intellecton/papers/project_paper_3_darwinism/references/Zurek2009.txt

Quantum Darwinism

Wojciech Hubert Zurek
Theory Division, MS B213, LANL Los Alamos, NM, 87545, U.S.A.

Quantum Darwinism describes the proliferation, in the environment, of multiple records of selected
states of a quantum system. It explains how the fragility of a state of a single quantum system can
lead to the classical robustness of states of their correlated multitude; shows how effective ‘wave-
packet collapse’ arises as a result of proliferation throughout the environment of imprints of the
states of quantum system; and provides a framework for the derivation of Born’s rule, which relates
probability of detecting states to their amplitude.
Taken together, these three advances mark
considerable progress towards settling the quantum measurement problem.

The quantum principle of superposition implies that
any combination of quantum states is also a legal state.
This seems to be in conflict with everyday reality: States
we encounter are localized. Classical objects can be ei-
ther here or there, but never both here and there. Yet, the
principle of superposition says that localization should be
a rare exception and not a rule for quantum systems.

Fragility of states is the second problem with quantum-
classical correspondence: Upon measurement, a general
preexisting quantum state is erased – it “collapses” into
an eigenstate of the measured observable. How is it then
possible that objects we deal with can be safely observed,
even though their basic building blocks are quantum?

To bypass these obstacles Bohr [1] followed Alexander
the Great’s example: Rather than try disentangling the
Gordian Knot at the beginning of his conquest, he cut
it.
The cut separates the quantum from the classical.
Bohr’s Universe consists of two realms, each governed by
its own laws. Fragile superpositions were banished from
the classical realm deemed more fundamental and indis-
pensable to interpret or even practice quantum theory.
Thus, instead of trying to understand Universe (includ-
ing “the classical”) in quantum terms one “quantized”
this and that, always starting from the classical base.

This was a brilliant tactical move: Physicists could
conquer the quantum realm without getting distracted by
interpretational worries. In those days only gedankenex-
periments like the famous Schr¨odinger cat [2] were truly
disturbing: Real experiments dealt with electrons, pho-
tons, atoms, or other microscopic systems. Bohr’s rule of
thumb – that the macroscopic is classical – was enough.
Moreover, many (including Einstein) believed that quan-
tum physics is just a step on a way to a deeper theory
that will solve or bypass interpretational conundrums.

That did not happen.
Instead, old gedankenexperi-
ments were carried out. They confirmed validity of quan-
tum laws on scales that have, of recent, begun to infringe
on “the macroscopic”. Quantum theory is here to stay.
It is also increasingly clear that its weirdest predictions
– superpositions and entanglement – are experimental
facts, in principle relevant also for macroscopic objects.
Therefore, questions about the origin of “the classical”,
with its restriction to localized states that are robust, un-
perturbed by measurements, can no longer be dismissed.

I.
DECOHERENCE AND EINSELECTION

Decoherence turns one of the two problems we noted
above – fragility of quantum states – into a solution of the
other. Environment-induced decoherence recognizes that
if a measurement can put a state at risk and re-prepare
it, so can accidental information transfers that happen
whenever a system interacts with its environment.
Decoherence is by now well understood [3, 4, 5]:
Fragility of states makes quantum systems very difficult
to isolate. Transfer of information (which has no effect on
classical states) has dramatic consequences in the quan-
tum realm. So, while fundamental problems of classical
physics were always solved in isolation (it sufficed to pre-
vent energy loss) this is not so in quantum physics (leaks
of information are much harder to plug).
When a quantum system gives up information, its own
state becomes consistent with the information that was
disseminated. “Collapse” in measurements is an extreme
example, but any interaction that leads to a correlation
can contribute to such re-preparation: Interactions that
depend on a certain observable correlate it with the en-
vironment, so its eigenstates are singled out, and phase
relations between such pointer states are lost [6].
Negative selection due to decoherence is the essence of
environment-induced superselection, or einselection [7]:
Under scrutiny of the environment, only pointer states
remain unchanged. Other states decohere into mixtures
of stable pointer states that can persist, and, in this sense,
exist: They are einselected.
These ideas can be made precise. The basic tool is the
reduced density matrix ρS. It represents the state of the
system that obtains from the composite state ΨSE of S
and E by tracing out the environment E:

ρS = TrE|ΨSE⟩⟨ΨSE| .
(1)

Evolution of ρS reveals preferred states: It is most pre-
dictable when the system starts in a pointer state. To
quantify this one can use (von Neumann) entropy, HS =
H(ρS) = −TrρS lg ρS, as a function of time.
Pointer
states result in smallest entropy increase. By contrast,
their superpositions produce entropy rapidly, at decoher-
ence rates, especially when S is macroscopic.
When pure states of the system are sorted by pre-
dictability, according to entropy of the evolved ρS,

arXiv:0903.5082v1  [quant-ph]  29 Mar 2009


2

pointer states are at the top. This criterion – the pre-
dictability sieve [4, 8, 9] – yields a short list of candidates
for effectively classical states: A cat can persist in one
of the two obvious stable states, but their superposition
would deteriorate into a mixture of |dead⟩ and |alive⟩
when initiated in a way envisaged by Schr¨odinger [2].
The special role of position is traced to the nature of
the SE interactions: They tend to depend on distance.
Hence, information about position is most readily passed
on to the environment. This is why localized states sur-
vive while nonlocal superpositions decay into their mix-
tures. For example, in a weakly damped harmonic os-
cillator the minimum uncertainty wavepackets – familiar
coherent states, best quantum approximation of classical
points in phase space – are einselected [9, 10, 11].

II.
ENVIRONMENT AS A WITNESS

Monitoring by the environment means that informa-
tion about S is deposited in E. What role does it play,
and what is its fate? Decoherence theory ignores it. En-
vironment is “traced out”.
Information it contains is
treated as inaccessible and irrelevant: E is a “rug to sweep
under” the data that might endanger classicality.
Quantum Darwinism recognizes that “tracing out” is
not what we do: Observers eavesdrop on the environ-
ment. Vast majority of our data comes from fragments
of E. Environment is a witness to the state of the system.
For example, this very moment you intercept a fraction
of the photon environment emitted by a screen or scat-
tered by a page. We never access all of E. Tiny fractions
suffice to reveal the state of various “systems of interest”.
This insight captures the essence of Quantum Darwin-
ism: Only states that produce multiple informational off-
spring – multiple imprints on the environment – can be
found out from small fragments of E. The origin of the
emergent classicality is then not just survival of the fittest
states (the idea already captured by einselection), but
their ability to “procreate”, to deposit multiple records
– copies of themselves – throughout E.
Proliferation of records allows information about S to
be extracted from many fragments of E (in the example
above, photon E). Thus, E acquires redundant records of
S. Now, many observers can find out the state of S in-
dependently, and without perturbing it. This is how pre-
ferred states of S become objective. Objective existence
– hallmark of classicality – emerges from the quantum
substrate as a consequence of redundancy.
Decoherence theory was focused on the system. Its aim
was to determine what states survive information leaks
to E. Now we ask: What information about the system
can be found out from fragments of E? This change of
focus calls for a more realistic model of the environment
(Fig. 1): Instead of a monolithic E we recognize that envi-
ronments consist of subsystems that comprise fragments
independently accessible to observers.
The reduced density matrix ρS representing the state

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FIG. 1: Quantum Darwinism and the structure of the envi-
ronment. Decoherence theory distinguishes between a system
(S) and its environment (E) as in (a), but makes no further
recognition of the structure of E; it could as well be mono-
lithic. In Quantum Darwinism the focus is on redundancy.
We recognize the subdivision of E into subsystems, as in (b).
The only requirement for a subsystem is that it should be
individually accessible to measurements; observables of dif-
ferent subsystems commute. To obtain information about S
from E one can then measure fragments F of the environ-
ment – non-overlapping collections of subsystems of E, (c).
ically, there are many copies of the information about S in E
– “progeny” of the “fittest observable” that survived monitor-
ing by E proliferates throughout E. This proliferation of the
multiple informational offspring defines Quantum Darwinism.
The environment becomes a witness with redundant copies of
information about the preferred observable. This leads to the
objective existence of pointer states: Many can find out the
state of the system independently, without prior information,
and they can do it indirectly, without perturbing S.

of the system was the basic tool of decoherence. To study
Quantum Darwinism we focus on correlations between
fragments of the environment and the system. The rele-
vant reduced density matrix ρSF is given by:

ρSF = TrE/F|ΨSE⟩⟨ΨSE| .
(2)

Above, trace is over “E less F”, or E/F – all of E except
for the fragment F. How much F knows about S can be
quantified using mutual information:

I(S : F) = HS + HF − HS,F ,
(3)

defined as the difference between entropies of two sys-
tems (here S and F) treated separately and jointly. For
example, the mutual information between an original and


3

FIG. 2: Information about S stored in E and its redundancy.
Mutual information is monotonic in f. When global state of
SE is pure, I(S : Ff) in a typical fraction f of the environ-
ment is antisymmetric around f = 0.5 [13]. For pure states
picked out at random from the combined Hilbert space HSE,
there is little mutual information between S and a typical F
smaller than half of E. However, once a threshold f = 1

2 is
attained, nearly all information is in principle at hand. Thus,
such random states (green line) exhibit no redundancy. By
contrast, states of SE created by decoherence (where the en-
vironment monitors preferred observable of S) contain almost
all (all but δ) of the information about S in small fractions
fδ of E. The corresponding I(S : Ff) (red line) quickly rises
to HS (entropy of S due to decoherence), which is all of the
information about S available from either E or S. (More, up
to 2HS, can be obtained only through global measurements
on S and nearly all E). HS is therefore the classically acces-
sible information. As (1 − δ)HS of information is contained
in fδ = 1/Rδ of E, there are Rδ such fragments in E: Rδ
is the redundancy of the information about S. Large redun-
dancy implies objectivity: The state of the system can be
found out indirectly and independently by many observers,
who will agree about their conclusions. Thus, Quantum Dar-
winism accounts for the emergence of objective existence.

a perfect copy (of, say, a book) is equal to the entropy of
the original, as either contains the same text. So, every
bit of information in the first copy reveals a bit of infor-
mation in the original. However, having extra copies does
not increase the information about the original. Yet, it
determines how many can independently access this in-
formation. The number of copies defines redundancy.
Similar ideas apply to the quantum case. Initially, ev-
ery bit of information gained from a fraction f ≪ 1 of
E that was pure before it monitored (and decohered) the
system is a bit about S. The red plot in Fig. 2 starts with
this steep “bit for bit” slope, but moderates as I(S : Ff)
approaches redundancy plateau at HS, where additional
bits only confirm what is already known.
Redundancy is the number of independent fragments
of the environment that supply almost all classical infor-
mation about S, i.e., (1 − δ)HS. In other words;

Rδ = 1/fδ .
(4)

Rδ is the number of times one can acquire (1 − δ) of the
information about S independently (from distinct F’s)

and indirectly – without perturbing S.
Rapid rise and gradual leveling of I(S : Ff), Fig. 2,
implies redundancy. The information in Ff allows one
to determine the state of S as it reaches redundancy
plateau.
Observables of different F’s commute – such
measurements are independent. Yet, underlying corre-
lations mean that their outcomes imply the same state
of the system, as if S were classical: The redundancy
plateau is a classical plateau. Its level HS is the classical
information accessible from a small fraction of E.
Redundancy allows for objective existence of the state
of S: It can be found out indirectly, so there is no danger
of perturbing S with a measurement. Error correction al-
lowed by redundancy is also important: Fragility of quan-
tum states means that copies in F’s are damaged by mea-
surements (we destroy photons!), and may be measured
in a “wrong” basis. One cannot access records in E with-
out endangering their existence.
But with many (Rδ)
copies, state of S can be found out by ∼ Rδ observers
who can get their information independently, and with-
out prior knowledge about S. Consensus between copies
suggests objective existence of the state of S.
The mutual information I(S : Ff) computed in mod-
els of decoherence exhibits behavior illustrated by the red
plot of Fig. 2. In the family of models representing spin
S surrounded by environments of many spins [12, 13, 14]
the same number of spins suffices to reach the plateau:
Adding more spins to E only extends length of the plateau
measured in “absolute units” – in the number of the en-
vironment spins. In this model (that can be viewed as
a simplified model of a photon environment) redundancy
is then proportional to the number of the environment
subsystems that interact with the system of interest.
Quantum Brownian motion – harmonic oscillator sur-
rounded by many environmental oscillators – is the other
well known model of decoherence. It is exactly solvable,
and the case of an underdamped oscillator yields sur-
prisingly simple results [15, 16]: (i) Mutual information
is approximately given by I(S : F) ≈ HS + 1

2 ln
f

(1−f),
and; (ii) Redundancy for an initially squeezed state of S
reaches Rδ ≈ s2δ, where s, the squeeze factor, quantifies
delocalization of the state. Similar equation should hold
for more general “Schr¨odinger cat” states, with s quan-
tifying the separation of the two localized alternatives.
These results confirm intuitions that originally moti-
vated Quantum Darwinism [4, 17]: Monitoring of the
system by the environment can deposit multiple records
of preferred states of S in E. States of SE that arise from
decoherence are special [13, 14], as I(S : Ff) for a typ-
ical pure state selected with Haar measure in the whole
Hilbert space of SE (green plot in Fig.
2) shows.
In
such random states small fragments reveal almost noth-
ing about the rest of the state. Only when half of E is
found out the whole state is suddenly revealed.
States that arise from decoherence are then far from
random. Roughly speaking, they have a branch structure.
This is why the rest of such a branch including the state
of the system – the “bud” from which this branch has


4

originated – can be deduced from its fragment. We shall
see how such branches grow in the next section.
Plots of I(S : Ff) for pure SE are antisymmetric
around the point {HS, f = 1

2} for typical fragments of
E [13]. Thus, rapid rise for small f must be matched at
the other end, for f ∼ 1. This is a signature of entan-
glement that allows state to be known “as the whole”,
while states of subsystems are unknown. The joint state
of SE is then pure, so that HS,F=E = 0, and I(S : Ff)
must rise to HS + HE = 2HS when f approaches 1.
This is a very quantum aspect of information. In clas-
sical physics knowing a composite object implies knowing
each of its subsystems. This is not so in quantum physics,
where composite states are given by tensor (rather than
Cartesian) products of their constituents. Thus, one can
know perfectly quantum state of the whole, but know
nothing about states of parts. We shall see in Section IV
how this feature can be used to derive Born’s rule [18]
that relates probabilities with wavefunctions.
To reveal this latent quantumness one would have to
measure the right global observable on all of SE.
For
example, when mutual information, Eq. (3), is defined
using Shannon entropy with probabilities corresponding
to optimal observables in S and in E, the resulting Shan-
non I(S : Ff) graph for small f would look very similar
to Fig. 2. However, using Shannon entropy involves lo-
cal probabilities (precluding global observables), so such
Shannon I(S : Ff) never exceeds HS, antisymmetry is
lost, and the plateau continues until the end at f ∼ 1.
Effective unattainability of the f ∼ 1 part of the plot
also shows why decoherence is so hard to undo: Correla-
tions that reveal coherence can be usually detected only
by such global measurements of whole SE. We intercept
small fractions of E, and never have the luxury of perfect
global measurements needed to undo decoherence. Yet,
because of redundancy, we get ∼ HS information with
“sloppy” measurements of f ≪ 1.
Quantum Darwinism does not require pure E. Mixed
environment is a noisy communication channel: Its initial
entropy of h per bit can still increase after interaction
with S, reflecting mutual information buildup. However,
now a bit gained from E yields only 1−h of a bit about S.
So, a completely mixed E (h = 1) is useless (even though
it can still induce decoherence!). For a partly mixed E
mutual information will increase more slowly, pure case
“bit per bit” rate tempered to ∼ 1 − h. Yet, it can still
climb the same redundancy plateau at HS [19].
These conclusions apply when E is initially mixed, but
are also relevant when this channel is noisy for other rea-
sons (e.g., imperfect measurements). In all such cases one
can still reach the same redundancy plateau, although
now a proportionally larger fragment of the environment
is needed to get the same information about S.
Suitability of the environment as a channel depends
on whether it provides a direct and easy access to the
records of the system.
This depends on the structure
and evolution of E.
Photons are ideal in this respect:
They interact with various systems, but, in effect, do

not interact with each other.
This is why light deliv-
ers most of our information. Moreover, photons emitted
by the usual sources (e.g., sun) are far from equilibrium
with our surroundings. Thus, even when decoherence is
dominated by other environments (e.g., air) photons are
much better in passing on information they acquire while
“monitoring the system of interest”: Air molecules scat-
ter from one another, so that whatever record they may
have gathered becomes effectively undecipherable.
Stability of the level of the redundancy plateau at HS,
even for mixed E’s, is a compelling reason to think of it as
“classical”. The question we shall now address concerns
the nature of that information – what does the environ-
ment know about the system, and why?

III.
FROM COPYING TO QUANTUM JUMPS

Quantum Darwinism leads to appearance, in the en-
vironment, of multiple copies of the state of the system.
However, the no-cloning theorem [20, 21] prohibits copy-
ing of unknown quantum states. If cloning is outlawed,
how can redundancy seen in Fig. 2 be possible?
Quick answer is that cloning refers to (unknown) quan-
tum states. So, copying of observables evades the theo-
rem. Nevertheless, the tension between the prohibition
on cloning and the need for copying is revealing: It leads
to breaking of unitary symmetry implied by the super-
position principle, accounts for quantum jumps, and sug-
gests origin of the “wavepacket collapse”, setting stage for
the study of quantum origins of probability in Section IV.
Quantum physics is based on several “textbook” pos-
tulates [22]. The first two; (i) States are represented by
vectors in Hilbert space, and; (ii) Evolutions are unitary –
give complete account of mathematics of quantum theory,
but make no connection with physics. For that one needs
to relate calculations made possible by the superposition
principle of (i) and unitarity of (ii) to experiments.
Postulate (iii) Immediate repetition of a measurement
yields the same outcome starts this task. This is the only
uncontroversial measurement postulate (even if it is diffi-
cult to approximate in the laboratory): Such repeatability
or predictability is behind the very idea of “a state”.
In contrast to (i)-(iii), collapse postulate (iv) Outcomes
correspond to eigenstates of the measured observable, and
only one of them is detected in any given run of the ex-
periment, is inconsistent with (i) and (ii). Conflict arises
for two reasons: Restriction to a preferred set of outcome
states seems at odds with with the egalitarian principle
of superposition, embodied in (i). This restriction pre-
vents one from finding out unknown quantum states, so
it is responsible for their fragility. And a single outcome
per run is at odds with unitarity (and, hence, linearity)
of quantum dynamics that preserves superpositions.
The last axiom; (v) Probability of an outcome is given
by the square of the associated amplitude, pk = |ψk|2,
is known as Born’s rule [18]. It completes the relation
between mathematics of (i) and (ii) and the experiments.


5

a

µ

R0.1(σ)

0

10

20

30

40

50

0
π/2

π/4

0

π/4

π/8
a

π/4
µ

0.6

0.8

π/4

π/2 0

π/8

1.0

0.4

0.2

0
0

ˆIN(σ)

m

0.4

0.8

1.0

π/2 0
10
20 30
40
50

µ

0.6

0.2

00

π/4

I(σ : e)

µ = 0.23

a)
b)
c)

FIG. 3: Quantum Darwinism in a simple model of decoherence [12]. The spin- 1

2 S interacts with N = 50 spin- 1

2 subsystems of E
with an Ising Hamiltonian HSE = PN
k=1 gkσS
z ⊗σEk
y . The initial state of S⊗E is
1
√

2(|0⟩+|1⟩)⊗|0⟩E1⊗. . .⊗|0⟩EN . Couplings gk are
distributed randomly in the interval (0,1]. All the plotted quantities are a function of the observable σ(µ) = cos(µ)σz +sin(µ)σx,
where µ is the angle between its eigenstates and the pointer states of S – eigenstates of σS
z . a) Information acquired by the
optimal measurement on the whole environment, ˆIN(σ), as a function of the inferred observable σ(µ) and the average interaction
action ⟨gkt⟩ = a. A lot of information is accessible in the whole E about any observable σ(µ) except when a is so small that
there was no decoherence. b) Redundancy of the information about S as a function of the inferred observable σ(µ) and the
average action ⟨gkt⟩ = a. Rδ=0.1(σ) counts the number of times 90% of the total information can be “read off” independently
by measuring distinct fragments of E. It is sharply peaked around the pointer observable: Redundancy is a very selective
criterion – the number of copies of relevant information is high only for the observables σ(µ) inside the theoretical bound (see
Ref.[12]) indicated by the dashed line. c) Information about σ(µ) extracted by local random measurements on m environmental
subsystems. Because of redundancy, pointer states – and only pointer states – can be found out through this far-from-optimal
strategy. Information about any other observable σ(µ) is restricted to what can be inferred from the pointer observable [12].

Bohr bypassed conflict of (i) and (ii) with (iv) by insist-
ing that apparatus is classical, so unitarity and the prin-
ciple of superposition need not apply to measurements.
But this is an excuse, not an explanation. We are dealing
with a quantum environment, and redundancy of previ-
ous section strengthened motivation for postulate (iii) –
repeatability. Let us see where this demand takes us in
a purely quantum setting of postulates (i), (ii), and (iii).
Suppose there are states of S (say, |u⟩ and |v⟩) that
produce an imprint in a subsystem of E (which plays a
role of an apparatus), but remain unperturbed (so they
can produce more imprints). This repeatability implies:
|u⟩|e0⟩ ⇒ |u⟩|eu⟩, |v⟩|e0⟩ ⇒ |v⟩|ev⟩ in obvious notation.
In a unitary process scalar product is preserved. Thus;

⟨u|v⟩ = ⟨u|v⟩⟨eu|ev⟩ ,
(5)

where we have set ⟨e0|e0⟩ = 1.
This simple equation
can be satisfied only when; (a) either ⟨eu|ev⟩ = 1 (which
means that copying was completely unsuccessful), or; (b)
⟨u|v⟩ = 0, i.e., they are orthogonal. In that case ⟨eu|ev⟩
is arbitrary – perfect record ⟨eu|ev⟩ = 0 is also possible.
It follows that multiple (perfect or imperfect) copies
of |u⟩ and |v⟩ can be imprinted in disjoint F’s.
As a
consequence of unitarity, only sets of orthogonal states
(that define Hermitean observables [22]) can be so copied,
explaining selection of a set of outcomes – terminal points
of quantum jumps [23]. Before, they had to be postulated
by the first part of axiom (iv). We emphasize that this
result relies on just two values of the scalar product – 0
and 1 – and, thus, does not appeal to Born’s rule.
This breaking of unitary symmetry (choice of preferred
states in an egalitarian Hilbert space) is induced by re-
peatability of the information transfer. It is a “nonlinear

demand”: As in cloning, one asks for “two (or more) of
the same”.
Its conflict with linearity of quantum the-
ory can be resolved only by restricting states that can
be copied.
Such pointer states then act as “buds” of
branches that grow by reproducing, in E, multiple copies
of the original in S. Interaction Hamiltonians do not per-
turb observables that commute with them. So, buds of
branches coincide with the einselected pointer states.

Evidence of such symmetry breaking is seen in Fig.
3. Mutual information and redundancy shown there are
obtained using Eq. (3), but with Shannon (rather than
von Neumann) entropies of specific observables of S and
F, i.e., using probabilities of their eigenstates. While von
Neumann-based I(S : Ff) and Rδ characterized total
information, Shannon-based counterparts are well suited
to enquire: What observable is this information about?

It turns out that the environment as a whole “knows”
many observables of S, as is seen in Fig. 3a. By contrast,
in Fig. 3b symmetry breaking is evident: The ridge of
redundancy appears abruptly only when test observable
σ(µ) and the preferred pointer observable σz (that re-
mains unperturbed by the environment) nearly coincide.

Why are pointer states favored? Commonsense says
that, to be reproduced, state must survive copying. This
leads to a theorem [12, 24] that only pointer states can be
discovered from fractions of E. Other observables (such
as σ(µ) in Fig. 3) can be deduced only to the extent they
are correlated with the pointer observable. So, fragments
of the environment offer a very narrow, projective point
of view. Redundant imprinting of some observables hap-
pens at the expense of their complements.

Structure of branching state betrays its origin and fore-


6

shadows “collapse”. Starting from |ψS⟩ = �n
k ψk|sk⟩,

|ΨSE⟩ =

n
�

k
ψk|sk⟩|e(1)
k ⟩ . . . |e(N)
k
⟩ =

n
�

k
ψk|sk⟩|εk⟩ (6)

branches grow to include N subsystems of E.
Branch
fragments can be nearly orthogonal; ΠJ
j=1⟨e(j)
k |e(j)
k′ ⟩ ≃
δkk′ for large enough J. This means that a pointer state
|sk⟩ of S can be determined (along with the rest of the
branch) from a sufficiently long fragment (which may still
be short compared to the length of the branch, J ≪ N).
In the huge Hilbert space HSE branching state is a
very atypical minimally entangled superposition of only
n product “branches” labelled by the pointer states of
the system. This is tiny compared to the dimension of
HSE that exceeds n by a factor exponential in N. This
is why the two plots in Fig. 2 are so different: Branch-
ing state is, to a good approximation, a multi-system
Schmidt decomposition, with long branch fragments con-
stituting “systems”. In a Schmidt decomposition, states
of partners are in one-to-one correspondence. Thus, in
Eq. (6), |sk⟩ implies |εk⟩ (and, vice versa), and measur-
ing a branch fragment F can reveal the whole branch.
Initial part of I(S : Ff), Fig. 2, represent buildup of
this correlation: When f = 0, observer is ignorant of
what branch he will find out, but the structure of the
correlations within |ΨSE⟩ leaves no doubt of what these
branches are. Using Born’s rule one could assign to them
probabilities pk = |ψk|2 and the corresponding entropy
HS. Next section shows how one can deduce these prob-
abilities without axiom (v) – how symmetries of entan-
glement imply Born’s rule.
When observer measures enough of E, he finds out
the branch (and what the state of S is).
Additional
data are redundant. They only confirm what is already
known. Probabilities associated with |ΨSE⟩ are replaced
with certainty of a branch. This transition from uncer-
tainty (initial presence of many branches – potential for
multiple outcomes) to certainty (once a sufficiently long
branch fragment becomes known) accounts for percep-
tion of “collapse”.
The initial, steeply rising, part of
I(S : Ff) “resolves” it: Collapse is brief compared to
the ensuing period of certainty about the outcome, as
fδ ≪ 1, but, nevertheless, not instantaneous.
Assumptions that lead from copying to preferred states
can be relaxed. Thus, E need not be initially pure [23].
Moreover, it suffices that the records (e.g., in the appara-
tus A) are “repeatably accessible”. Transfer of responsi-
bility for repeatability from a quantum S to a (still quan-
tum) A allows one to model non-orthogonal measurement
outcomes (POVM’s): A entangles with the system, and
then acts as ancilla. Its orthogonal pointer states |Ak⟩
correlate with non-orthogonal |ςk⟩ of S, �

k ˜ψk|ςk⟩|Ak⟩.
Interaction of A with the environment results in multiple
copies of |Ak⟩. The usual projective measurement imple-
mentation of POVM’s (see e.g. [25]) is now straightfor-
ward. Branches are labelled by |Ak⟩. Indeed, we usually
experience “quantum jumps” via an apparatus pointer.

Selection of the set of outcomes by the proliferation of
information essential for Quantum Darwinism parallels
Bohr’s insistence [1] that a “classical apparatus” should
determine the outcomes. However, it follows from purely
quantum Eq. (5), and is caused by a unitary evolution
responsible for the information transfer. Nevertheless, as
classical apparatus would, preferred pointer states desig-
nate possible future outcomes, precluding measurements
of complementary observables or determining preexist-
ing state of the system. Thus, information acquisition –
a copying process – results in preferred states.
Consensus between records deposited in fragments of
E looks like “collapse”. In this sense we have accounted
for postulate (iv) using only very quantum postulates (i)-
(iii). In particular, in deriving and analyzing Eq. (5) we
have not employed Born’s rule, axiom (v). We shall be
therefore able to use our results as a starting point for
such a derivation in the next section.
There was nothing nonunitary above – unitarity was
the crux of our argument, and orthogonality of branch
seeds our main result. Relative states of Everett [26, 27,
28] come to mind. One could speculate about reality of
branches with other outcomes. We abstain from this –
our discussion is interpretation-free, and this is a virtue.
Indeed, “reality” or “existence” of universal state vector
seems problematic.
Quantum states acquire objective
existence when reproduced in many copies. Individual
states – one might say with Bohr – are mostly informa-
tion, too fragile for objective existence. And there is only
one copy of the Universe. Treating its state as if it really
existed [26, 27, 28] seems unwarranted and “classical”.

IV.
PROBABILITIES FROM ENTANGLEMENT

Observer prepared S in a state |ψS⟩, but wants to mea-
sure observable with eigenstates {|sk⟩}. This will lead to
entangled |ΨSE⟩ with branch structure, Eq. (6). Pointer
states {|sk⟩} define the outcomes, but, as yet, observer
has not measured E, and does not know the result. Given
|ΨSE⟩, what is the probability of, say, |s17⟩?
To derive it we cannot use reduced density matrices,
Eqs. (1,2). Tracing out is averaging [25, 29, 30] – it relies
on pk = |ψk|2, Born’s rule we want to derive. We have
imposed that ban while deriving and analyzing Eq. (5),
but relaxed it to plot Fig. 3. Now we reimpose it again.
So, Born’s rule and standard tools of decoherence are
off limits – using them courts circularity. Our derivation
will rest instead on certainty and symmetry, cornerstones
that mark two extremal cases of probability.
The case of certainty was just settled without Born’s
rule using Eq. (5). When one re-measures an observable,
the same outcome will be seen again. Thus, when {|sk⟩}
includes |ψS⟩ (e.g., |ψS⟩ = |s17⟩), newly added copies
just extend the branch already correlated with observer’s
state, and the outcome is certain; p17 = 1. Certainty of
correlations between partners in Schmidt decomposition,
Eq. (6) is another important example.


7

a)

b)

c)

+
| >S| >E | >S| >E

+
| >S| >E | >S| >E

+
| >S| >E | >S| >E

+
| >S| >E | >S| >E

+
| >S| >E | >S| >E
=

~~

=

FIG. 4: Probabilities and symmetry: (a) Laplace used subjective ignorance to define probability. Player who does not know face
values of the cards, but knows that one of them is a spade will infer probability p♠ = 1

2 for the top card. (b) The real physical
state of the system is however altered by the swap, illustrating subjective nature of Laplace’s approach, and demonstrating its
unsuitability for physics. (c) Perfectly known entangled states have objective symmetries that allow one to rigorously deduce
probabilities. When two systems are maximally entangled as above, probabilities of Schmidt partners are equal, p♥ = p♦, and
p♠ = p♣. After a swap uS = |♠⟩⟨♥| + |♥⟩⟨♠| in S, the resulting state |♠⟩|♦⟩ + |♥⟩|♣⟩ must have p′
♠ = p♦, and p′
♥ = p♣. (We
‘primed’ probabilities in S, as it was acted upon by a swap, so they might have changed.) A counterswap uE = |♦⟩⟨♣| + |♣⟩⟨♦|
in E restores the original entangled state, proving that p′
♥ = p♥ and p′
♠ = p♠, after all (as counterswap uE leaves S untouched).
This sequence of equalities implies p♠ = p♦ = p♥, so that p♠ = p♥ = 1

2, as probabilities in S must add up to 1.

Certainty seems trivial but is important. Confirmation
that a state “is what it is” – postulate (iii) – is a part of
standard quantum lore [22]. We re-affirmed it, but with
a key insight: Redundancy allows observers to discover
(and not just confirm) that S is in a certain pointer state.

We now turn to the opposite case of complete inde-
terminacy. Its connection with symmetry was noted by
Laplace. He wrote: “The theory of chance consists in re-
ducing all the events ... to a certain number of cases that
are equally possible... The ratio of this number to that of
all the cases possible is the measure of probability” [31].

Figure 4 illustrates how this classical intuition yields –
far more convincingly — quantum probabilities.
Symmetry is probed by invariance. Transformations
that respect it take system between states that exhibit
no measurable differences. For example, change of phase
in the coefficients in the Schmidt decomposition |ΨSE⟩ =
�n
k ψk|sk⟩|εk⟩ cannot influence the state of S: It is in-
duced by uS = eiφk|sk⟩⟨sk|, local unitary on S, that can
be “undone” by uE = e−iφk|εk⟩⟨εk| on E, or;

uS ⊗ 1E|ΨSE⟩ = |ΦSE⟩; 1S ⊗ uE|ΦSE⟩ = |ΨSE⟩
(7)


8

So, phases of ψk cannot matter for a local state or influ-
ence probabilities in S. This symmetry, Eq. (7), is the
entanglement-assisted invariance or envariance [32, 33].
Such loss of phase significance for S entangled with E
implies decoherence [33]. We arrived at its essence using
envariance, without reduced density matrices, trace, etc.
We now use phase envariance to show that equal ab-
solute values of the coefficients ψk imply equal prob-
abilities.
For equal |ψk| any orthogonal basis of S
is “Schmidt” (i.e., has an orthogonal partner in E).
Thus, | ¯ϕSE⟩ =
|0⟩S|0⟩E+|1⟩S|1⟩E
√

2
=
|+⟩S|+⟩E+|−⟩S|−⟩E
√

2
,

where |±⟩ = |0⟩±|1⟩
√

2
. Sign change induced by eiπ|−⟩⟨−|

acting on S produces |¯ηSE⟩ =
|+⟩S|+⟩E−|−⟩S|−⟩E
√

2
=

|1⟩S|0⟩E+|0⟩S|1⟩E
√

2
. In other words, one can swap |0⟩S with
|1⟩S by rotating phase in a |±⟩ basis by π. Yet, we just
saw that phases of Schmidt coefficients do not matter for
the state of S, so probabilities of 0 and 1 in S must have
remained the same. Moreover, probabilities of paired up
Schmidt states are equal, so that pS(0) = pE(0) in | ¯ϕSE⟩
and pS(1) = pE(0) in |¯ηSE⟩. Hence, pS(0) = pS(1) = 1

2,
where we assumed that probabilities add up to 1.
In contrast to Laplace’s subjective “ignorance-based”
approach, we obtained objective probabilities for a com-
pletely known entangled state. Phase envariance implied
equiprobability in S.
To paraphrase Beatles, “All you
need is phase...”. We rotated phases of the coefficients to
induce a swap in a complementary basis. Another proof
(that implements swap more directly) is given in Fig. 4.
This equiprobability case is the difficult part of the
proof. Instead of subjectivity (that undermined appli-
cability of Laplace’s approach to physics) we relied on
objective symmetries of entangled quantum states. This
was made possible by the nature of quantum states of
composite systems. Classically, pure states have struc-
ture of a Cartesian product – knowing the whole implies
knowledge of each subsystem. In quantum theory they
are tensor products – one can know state of the whole,
and thus know nothing about parts, as envariance shows.
This was the basis of our proof of equiprobability. We
assumed unitarity. Moreover, we assumed; (1) When a
system is not acted upon by a unitary transformation, its
state remains unaffected.
This state is a property of
S alone, so; (2) Predictions regarding measurement out-
comes on S (including their probabilities) can be inferred
from the state of S. Last not least; (3) When S is entan-
gled with other systems (e.g., the environment) the state
of S alone is determined by the state of the whole SE.
These “facts of life” are accepted properties of systems
and states, but given the fundamental nature of our dis-
cussion it seems a good idea to make them explicit [33].
For instance, to establish independence from phases of
the coefficients ψk we noted that the state of S is un-
affected by the unitaries uS diagonal in Schmidt basis
acting on S (like changes of Schmidt coefficient phases)
that would normally affect isolated S: The global state
ΨSE is restored by uE. Thus, by fact (3), so is local state

of S. However, this is done by a unitary “countertrans-
formation” acting solely on E. Hence, by fact (1), state
of S must have been unaffected by uS in the first place.
So, by fact (2), phases of ψk cannot change outcomes of
any measurement on S. Equiprobability follows.
One can now derive Born’s rule, pk = |ψk|2, with
straightforward algebra from the above two simple cases
of complete certainty (pk = 1) and equiprobability (pk =
1
n): The general case can be always reduced to the case
case of equal coefficients by “finegraining” (see Box).
The origin of probability is a fascinating problem that
is older than quantum measurement problem, and is for-
gotten primarily because it is so old. We have seen how
quantum physics sheds a new, very fundamental, light
on probability. We cannot do justice to the history of
this subject here, but Ref. [34] provides a basic overview
and exhaustive set of references. In particular, envariant
derivation is very different from the classic proof of Glea-
son [35] in that it sheds light on the physical significance
of the resulting measure. Moreover, it does not assume
probabilities are additive (except to posit that probabil-
ity of an event and its complement are certain, i.e., to
establish normalization; see Box and Ref. [33, 38]). By-
passing additivity of probabilities is essential when deal-
ing with a theory with another principle of additivity
– the quantum superposition principle – which trumps
additivity of probabilities or at least classical intuitiions
about it (e.g., in the double-slit experiment).
Discus-
sion of the implications of envariance has already started,
with [36, 37], and [5] providing insightful commentary.

BOX
We show here how “finegraining” reduces the case of
arbitrary ψk to equiprobability.
To illustrate general
strategy consider state in a 2D Hilbert space HS of S
spanned by orthonormal {|0⟩, |2⟩} and (at least) 3D HE:

|ψSE⟩ ∝
�

2
3 |0⟩S|+⟩E
+
�

1
3 |2⟩S|2⟩E .

The state |+⟩E = |0⟩E+|1⟩E
√

2
exists in (at least 2D) sub-
space of E orthogonal to |2⟩E, i.e., ⟨0|1⟩ = ⟨0|2⟩ = ⟨1|2⟩ =
⟨+|2⟩ = 0. We know we can ignore phases.
To reduce |ψSE⟩ to equal coefficients case we “extend
it” to a state |¯ΨSEC⟩ by letting E act on an ancilla C.
(S is not acted upon, so, by fact (1), probabilities for S
cannot change.) This can be done by a generalization of
controlled-not acting between E (control) and C (target),
so that (in obvious notation) |k⟩|0′⟩ ⇒ |k⟩|k′⟩, leading to

√

2|0⟩|+⟩|0′⟩+|2⟩|2⟩|0′⟩ ⇒
√

2|0⟩ |0⟩|0′⟩+|1⟩|1′⟩
√

2
+|2⟩|2⟩|2′⟩.

Above, and from now on we skip subscripts: The state of
S will be listed first, and the state of C will be primed.
The cancellation of
√

2 yields an equal coefficient state:

|¯ΨSCE⟩ ∝ |0, 0′⟩|0⟩ + |0, 1′⟩|1⟩ + |2, 2′⟩|2⟩ .

We have combined S and C in a single ket and (below)
we shall swap states of SC as if it was a single system.


9

Clearly, this is a Schmidt decomposition of (SC)E.
Three orthonormal product states have coefficients with
the same absolute value.
Therefore, they can be en-
variantly swapped.
Thus, the probabilities of states
|0⟩|0′⟩, |0⟩|1′⟩, and |2⟩|2′⟩ are all equal. By normalization
they are 1

3. So, probability of detecting state |2⟩ of S is
1
3. Moreover, |0⟩ and |2⟩ are the only two outcome states
for S. It follows that probability of |0⟩ must be 2

3;
p0 = 2

3;
p2 = 1

3 .
This is Born’s rule. We have just seen why the amplitudes
in the initial |ψSE⟩ “get squared” to yield probabilities.
Note that we have avoided assuming additivity of prob-
abilities: p0 =
2
3 not because it is a sum of two fine-
grained alternatives for SE, each with probability of 1

3,
but rather because there are only two (mutually exclu-
sive and exhaustive) alternatives for S; |0⟩ and |2⟩, and
p2 = 1

3. Therefore, by normalization, p0 = 1 − 1

3. Prob-
abilities of Schmidt states can be added because of the
loss of phase coherence that follows directly from phase
envariance established earlier (see also Ref. [32, 33]).
Extension of this proof to the case where proba-
bilities are commensurate is conceptually straightfor-
ward but notationally cumbersome.
The case of non-
commensurate probabilities is settled with an appeal to
continuity. Frequency of the outcomes can be also de-
duced, allowing one to establish connection with the fa-
miliar relative frequency approach to probabilities [32,
33, 38], but in a quantum setting probability arises as a
consequence of symmetries of a single entangled state.
We end by noting that the finegraining discussed above
does not need to be carried out experimentally each time
probabilities are discussed: Rather, it is a way to de-
duce a measure that is consistent with the geometry of
the Hilbert spaces using entanglement as a tool. Still,
given fundamental implications of envariance experimen-
tal tests would be most useful.

V.
DISCUSSION

We derived the two controversial quantum postulates
from the first three. We have thus seen how classical do-
main of the Universe arises from the superposition princi-
ple (postulate (i)) and unitarity (postulate (ii)) as well as
rudimentary assumptions about information flows (pos-
tulate (iii)), and a few basic facts about states of com-
posite quantum systems (including their tensor nature,
often cited as additional “axiom (0)”).
The essence of the measurement problem – accounting
for axioms (iv) and (v) – has been largely settled. It is of
course likely one may be able to clarify assumptions and
simplify proofs. Much work remains to be done on Quan-
tum Darwinism and envariance. Nevertheless, nature of
the quantum-classical correspondence has been clarified.
Physicists take it for granted that even hard problems
are solved by a single good idea. Therefore, when a single
idea does not do the whole job, often our first instinct is to
dismiss it. Measurement problem does not fall into this

“single idea” category. Several ideas, applied in the right
order, led to advances described here. Logically, we may
well have started with the derivation of Eq. (5) and the
analysis of quantum jumps. Their randomness leads to
probabilities. And symmetries of entangled states (that
arise in decoherence and Quantum Darwinism) allow one
to derive Born’s rule. As we have seen, phase envariance
is (nearly) “all you need”. With probabilities at hand
one has then every right to use reduced density matrices
to analyze Quantum Darwinism and decoherence.
Our presentation was “historical”. We started with de-
coherence, and used it to introduce Quantum Darwinism.
Analysis of copying essential to information flows in both
of these phenomena led to quantum jumps. This in turn
motivated entangelment-based derivation of Born’s rule.
Quantum Darwinism – upgrade of E to a communication
channel from a mundane role it played in decoherence –
tied together all of the other developments. This order
had the advantage of making motivations clear, but it is
different from more logical presentation where postulates
(i)-(iii) are the starting point (strategy followed in [38]).
The collection of ideas discussed here allows one to un-
derstand how “the classical” emerges from the quantum
substrate staring from more basic assumptions than de-
coherence. We have bypassed a related question of why is
our Universe quantum to the core. The nature of quan-
tum state vectors is a part of this larger mystery. Our
focus was not on what quantum states are, but on what
they do. Our results encourage a view one might describe
(with apologies to Bohr) as “complementary”. Thus, |ψ⟩
is in part information (as, indeed, Bohr thought), but
also the obvious quantum object to explain “existence”.
We have seen how Quantum Darwinism accounts for the
transition from quantum fragility (of information) to the
effectively classical robustness.
One can think of this
transition as “It from bit” of John Wheeler [39].
In the end one might ask: “How Darwinian is Quan-
tum Darwinism?”. Clearly, there is survival of the fittest,
and fitness is defined as in natural selection – through
the ability to procreate. The no-cloning theorem implies
competition for resources – space in E – so that only
pointer states can multiply (at the expense of their com-
plementary competition). There is also another aspect
of this competition: Huge memory available in the Uni-
verse as a whole is nevertheless limited. So the question
arises: What systems get to be “of interest”, and imprint
their state on their obliging environments, and what are
the environments? Moreover, as the Universe has a finite
memory, old events will be eventually “overwritten” by
new ones, so that some of the past will gradually cease
to be reflected in the present record. And if there is no
record of an event, has it really happened? These ques-
tions seem far more interesting than deciding closeness
of the analogy with natural selection [40]. They suggest
one more question: Is Quantum Darwinism (a process of
multiplication of information about certain favored states
that seems to be a “fact of quantum life”) in some way
behind the familiar natural selection? I cannot answer


10

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Acknowledgments:
I am grateful to Robin Blume-
Kohout, Fernando Cucchietti, Juan Pablo Paz, David
Poulin, Hai-Tao Quan, Michael Zwolak for stimulating
discussions. This research was supported by an LDRD
grant at Los Alamos and, in part, by FQXi.