopic correlations remain, of course.
These considerations motivate the picture of an idealized frequency experiments of an equilibrated quantum
process, where |ψ(m)⟩ is associated with a record of m
‘ones’ after L independent trials—provided the branch
decoheres to allow the formation of a record. These statements will be precisely quantified below. Here, we only
remark that, of course, our toy model is unable to capture
any realistic experimental setup owing to obvious numerical limitations. We use the convenient metaphor of a frequency experiments because we capture the key aspect
of having a binary degree of freedom (which, as noted
above, could label the states of an open quantum system coupled to a complicated laboratory environment)
probed and (re)prepared many times in a seemingly independent way. Moreover, our main point—the emergence of a non-trivial and non-classical branch structure
of the many-worlds tree—does not rely on this purported
connection.
In particular, we like to counteract sceptical voices announcing that the random Hamiltonian or the random
initial state introduce classical noise or probabilities from
3
1.0 0.5 0.0 0.5 1.0
2(n ¡
n
®
)=L
0.0
0.2
0.4
0.6
0.8
1.0
(c) D = 250; L = 250
0.5 0.0 0.5 1.0
2(n ¡
n
®
)=L
(d) D = 25000; L = 250
0.0
0.2
0.4
0.6
0.8
1.0
(a) D = 250; L = 25 (b) D = 25000; L = 25 ²
p=pmax
q=pmax
Figure 2. Coherence measure ϵ(n) (blue circles) and probabilities p(n) (solid purple line) and q(n) (black stars) as a
function of n. Here and in all figures: ⟨n⟩ = Ld1/D is the
expected number of ones according to p(n), we always rescale
p(n) and q(n) by dividing by pmax = maxn p(n), the x axis is
rescaled to display n on an interval of size two, and the gray
area corresponds to one standard deviation of p(n).
the outside into our treatment. This is wrong because
all presented numerical results are obtained for a single
choice of H (kept fixed throughout the dynamics) and
a single choice of |ψ(t0)⟩, we do not perform any averages. The global state remains pure at all times and all
uncertainties are entirely of quantum origin. Nevertheless, and very importantly, we have observed the results
to be generic, i.e., different choices for H and |ψ(t0)⟩
give rise to similar behaviour. Backed up by the success
of random matrix theory [39–45] and typicality [52, 54],
this motivates the conclusion that our results are more
widely applicable.
Numerical evidence. In all plots we quantify the
amount of (de)coherence by considering
ϵ(n) = max
m̸=n
|⟨ψ(m)|ψ(n)⟩|
p
⟨ψ(m)|ψ(m)⟩
p
⟨ψ(n)|ψ(n)⟩
∈ [0, 1]. (6)
If ϵ(n) is close to zero, the branch giving rise to n ‘ones’ is
decohered from all other branches and allows the formation of stable, classical records; whereas ϵ(n) close to one
implies strong interference with other branches m ̸= n,
preventing a formation of reliable records. Note that
strong coherence between the branches |ψ(n)⟩ implies
also strong coherence between the fine-grained branches
|ψ(x)⟩ as shown in the Appendix.
1.5 1.0 0.5 0.0
2(n ¡
n
®
)=L
0.0
0.2
0.4
0.6
0.8
1.0
(a) L = 25
1.5 1.0 0.5 0.0
2(n ¡
n
®
)=L
(b) L = 250
² (D = 25000)
² (D = 250)
p=pmax
q=pmax
Figure 3. Coherence measure ϵ(n) for D = 250 (orange
squares) and D = 25000 (blue circles) and probabilities p(n)
and q(n) (as in Fig. 2) for D = 25000 as a function of n.
Moreover, we also compare the two probabilities
q(n) ≡ ⟨ψ(n)|ψ(n)⟩, p(n) ≡
L
n
p
n
1
(1 − p1)
L−n
, (7)
where q(n) is the exact probability to observe n times
‘one’, whereas p(n) is obtained by applying Born’s rule
with a single-time probability p1 to observe x = 1 for
L independent trials. Thus, p(n) is a binomial distribution and from what we said above we expect (and verify
below) that p1 = d1/D.
We start with equal subspace dimensions d0 = d1 and
ensure weak coupling by setting c = 0.0025. The time
evolution of a single history along a particular branch for
a few steps then looks as in Fig. 1 (b) in unison with our
expectations. Turning to longer histories, Fig. 2 displays
ϵ(n), q(n) and p(n). First, in Figs. 2(a) and (b) we consider a history of length L = 25 and compare two system
sizes: D = 250 and D = 25000. We confirm that decoherence is much stronger for larger system size, in unison
with the scaling laws of Refs. [28, 29, 55]. Moreover, we
have q(n) ≈ p(n) for D = 25000, but see significant deviations for D = 250 owing to the strong influence of finite
size effects. Thus, we see probabilities q(n) emerging that
start to behave as in an ideal frequency experiment for
large D. However, the coherences ϵ(n) do not yet show
any clear pattern, but L = 25 is still quite short.
Things start to change drastically for long histories
with L = 250 as shown in Figs. 2(c) and (d). Suddenly,
there is a very clear minimum around n ≈ ⟨n⟩. Remarkably, away from it ϵ(n) shoots up close to its maximum
value for both D = 250 and D = 25000, signifying the
strongest possible coherences. Moreover, we see now that
q(n) perfectly fits p(n) for D = 25000, whereas strong
deviations continue to exist for D = 250 (as expected).
Thus, for D = 25000 we are close to an ideal frequency
experiment, and coherent effects start growing approximately when |n − ⟨n⟩| exceeds one standard deviation.
However, the 50/50 splitting of the subspaces is special,
and to challenge our approach we continue by considering
a 20/80 splitting with d1 = 4d0, implying p1 = 0.8. The
results are shown in Fig. 3, where we display ϵ(n) for
4
0.5 0.0 0.5 1.0
2(n ¡
n
®
)=L
0.0
0.2
0.4
0.6
0.8
1.0
(a) c = 0:25
0.5 0.0 0.5 1.0
2(n ¡
n
®
)=L
(b) c = 25:0
²
p=pmax
q=pmax
Figure 4. Plot of ϵ(n), p(n) and q(n) for L = 250 and D =
25000 (legend as in Fig. 2). For (b) we rescaled τ to 10τ to
ensure equilibration between different trials.
D = 250 and D = 25000 jointly in one plot, whereas we
plot q(n) and p(n) only for D = 25000. In Fig. 3(a) we
can now see some clear signatures for large coherences
ϵ(n) if n deviates significantly from ⟨n⟩ even for short
histories of length L = 25. Moreover, in both Figs. 3(a)
and (b) we see that ϵ(n) for D = 25000 can be larger than
ϵ(n) for D = 250, contrary to what one would expect
from the scaling behaviour for small L [28, 29, 55]. Apart
from these additional observations, the evidence in Fig. 3
matches the conclusions from Fig. 2.
Finally, we consider the strong coupling case as a counterexample. Indeed, decoherent histories are expected to
emerge only for slow and coarse observables of a large dimensional system [20, 28, 29, 55–57], but at strong coupling the observable is no longer slow. The numerical
results for L = 250 and D = 25000 with d0 = 3d1/2 are
shown in Fig. 4 (a) for c = 0.25 (moderate coupling) and
(b) for c = 25.0 (very strong coupling). Figure 4(a) still
admits an interpretation along the lines of the previous
figures, showing a certain robustness of our results. Nevertheless, deviations between q(n) and p(n) become noticably visible, in particular around n ≈ ⟨n⟩. The picture
drastically changes for very strong coupling in Fig. 4(b).
While ϵ(n) still shows some non-trivial structure, it seems
hard to relate it to any physical properties. Moreover,
q(n) and p(n) are completely distinct, indicating strong
correlations between the trials.
Before turning to a broader discussion, it is worth to
summarize our numerical results within the MWI. First,
for L ≫ 1 the decohering branches of the universal wave
function of our toy model are only those sequences where
m/L ≈ p1. Observers inside it (which, admittably, are
hard to picture here) and without prior knowledge of
Born’s rule would therefore (self)locate themselves in
worlds obeying Born’s rule, i.e., they could infer the validity of the purple solid line in Fig. 2 and 3. Moreover,
the fact that p(n) ≈ q(n) (which is something an observer
inside the Multiverse could not confirm) proves that the
sampled frequencies correspond to independent trials.
Discussion. To the best of our knowledge, this is
the first study investigating decoherence properties for
L ≫ 1. In contrast to the widely held believe that singletime decoherence proliferates together with the branches,
we have shown that the many worlds tree possesses a nontrivial structure charactrized by branches that show the
highest possible degree of coherence, i.e., ϵ(n) ≈ 1.
Moreover, we established a connection to another important result. If one consider a unitary protocol where
L decorrelated subsystems interact with a measurement
apparatus (either in parallel at the same time or sequentially), then it can be shown that for any fixed δ > 0
lim
L→∞
Π (L) Born(δ)|ψ(tf )⟩ − |ψ(tf )⟩
= 0, (8)
where Π(L)
Born(δ) projects the final universal wave function
|ψ(tf )⟩ onto a subspace compatible with Born’s rule up
to an error δ. This result is in essence the law of large
numbers and was already used by Everett [1] (see also
Refs. [58–60]), but it relies crucially on the assumed independence of the measured systems. Obviously, if the
Universe is only a single wave function, it is full of microscopic correlations that have formed during its evolution or were already present initially. Here, we showed
that Eq. (8) emerges also in our equilibrated model without assuming independence or decorrelated subsystems.
This follows from q(n) ≈ p(n) and the concentration of
p(n) around its mean, which clearly calls for a rigorous
analytical result in the future.
Whereas the previous two paragraphs summarized two
unambiguous main results of our study, a remaining big
question is whether the structure of (de)coherence together with Eq. (8) can be used to explain Born’s rule
within the MWI in general. The short and honest answer
is: We are sceptical, but we do not know!
To put this question in context, we recall the attempts
to derive Born’s rule within the MWI [1, 3, 6–8, 58–72],
but all rely on additional postulates and none could convince its opponents [6, 73]—not to mention that even the
proposed solutions show a remarkable lack of consensus
among themselves. In particular, the so-called theory
confirmation problem asks why do we find ourselves in
a world compatible with Born’s rule if the vast majority
of branches is incompatible with it? Indeed, the number
of worlds is, by simple counting, always given by a binomial coefficient L
n
centered around ⟨n⟩ = L/2. Thus, for
p1 ∈ (0, 1) \ {1/2} the term in Eq. (8), while small with
respect to the Hilbert space norm, contains the majority
of worlds and, as Kent emphasizes [74], “It’s no more scientifically respectable to declare that we can [...] confirm
Everettian quantum theory by neglecting the observations
made on selected low Born weight branches [...] unless
we add further structure to the theory [...].”
In our toy model, we clearly see such further structure emerging from the Schr¨odinger equation itself, i.e.,
we find an unambiguous resolution of the quantum measurement problem for an arguably irrelevant model of the
Multiverse. Unfortunately, there is no evidence that this
could solve the theory confirmation problem in general,
but in light of the variety of attempts we believe this
5
novel direction deserves attention.
Finally, we try to give some physical intuition for our
results. Recall that decoherence requires coarse-graining
to “hide” coherences in inaccessible microscopic degrees
of freedom, but if these microscopic degrees of freedom
are restricted to evolve in a small and non-generic subspace, they no longer can induce decoherence effectively.
We believe that we observe this effect here: as L becomes large, some sequences restrict the relative state on
this branch to a very small and non-generic subspace,
thus preventing decoherence. This effect might be related to a recent result in pure state statistical mechanics [75, 76], where the authors showed that the distinguishability between an arbitrary quantum process sampled at L random times from a corresponding equilibrium
process is bounded by a number that scales as 22L/D
in the worst case, thus revealing a competition between
L and D. However, the effect seems subtle as already
Fig. 3 shows that increasing the Hilbert space dimension does not necessarily decrease the coherence among
all branches. Moreover, it does not matter whether we
humans are able to practically use the information contained in a sequence x to infer the true relative state.
Within the realist stance behind the MWI it is a matter
of principle whether the relative state compatible with
all the information out there looks generic or not.
Concluding perspectives. We demonstrated that
the many worlds tree can have a non-trivial structure
conflicting with the naive branch realism that is found
behind many arguments of proponents and opponents of
the MWI alike. We further showed that Eq. (8) holds in a
unitarily evolving and quantum correlated Universe. Finally, we observed the emergence of Born’s rule (which,
interestingly, is also a feature of de Broglie-Bohm theory [77, 78]) within our toy model, but we have no evidence that this is true in general.
While many open questions remain, the present approach demonstrates that fundamental aspects of the
MWI can be studied by using nothing but Schr¨odinger’s
equation (in a non-relativstic context) without approximations or additional metaphysical postulates. We have
the tools to rigorously access the decoherence of (long)
histories and the structure of the (potential) quantum
mechanical Multiverse (see also Ref. [29]), and it might
be full of marvelous wonders. Whether they speak in
favour or against the MWI needs to be found out.
Acknowledgements. We gratefully acknowledge discussions with Teresa E. Reinhard and Giulio Gasbarri.
Finanical support by MICINN with funding from European Union NextGenerationEU (PRTR-C17.I1) and
by the Generalitat de Catalunya (project 2017-SGR1127) are acknowledged. PS is further supported by
“la Caixa” Foundation (ID 100010434, fellowship code
LCF/BQ/PR21/11840014), the European Commission
QuantERA grant ExTRaQT (Spanish MICIN project
PCI2022-132965), and the Spanish MINECO (project
PID2019-107609GB-I00) with the support of FEDER
funds.
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Appendix
We show that
ϵ(m, n)
2 =
|⟨ψ(m)|ψ(n)⟩|2
⟨ψ(m)|ψ(m)⟩⟨ψ(n)|ψ(n)⟩
= 1 ⇒ ϵ(x, y)
2 =
|⟨ψ(x)|ψ(y)⟩|2
⟨ψ(x)|ψ(x)⟩⟨ψ(y)|ψ(y)⟩
= 1 (9)
for all x and y such that n1(x) ̸= n1(y). If instead m ≡ n1(x) = n1(y), it follows by definition that ϵ(m, m) = 1.
Moreover, we also assume that ⟨ψ(x)|ψ(x)⟩ > 0 and ⟨ψ(y)|ψ(y)⟩ > 0, i.e., we exclude all branches that do not exist.
Note that in our example all branches have non-zero weight (even though the weight might be extremely small).
To verify Eq. (9), we note that ϵ(m, n)
2 = 1 is equivalent to
0 = X
x,x′
,y,y′
A(x, x
′
, y, y
′
, m, n)[ϵ(x, y)ϵ(y
′
, x
′
) − ϵ(x, x
′
)ϵ(y
′
, y)], (10)
where
A(x, x
′
, y, y
′
, m, n) ≡ ⟨ψ(x)|ψ(x)⟩⟨ψ(x
′
)|ψ(x
′
)⟩⟨ψ(y)|ψ(y)⟩⟨ψ(y
′
)|ψ(y
′
)⟩δm,n1(x)δn,n1(y)δm,n1(x′)δn,n1(y′) (11)
Since A(x, x
′
, y, y
′
, m, n) > 0 if m ≡ n1(x) = n1(x
′
) ̸= n ≡ n1(y) = n1(y
′
), we find that the condition ϵ(m, n)
2 = 1
for m ̸= n implies
ϵ(x, y)ϵ(y
′
, x
′
) = ϵ(x, x
′
)ϵ(y
′
, y). (12)
We now choose x = x
′ and y = y
′ with n1(x) = m ̸= n1(y) = n, which implies ϵ(x, y) = 1 since ϵ(x, x) = 1 by
definition