# Quantum Darwinism

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

###### Abstract

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 either
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 11  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 indispensable to interpret or even practice quantum theory. Thus, instead
of trying to understand Universe (including “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
gedankenexperiments like the famous Schrödinger cat 47  were truly disturbing:
Real experiments dealt with electrons, photons, atoms, or other microscopic
systems. Bohr’s rule of thumb – that the macroscopic is classical – was
enough. Moreover, many (including Einstein) believed that quantum 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 gedankenexperiments were carried out. They
confirmed validity of quantum 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, unperturbed 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 36 ; 75 ; 52 : 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 quantum realm.
So, while fundamental problems of classical physics were always solved in
isolation (it sufficed to prevent 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 environment, so its eigenstates are singled
out, and phase relations between such pointer states are lost 69 .

Negative selection due to decoherence is the essence of environment-induced
superselection, or einselection 70 : 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
$\rho_{{\mathcal{S}}}$. It represents the state of the system that obtains
from the composite state ${\Psi_{{\mathcal{S}}{\mathcal{E}}}}$ of
${\mathcal{S}}$ and ${\mathcal{E}}$ by tracing out the environment
${\mathcal{E}}$:

$\rho_{\mathcal{S}}=Tr_{{\mathcal{E}}}|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle\\!\langle\Psi_{{\mathcal{S}}{\mathcal{E}}}|\
.$ $None$

Evolution of $\rho_{\mathcal{S}}$ reveals preferred states: It is most
predictable when the system starts in a pointer state. To quantify this one
can use (von Neumann) entropy,
$H_{{\mathcal{S}}}=H(\rho_{\mathcal{S}})=-Tr\rho_{\mathcal{S}}\lg\rho_{\mathcal{S}}$,
as a function of time. Pointer states result in smallest entropy increase. By
contrast, their superpositions produce entropy rapidly, at decoherence rates,
especially when ${\mathcal{S}}$ is macroscopic.

When pure states of the system are sorted by predictability, according to
entropy of the evolved $\rho_{\mathcal{S}}$, pointer states are at the top.
This criterion – the predictability sieve 75 ; 45 ; 80  – 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 $|\tt dead\rangle$ and $|\tt alive\rangle$ when initiated in a
way envisaged by Schrödinger 47 .

The special role of position is traced to the nature of the
${\mathcal{S}}{\mathcal{E}}$ interactions: They tend to depend on distance.
Hence, information about position is most readily passed on to the
environment. This is why localized states survive while nonlocal
superpositions decay into their mixtures. For example, in a weakly damped
harmonic oscillator the minimum uncertainty wavepackets – familiar coherent
states, best quantum approximation of classical points in phase space – are
einselected 80 ; 30 ; 55 .

##  II Environment as a witness

Monitoring by the environment means that information about ${\mathcal{S}}$ is
deposited in ${\mathcal{E}}$. What role does it play, and what is its fate?
Decoherence theory ignores it. Environment is “traced out”. Information it
contains is treated as inaccessible and irrelevant: ${\mathcal{E}}$ 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 environment. Vast majority of our data comes from fragments
of ${\mathcal{E}}$. 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 scattered by a page. We never access all of
${\mathcal{E}}$. Tiny fractions suffice to reveal the state of various
“systems of interest”.

This insight captures the essence of Quantum Darwinism: Only states that
produce multiple informational offspring – multiple imprints on the
environment – can be found out from small fragments of ${\mathcal{E}}$. 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
${\mathcal{E}}$.

---
Figure 1: _Quantum Darwinism and the structure of the environment_.
Decoherence theory distinguishes between a system ($\mathcal{S}$) and its
environment ($\mathcal{E}$) as in (a), but makes no further recognition of the
structure of ${\mathcal{E}}$; it could as well be monolithic. In Quantum
Darwinism the focus is on redundancy. We recognize the subdivision of
$\mathcal{E}$ into subsystems, as in (b). The only requirement for a subsystem
is that it should be individually accessible to measurements; observables of
different subsystems commute. To obtain information about ${\mathcal{S}}$ from
${\mathcal{E}}$ one can then measure _fragments_ ${\mathcal{F}}$ of the
environment – non-overlapping collections of subsystems of ${\mathcal{E}}$,
(c). ically, there are many copies of the information about ${\mathcal{S}}$ in
${\mathcal{E}}$ – “progeny” of the “fittest observable” that survived
monitoring by ${\mathcal{E}}$ proliferates throughout ${\mathcal{E}}$. 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 ${\mathcal{S}}$.

Proliferation of records allows information about ${\mathcal{S}}$ to be
extracted from many fragments of ${\mathcal{E}}$ (in the example above, photon
${\mathcal{E}}$). Thus, ${\mathcal{E}}$ acquires redundant records of
${\mathcal{S}}$. Now, many observers can find out the state of ${\mathcal{S}}$
independently, and without perturbing it. This is how preferred states of
${\mathcal{S}}$ 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 ${\mathcal{E}}$. Now we ask: What
information about the system can be found out from fragments of
${\mathcal{E}}$? This change of focus calls for a more realistic model of the
environment (Fig. 1): Instead of a monolithic ${\mathcal{E}}$ we recognize
that environments consist of subsystems that comprise fragments independently
accessible to observers.

The reduced density matrix $\rho_{\mathcal{S}}$ representing the state 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
relevant reduced density matrix $\rho_{{\mathcal{S}}{\mathcal{F}}}$ is given
by:

$\rho_{{\mathcal{S}}{\mathcal{F}}}=Tr_{{\mathcal{E}}/{\mathcal{F}}}|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle\\!\langle\Psi_{{\mathcal{S}}{\mathcal{E}}}|\
.$ $None$

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

$I({\mathcal{S}}:{\mathcal{F}})=H_{{\mathcal{S}}}+H_{{\mathcal{F}}}-H_{{\mathcal{S}},{\mathcal{F}}}\
,$ $None$

defined as the difference between entropies of two systems (here
${\mathcal{S}}$ and ${\mathcal{F}}$) treated separately and jointly. For
example, the mutual information between an original and 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
information in the original. However, having extra copies does not increase
the information about the original. Yet, it determines how many can
independently access this information. The number of copies defines
redundancy.

---
Figure 2: _Information about ${\mathcal{S}}$ stored in ${\mathcal{E}}$ and its
redundancy_. Mutual information is monotonic in $f$. When global state of
${\mathcal{S}}{\mathcal{E}}$ is pure, $I({\mathcal{S}}:{\mathcal{F}}_{f})$ in
a typical fraction $f$ of the environment is antisymmetric around $f=0.5$ 8 .
For pure states picked out at random from the combined Hilbert space
${\mathcal{H}}_{{\mathcal{S}}{\mathcal{E}}}$, there is little mutual
information between ${\mathcal{S}}$ and a typical ${\mathcal{F}}$ smaller than
half of ${\mathcal{E}}$. However, once a threshold $f=\frac{1}{2}$ is
attained, nearly all information is in principle at hand. Thus, such random
states (green line) exhibit no redundancy. By contrast, states of
${\mathcal{S}}{\mathcal{E}}$ created by decoherence (where the environment
monitors preferred observable of ${\mathcal{S}}$) contain almost all (all but
$\delta$) of the information about ${\mathcal{S}}$ in small fractions
$f_{\delta}$ of ${\mathcal{E}}$. The corresponding
$I({\mathcal{S}}:{\mathcal{F}}_{f})$ (red line) quickly rises to
$H_{\mathcal{S}}$ (entropy of ${\mathcal{S}}$ due to decoherence), which is
all of the information about ${\mathcal{S}}$ available from either
${\mathcal{E}}$ or ${\mathcal{S}}$. (More, up to $2H_{\mathcal{S}}$, can be
obtained only through global measurements on ${\mathcal{S}}$ and nearly all
${\mathcal{E}}$). $H_{\mathcal{S}}$ is therefore the _classically accessible
information_. As $(1-\delta)H_{\mathcal{S}}$ of information is contained in
$f_{\delta}=1/R_{\delta}$ of ${\mathcal{E}}$, there are $R_{\delta}$ such
fragments in ${\mathcal{E}}$: $R_{\delta}$ is the redundancy of the
information about ${\mathcal{S}}$. Large redundancy 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 Darwinism
accounts for the emergence of objective existence_.

Similar ideas apply to the quantum case. Initially, every bit of information
gained from a fraction $f\ll 1$ of ${\mathcal{E}}$ that was pure before it
monitored (and decohered) the system is a bit about ${\mathcal{S}}$. The red
plot in Fig. 2 starts with this steep “bit for bit” slope, but moderates as
$I({\mathcal{S}}:{\mathcal{F}}_{f})$ approaches redundancy plateau at
$H_{{\mathcal{S}}}$, where additional bits only confirm what is already known.

Redundancy is the number of independent fragments of the environment that
supply almost all classical information about ${\mathcal{S}}$, i.e.,
$(1-\delta)H_{\mathcal{S}}$. In other words;

$R_{\delta}=1/f_{\delta}\ .$ $None$

$R_{\delta}$ is the number of times one can acquire $(1-\delta)$ of the
information about ${\mathcal{S}}$ independently (from distinct
${\mathcal{F}}$’s) and indirectly – without perturbing ${\mathcal{S}}$.

Rapid rise and gradual leveling of $I({\mathcal{S}}:{\mathcal{F}}_{f})$, Fig.
2, implies redundancy. The information in ${\mathcal{F}}_{f}$ allows one to
determine the state of ${\mathcal{S}}$ as it reaches redundancy plateau.
Observables of different ${\mathcal{F}}$’s commute – such measurements are
independent. Yet, underlying correlations mean that their outcomes imply the
same state of the system, as if ${\mathcal{S}}$ were classical: The redundancy
plateau is a classical plateau. Its level $H_{{\mathcal{S}}}$ is the classical
information accessible from a small fraction of ${\mathcal{E}}$.

Redundancy allows for objective existence of the state of ${\mathcal{S}}$: It
can be found out indirectly, so there is no danger of perturbing
${\mathcal{S}}$ with a measurement. Error correction allowed by redundancy is
also important: Fragility of quantum states means that copies in
${\mathcal{F}}$’s are damaged by measurements (we destroy photons!), and may
be measured in a “wrong” basis. One cannot access records in ${\mathcal{E}}$
without endangering their existence. But with many ($R_{\delta}$) copies,
state of ${\mathcal{S}}$ can be found out by $\sim R_{\delta}$ observers who
can get their information independently, and without prior knowledge about
${\mathcal{S}}$. Consensus between copies suggests objective existence of the
state of ${\mathcal{S}}$.

The mutual information $I({\mathcal{S}}:{\mathcal{F}}_{f})$ computed in models
of decoherence exhibits behavior illustrated by the red plot of Fig. 2. In the
family of models representing spin ${\mathcal{S}}$ surrounded by environments
of many spins 42 ; 8 ; 9  the same number of spins suffices to reach the
plateau: Adding more spins to ${\mathcal{E}}$ only extends length of the
plateau measured in “absolute units” – in the number of the environment 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 surrounded by many environmental
oscillators – is the other well known model of decoherence. It is exactly
solvable, and the case of an underdamped oscillator yields surprisingly simple
results 10 ; PR : (i) Mutual information is approximately given by
$I({\mathcal{S}}:{\mathcal{F}})\approx
H_{{\mathcal{S}}}+\frac{1}{2}\ln{\frac{f}{(1-f)}}$, and; (ii) Redundancy for
an initially squeezed state of ${\mathcal{S}}$ reaches $R_{\delta}\approx
s^{2\delta}$, where $s$, the squeeze factor, quantifies delocalization of the
state. Similar equation should hold for more general “Schrödinger cat” states,
with $s$ quantifying the separation of the two localized alternatives.

These results confirm intuitions that originally motivated Quantum Darwinism
75 ; Z2000 : Monitoring of the system by the environment can deposit multiple
records of preferred states of ${\mathcal{S}}$ in ${\mathcal{E}}$. States of
${\mathcal{S}}{\mathcal{E}}$ that arise from decoherence are special 8 ; 9 ,
as $I({\mathcal{S}}:{\mathcal{F}}_{f})$ for a typical pure state selected with
Haar measure in the whole Hilbert space of ${\mathcal{S}}{\mathcal{E}}$ (green
plot in Fig. 2) shows. In such random states small fragments reveal almost
nothing about the rest of the state. Only when half of ${\mathcal{E}}$ 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 originated –
can be deduced from its fragment. We shall see how such branches grow in the
next section.

Plots of $I({\mathcal{S}}:{\mathcal{F}}_{f})$ for pure
${\mathcal{S}}{\mathcal{E}}$ are antisymmetric around the point
$\\{H_{{\mathcal{S}}},f=\frac{1}{2}\\}$ for typical fragments of
${\mathcal{E}}$ 8 . Thus, rapid rise for small $f$ must be matched at the
other end, for $f\sim 1$. This is a signature of entanglement that allows
state to be known “as the whole”, while states of subsystems are unknown. The
joint state of ${\mathcal{S}}{\mathcal{E}}$ is then pure, so that
$H_{{\mathcal{S}},{\mathcal{F}}={\mathcal{E}}}=0$, and
$I({\mathcal{S}}:{\mathcal{F}}_{f})$ must rise to
$H_{{\mathcal{S}}}+H_{{\mathcal{E}}}=2H_{{\mathcal{S}}}$ when $f$ approaches
1.

This is a very quantum aspect of information. In classical 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 12  that
relates probabilities with wavefunctions.

To reveal this latent quantumness one would have to measure the right global
observable on all of ${\mathcal{S}}{\mathcal{E}}$. For example, when mutual
information, Eq. (3), is defined using Shannon entropy with probabilities
corresponding to optimal observables in ${\mathcal{S}}$ and in
${\mathcal{E}}$, the resulting Shannon $I({\mathcal{S}}:{\mathcal{F}}_{f})$
graph for small $f$ would look very similar to Fig. 2. However, using Shannon
entropy involves local probabilities (precluding global observables), so such
Shannon $I({\mathcal{S}}:{\mathcal{F}}_{f})$ never exceeds
$H_{{\mathcal{S}}}$, antisymmetry is lost, and the plateau continues until the
end at $f\sim 1$.

Effective unattainability of the $f\sim 1$ part of the plot also shows why
decoherence is so hard to undo: Correlations that reveal coherence can be
usually detected only by such global measurements of whole
${\mathcal{S}}{\mathcal{E}}$. We intercept small fractions of ${\mathcal{E}}$,
and never have the luxury of perfect global measurements needed to undo
decoherence. Yet, because of redundancy, we get $\sim H_{{\mathcal{S}}}$
information with “sloppy” measurements of $f\ll 1$.

Quantum Darwinism does not require pure ${\mathcal{E}}$. Mixed environment is
a noisy communication channel: Its initial entropy of $h$ per bit can still
increase after interaction with ${\mathcal{S}}$, reflecting mutual information
buildup. However, now a bit gained from ${\mathcal{E}}$ yields only $1-h$ of a
bit about ${\mathcal{S}}$. So, a completely mixed ${\mathcal{E}}$ ($h=1$) is
useless (even though it can still induce decoherence!). For a partly mixed
${\mathcal{E}}$ mutual information will increase more slowly, pure case “bit
per bit” rate tempered to $\sim 1-h$. Yet, it can still climb the same
redundancy plateau at $H_{{\mathcal{S}}}$ QZZ .

These conclusions apply when ${\mathcal{E}}$ is initially mixed, but are also
relevant when this channel is noisy for other reasons (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 ${\mathcal{S}}$.

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 ${\mathcal{E}}$. Photons are ideal in this respect:
They interact with various systems, but, in effect, do not interact with each
other. This is why light delivers 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 scatter from one
another, so that whatever record they may have gathered becomes effectively
undecipherable.

Stability of the level of the redundancy plateau at $H_{{\mathcal{S}}}$, even
for mixed ${\mathcal{E}}$’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 environment know about the system, and why?

##  III From Copying to Quantum Jumps

Quantum Darwinism leads to appearance, in the environment, of multiple copies
of the state of the system. However, the no-cloning theorem 24 ; 65  prohibits
copying of unknown quantum states. If cloning is outlawed, how can redundancy
seen in Fig. 2 be possible?

Figure 3: Quantum Darwinism in a simple model of decoherence 42 . The
spin-$\frac{1}{2}$ ${\mathcal{S}}$ interacts with $N=50$ spin-$\frac{1}{2}$
subsystems of ${\mathcal{E}}$ with an Ising Hamiltonian ${\bf
H_{{\mathcal{S}}{\mathcal{E}}}}=\sum_{k=1}^{N}g_{k}\sigma_{z}^{{\mathcal{S}}}\otimes\sigma_{y}^{{\mathcal{E}}_{k}}$.
The initial state of $\mathcal{S}\otimes{\mathcal{E}}$ is
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes|0\rangle^{{\mathcal{E}}_{1}}\otimes\ldots\otimes|0\rangle^{{\mathcal{E}}_{N}}$.
Couplings $g_{k}$ are distributed randomly in the interval (0,1]. All the
plotted quantities are a function of the observable
$\sigma(\mu)=\cos(\mu)\sigma_{z}+\sin(\mu)\sigma_{x}$, where $\mu$ is the
angle between its eigenstates and the pointer states of ${\mathcal{S}}$ –
eigenstates of $\sigma_{z}^{\mathcal{S}}$. a) Information acquired by the
optimal measurement on the whole environment, $\hat{I}_{N}(\sigma)$, as a
function of the inferred observable $\sigma(\mu)$ and the average interaction
action $\langle g_{k}t\rangle=a$. A lot of information is accessible in the
whole ${\mathcal{E}}$ about any observable $\sigma(\mu)$ except when $a$ is so
small that there was no decoherence. b) Redundancy of the information about
${\mathcal{S}}$ as a function of the inferred observable $\sigma(\mu)$ and the
average action $\langle g_{k}t\rangle=a$. $R_{\delta=0.1}(\sigma)$ counts the
number of times 90% of the total information can be “read off” independently
by measuring distinct fragments of ${\mathcal{E}}$. 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
$\sigma(\mu)$ inside the theoretical bound (see Ref.42 ) indicated by the
dashed line. c) Information about $\sigma(\mu)$ 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 $\sigma(\mu)$ is
restricted to what can be inferred from the pointer observable 42 .

Quick answer is that cloning refers to (unknown) quantum states. So, copying
of observables evades the theorem. 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 superposition principle, accounts
for quantum jumps, and suggests 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” postulates 23 . 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 difficult 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 experiment, 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 prevents 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, $p_{k}=|\psi_{k}|^{2}$, is known as Born’s rule 12 . It
completes the relation between mathematics of (i) and (ii) and the
experiments.

Bohr bypassed conflict of (i) and (ii) with (iv) by insisting that apparatus
is classical, so unitarity and the principle 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 previous 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 ${\mathcal{S}}$ (say, $|u\rangle$ and $|v\rangle$)
that produce an imprint in a subsystem of ${\mathcal{E}}$ (which plays a role
of an apparatus), but remain unperturbed (so they can produce more imprints).
This repeatability implies:
$|u\rangle|e_{0}\rangle\Rightarrow|u\rangle|e_{u}\rangle$,
$|v\rangle|e_{0}\rangle\Rightarrow|v\rangle|e_{v}\rangle$ in obvious notation.
In a unitary process scalar product is preserved. Thus;

$\langle u|v\rangle=\langle u|v\rangle\langle e_{u}|e_{v}\rangle\ ,$ $None$

where we have set $\langle e_{0}|e_{0}\rangle=1$. This simple equation can be
satisfied only when; (a) either $\langle e_{u}|e_{v}\rangle=1$ (which means
that copying was completely unsuccessful), or; (b) $\langle u|v\rangle=0$,
i.e., they are orthogonal. In that case $\langle e_{u}|e_{v}\rangle$ is
arbitrary – perfect record $\langle e_{u}|e_{v}\rangle=0$ is also possible.

It follows that multiple (perfect or imperfect) copies of $|u\rangle$ and
$|v\rangle$ can be imprinted in disjoint ${\mathcal{F}}$’s. As a consequence
of unitarity, only sets of orthogonal states (that define Hermitean
observables 23 ) can be so copied, explaining selection of a set of outcomes –
terminal points of quantum jumps 79 . 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 repeatability 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 theory 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 ${\mathcal{E}}$,
multiple copies of the original in ${\mathcal{S}}$. Interaction Hamiltonians
do not perturb 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 ${\mathcal{S}}$ and
${\mathcal{F}}$, i.e., using probabilities of their eigenstates. While von
Neumann-based $I({\mathcal{S}}:{\mathcal{F}}_{f})$ and $R_{\delta}$
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
${\mathcal{S}}$, 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 $\sigma(\mu)$ and the preferred pointer observable $\sigma_{z}$
(that remains 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 42 ; 43  that only pointer
states can be discovered from fractions of ${\mathcal{E}}$. Other observables
(such as $\sigma(\mu)$ 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 happens at the expense of their complements.

Structure of branching state betrays its origin and foreshadows “collapse”.
Starting from
$|\psi_{{\mathcal{S}}}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle$,

$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle|e^{(1)}_{k}\rangle\dots|e^{({\mathcal{N}})}_{k}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle|\varepsilon_{k}\rangle\
\ (6)$

branches grow to include $\cal N$ subsystems of ${\mathcal{E}}$. Branch
fragments can be nearly orthogonal; $\Pi_{j=1}^{J}\langle
e_{k}^{(j)}|e_{k^{\prime}}^{(j)}\rangle\simeq\delta_{kk^{\prime}}$ for large
enough $J$. This means that a pointer state $|s_{k}\rangle$ of ${\mathcal{S}}$
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,
$J\ll\cal N$).

In the huge Hilbert space ${\mathcal{H}}_{{\mathcal{S}}{\mathcal{E}}}$
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 ${\mathcal{H}}_{{\mathcal{S}}{\mathcal{E}}}$
that exceeds $n$ by a factor exponential in ${\mathcal{N}}$. This is why the
two plots in Fig. 2 are so different: Branching state is, to a good
approximation, a multi-system Schmidt decomposition, with long branch
fragments constituting “systems”. In a Schmidt decomposition, states of
partners are in one-to-one correspondence. Thus, in Eq. (6), $|s_{k}\rangle$
implies $|\varepsilon_{k}\rangle$ (and, vice versa), and measuring a branch
fragment ${\mathcal{F}}$ can reveal the whole branch.

Initial part of $I({\mathcal{S}}:{\mathcal{F}}_{f})$, 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
$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$ leaves no doubt of what these
branches are. Using Born’s rule one could assign to them probabilities
$p_{k}=|\psi_{k}|^{2}$ and the corresponding entropy $H_{{\mathcal{S}}}$. Next
section shows how one can deduce these probabilities without axiom (v) – how
symmetries of entanglement imply Born’s rule.

When observer measures enough of ${\mathcal{E}}$, he finds out the branch (and
what the state of ${\mathcal{S}}$ is). Additional data are redundant. They
only confirm what is already known. Probabilities associated with
$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$ are replaced with certainty of a
branch. This transition from uncertainty (initial presence of many branches –
potential for multiple outcomes) to certainty (once a sufficiently long branch
fragment becomes known) accounts for perception of “collapse”. The initial,
steeply rising, part of $I({\mathcal{S}}:{\mathcal{F}}_{f})$ “resolves” it:
Collapse is brief compared to the ensuing period of certainty about the
outcome, as $f_{\delta}\ll 1$, but, nevertheless, not instantaneous.

Assumptions that lead from copying to preferred states can be relaxed. Thus,
${\mathcal{E}}$ need not be initially pure 79 . Moreover, it suffices that the
records (e.g., in the apparatus ${\cal A}$) are “repeatably accessible”.
Transfer of responsibility for repeatability from a quantum ${\mathcal{S}}$ to
a (still quantum) ${\cal A}$ allows one to model non-orthogonal measurement
outcomes (POVM’s): ${\cal A}$ entangles with the system, and then acts as
ancilla. Its orthogonal pointer states $|A_{k}\rangle$ correlate with non-
orthogonal $|\varsigma_{k}\rangle$ of ${{\mathcal{S}}}$,
$\sum_{k}\tilde{\psi}_{k}|\varsigma_{k}\rangle|A_{k}\rangle$. Interaction of
$\cal A$ with the environment results in multiple copies of $|A_{k}\rangle$.
The usual projective measurement implementation of POVM’s (see e.g. NC ) is
now straightforward. Branches are labelled by $|A_{k}\rangle$. 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 11  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 designate possible future outcomes, precluding measurements of
complementary observables or determining preexisting state of the system.
Thus, information acquisition – a copying process – results in preferred
states.

Consensus between records deposited in fragments of ${\mathcal{E}}$ 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
25 ; 26 ; 22  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 information, too fragile for objective existence. And there is only one
copy of the Universe. Treating its state as if it really existed 25 ; 26 ; 22
seems unwarranted and “classical”.

##  IV Probabilities from Entanglement

---
Figure 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
$p_{\spadesuit}={\frac{1}{2}}$ 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,
$p_{\heartsuit}=p_{\diamondsuit}$, and $p_{\spadesuit}=p_{\clubsuit}$. After a
swap
$u_{{\mathcal{S}}}=|\spadesuit\rangle\langle\heartsuit|+|\heartsuit\rangle\langle\spadesuit|$
in ${\mathcal{S}}$, the resulting state
$|\spadesuit\rangle|\diamondsuit\rangle+|\heartsuit\rangle|\clubsuit\rangle$
must have $p^{\prime}_{\spadesuit}=p_{\diamondsuit}$, and
$p^{\prime}_{\heartsuit}=p_{\clubsuit}$. (We ‘primed’ probabilities in
${\mathcal{S}}$, as it was acted upon by a swap, so they might have changed.)
A counterswap
$u_{{\mathcal{E}}}=|\diamondsuit\rangle\\!\langle\clubsuit|+|\clubsuit\rangle\\!\langle\diamondsuit|$
in ${\mathcal{E}}$ restores the original entangled state, proving that
$p^{\prime}_{\heartsuit}=p_{\heartsuit}$ and
$p^{\prime}_{\spadesuit}=p_{\spadesuit}$, after all (as counterswap
$u_{{\mathcal{E}}}$ leaves ${\mathcal{S}}$ untouched). This sequence of
equalities implies $p_{\spadesuit}=p_{\diamondsuit}=p_{\heartsuit}$, so that
$p_{\spadesuit}=p_{\heartsuit}=\frac{1}{2}$, as probabilities in
${\mathcal{S}}$ must add up to 1.

Observer prepared ${\mathcal{S}}$ in a state $|{\psi}_{{\mathcal{S}}}\rangle$,
but wants to measure observable with eigenstates $\\{|s_{k}\rangle\\}$. This
will lead to entangled $|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$ with branch
structure, Eq. (6). Pointer states $\\{|s_{k}\rangle\\}$ define the outcomes,
but, as yet, observer has not measured ${\mathcal{E}}$, and does not know the
result. Given $|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$, what is the
probability of, say, $|s_{17}\rangle$?

To derive it we cannot use reduced density matrices, Eqs. (1,2). Tracing out
is averaging Landau ; 59 ; NC  – it relies on $p_{k}=|\psi_{k}|^{2}$, 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
$\\{|s_{k}\rangle\\}$ includes $|{\psi}_{{\mathcal{S}}}\rangle$ (e.g.,
$|{\psi}_{{\mathcal{S}}}\rangle=|s_{17}\rangle$), newly added copies just
extend the branch already correlated with observer’s state, and the outcome is
certain; $p_{17}=1$. Certainty of correlations between partners in Schmidt
decomposition, Eq. (6) is another important example.

Certainty seems trivial but is important. Confirmation that a state “is what
it is” – postulate (iii) – is a part of standard quantum lore 23 . We re-
affirmed it, but with a key insight: Redundancy allows observers to discover
(and not just confirm) that ${\mathcal{S}}$ is in a certain pointer state.

We now turn to the opposite case of complete indeterminacy. Its connection
with symmetry was noted by Laplace. He wrote: “The theory of chance consists
in reducing 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 ” 40 . 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
$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle|\varepsilon_{k}\rangle$
cannot influence the state of ${\mathcal{S}}$: It is induced by
$u_{{\mathcal{S}}}=e^{i\phi_{k}}|s_{k}\rangle\\!\langle s_{k}|$, local unitary
on ${\mathcal{S}}$, that can be “undone” by
$u_{{\mathcal{E}}}=e^{-i\phi_{k}}|\varepsilon_{k}\rangle\\!\langle\varepsilon_{k}|$
on ${\mathcal{E}}$, or;

$u_{{\mathcal{S}}}\otimes{\bf
1}_{{\mathcal{E}}}|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle=|\Phi_{{\mathcal{S}}{\mathcal{E}}}\rangle;\
{\bf 1}_{{\mathcal{S}}}\otimes
u_{{\mathcal{E}}}|\Phi_{{\mathcal{S}}{\mathcal{E}}}\rangle=|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$
$None$

So, phases of $\psi_{k}$ cannot matter for a local state or influence
probabilities in ${\mathcal{S}}$. This symmetry, Eq. (7), is the entanglement-
assisted invariance or envariance 76 ; 78 .

Such loss of phase significance for ${\mathcal{S}}$ entangled with
${\mathcal{E}}$ implies decoherence 78 . We arrived at its essence using
envariance, without reduced density matrices, trace, etc.

We now use phase envariance to show that equal absolute values of the
coefficients $\psi_{k}$ imply equal probabilities. For equal $|\psi_{k}|$ any
orthogonal basis of ${\mathcal{S}}$ is “Schmidt” (i.e., has an orthogonal
partner in ${\mathcal{E}}$). Thus,
$|\bar{\varphi}_{{\mathcal{S}}{\mathcal{E}}}\rangle=\frac{{|0\rangle}_{{\mathcal{S}}}{|0\rangle}_{{\mathcal{E}}}+{|1\rangle}_{{\mathcal{S}}}{|1\rangle}_{{\mathcal{E}}}}{\sqrt{2}}=\frac{{|+\rangle}_{{\mathcal{S}}}{|+\rangle}_{{\mathcal{E}}}+{|-\rangle}_{{\mathcal{S}}}{|-\rangle}_{{\mathcal{E}}}}{\sqrt{2}}$,
where $|\pm\rangle=\frac{|0\rangle\pm|1\rangle}{\sqrt{2}}$. Sign change
induced by $e^{i\pi}|-\rangle\\!\langle-|$ acting on ${\mathcal{S}}$ produces
$|\bar{\eta}_{{\mathcal{S}}{\mathcal{E}}}\rangle=\frac{{|+\rangle}_{{\mathcal{S}}}{|+\rangle}_{{\mathcal{E}}}-{|-\rangle}_{{\mathcal{S}}}{|-\rangle}_{{\mathcal{E}}}}{\sqrt{2}}=\frac{{|1\rangle}_{{\mathcal{S}}}{|0\rangle}_{{\mathcal{E}}}+{|0\rangle}_{{\mathcal{S}}}{|1\rangle}_{{\mathcal{E}}}}{\sqrt{2}}$.
In other words, one can swap ${|0\rangle}_{{\mathcal{S}}}$ with
${|1\rangle}_{{\mathcal{S}}}$ by rotating phase in a $|\pm\rangle$ basis by
$\pi$. Yet, we just saw that phases of Schmidt coefficients do not matter for
the state of ${\mathcal{S}}$, so probabilities of 0 and 1 in ${\mathcal{S}}$
must have remained the same. Moreover, probabilities of paired up Schmidt
states are equal, so that $p_{{\mathcal{S}}}(0)=p_{{\mathcal{E}}}(0)$ in
$|\bar{\varphi}_{{\mathcal{S}}{\mathcal{E}}}\rangle$ and
$p_{{\mathcal{S}}}(1)=p_{{\mathcal{E}}}(0)$ in
$|\bar{\eta}_{{\mathcal{S}}{\mathcal{E}}}\rangle$. Hence,
$p_{{\mathcal{S}}}(0)=p_{{\mathcal{S}}}(1)=\frac{1}{2}$, where we assumed that
probabilities add up to 1.

In contrast to Laplace’s subjective “ignorance-based” approach, we obtained
objective probabilities for a completely known entangled state. Phase
envariance implied equiprobability in ${\mathcal{S}}$. 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 applicability 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 structure 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
${\mathcal{S}}$ alone, so; (2) Predictions regarding measurement outcomes on
${\mathcal{S}}$ (including their probabilities) can be inferred from the state
of ${\mathcal{S}}$. Last not least; (3) When ${\mathcal{S}}$ is entangled with
other systems (e.g., the environment) the state of ${\mathcal{S}}$ alone is
determined by the state of the whole ${\mathcal{S}}{\mathcal{E}}$.

These “facts of life” are accepted properties of systems and states, but given
the fundamental nature of our discussion it seems a good idea to make them
explicit 78 .

For instance, to establish independence from phases of the coefficients
$\psi_{k}$ we noted that the state of ${\mathcal{S}}$ is unaffected by the
unitaries $u_{{\mathcal{S}}}$ diagonal in Schmidt basis acting on
${\mathcal{S}}$ (like changes of Schmidt coefficient phases) that would
normally affect isolated ${\mathcal{S}}$: The global state
$\Psi_{{\mathcal{S}}{\mathcal{E}}}$ is restored by $u_{{\mathcal{E}}}$. Thus,
by fact (3), so is local state of ${\mathcal{S}}$. However, this is done by a
unitary “countertransformation” acting solely on ${\mathcal{E}}$. Hence, by
fact (1), state of ${\mathcal{S}}$ must have been unaffected by
$u_{{\mathcal{S}}}$ in the first place. So, by fact (2), phases of $\psi_{k}$
cannot change outcomes of any measurement on ${\mathcal{S}}$. Equiprobability
follows.

One can now derive Born’s rule, $p_{k}=|\psi_{k}|^{2}$, with straightforward
algebra from the above two simple cases of complete certainty ($p_{k}=1$) and
equiprobability ($p_{k}=\frac{1}{n}$): 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 forgotten 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. Aul
provides a basic overview and exhaustive set of references. In particular,
envariant derivation is very different from the classic proof of Gleason Gle
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
probability of an event and its complement are certain, i.e., to establish
normalization; see Box and Ref. 78 ; Z07 ). Bypassing additivity of
probabilities is essential when dealing 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). Discussion of the implications of envariance has
already started, with SF ; Bar , and 52  providing insightful commentary.

BOX

We show here how “finegraining” reduces the case of arbitrary $\psi_{k}$ to
equiprobability. To illustrate general strategy consider state in a 2D Hilbert
space ${\cal H}_{{\mathcal{S}}}$ of ${\mathcal{S}}$ spanned by orthonormal
$\\{|0\rangle,|2\rangle\\}$ and (at least) 3D ${\cal H}_{{\mathcal{E}}}$:

$\ \ \ \ \ \ |\psi_{\cal SE}\rangle\ \propto\
\sqrt{\frac{2}{3}}~{}|0\rangle_{{\mathcal{S}}}|+\rangle_{{\mathcal{E}}}\ \ +\
\ \sqrt{\frac{1}{3}}~{}|2\rangle_{{\mathcal{S}}}|2\rangle_{{\mathcal{E}}}\ .$

The state
$|+\rangle_{{\mathcal{E}}}=\frac{|0\rangle_{{\mathcal{E}}}+|1\rangle_{{\mathcal{E}}}}{\sqrt{2}}$
exists in (at least 2D) subspace of ${\cal E}$ orthogonal to
$|2\rangle_{{\mathcal{E}}}$, i.e., $\langle 0|1\rangle=\langle
0|2\rangle=\langle 1|2\rangle=\langle+|2\rangle=0$. We know we can ignore
phases.

To reduce $|\psi_{\cal SE}\rangle$ to equal coefficients case we “extend it”
to a state $|\bar{\Psi}_{\cal SEC}\rangle$ by letting ${\mathcal{E}}$ act on
an ancilla ${\cal C}$. (${\mathcal{S}}$ is not acted upon, so, by fact (1),
probabilities for ${\mathcal{S}}$ cannot change.) This can be done by a
generalization of controlled-not acting between ${\cal E}$ (control) and
${\cal C}$ (target), so that (in obvious notation)
$|k\rangle|0^{\prime}\rangle\Rightarrow|k\rangle|k^{\prime}\rangle$, leading
to

$\sqrt{2}|0\rangle|+\rangle|0^{\prime}\rangle+|2\rangle|2\rangle|0^{\prime}\rangle\Rightarrow\sqrt{2}|0\rangle{{|0\rangle|0^{\prime}\rangle+|1\rangle|1^{\prime}\rangle}\over\sqrt{2}}+|2\rangle|2\rangle|2^{\prime}\rangle.$

Above, and from now on we skip subscripts: The state of ${\cal S}$ will be
listed first, and the state of ${\cal C}$ will be primed.

The cancellation of $\sqrt{2}$ yields an equal coefficient state:

$|\bar{\Psi}_{\cal
SCE}\rangle\propto|0,0^{\prime}\rangle|0\rangle+|0,1^{\prime}\rangle|1\rangle+|2,2^{\prime}\rangle|2\rangle\
.$

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

Clearly, this is a Schmidt decomposition of (${\mathcal{S}}\cal C){\cal E}$.
Three orthonormal product states have coefficients with the same absolute
value. Therefore, they can be envariantly swapped. Thus, the probabilities of
states $|0\rangle|0^{\prime}\rangle,\ |0\rangle|1^{\prime}\rangle,$ and
$|2\rangle|2^{\prime}\rangle$ are all equal. By normalization they are
$\frac{1}{3}$. So, probability of detecting state $|2\rangle$ of
${\mathcal{S}}$ is $\frac{1}{3}$. Moreover, $|0\rangle$ and $|2\rangle$ are
the only two outcome states for ${\mathcal{S}}$. It follows that probability
of $|0\rangle$ must be $\frac{2}{3}$;

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p_{0}=\frac{2}{3};\ \
p_{2}=\frac{1}{3}\ .$

This is Born’s rule. We have just seen why the amplitudes in the initial
$|\psi_{\cal SE}\rangle$ “get squared” to yield probabilities.

Note that we have avoided assuming additivity of probabilities:
$p_{0}=\frac{2}{3}$ not because it is a sum of two fine-grained alternatives
for ${\mathcal{S}}{\mathcal{E}}$, each with probability of $\frac{1}{3}$, but
rather because there are only two (mutually exclusive and exhaustive)
alternatives for ${\mathcal{S}}$; $|0\rangle$ and $|2\rangle$, and
$p_{2}=\frac{1}{3}$. Therefore, by normalization, $p_{0}=1-\frac{1}{3}$.
Probabilities of Schmidt states can be added because of the loss of phase
coherence that follows directly from phase envariance established earlier (see
also Ref. 76 ; 78 ).

Extension of this proof to the case where probabilities are commensurate is
conceptually straightforward but notationally cumbersome. The case of non-
commensurate probabilities is settled with an appeal to continuity. Frequency
of the outcomes can be also deduced, allowing one to establish connection with
the familiar relative frequency approach to probabilities 76 ; 78 ; Z07 , 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 deduce a measure that is consistent with the geometry of the
Hilbert spaces using entanglement as a tool. Still, given fundamental
implications of envariance experimental tests would be most useful.

##  V Discussion

We derived the two controversial quantum postulates from the first three. We
have thus seen how classical domain of the Universe arises from the
superposition principle (postulate (i)) and unitarity (postulate (ii)) as well
as rudimentary assumptions about information flows (postulate (iii)), and a
few basic facts about states of composite 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 Quantum
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 decoherence, 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
${\mathcal{E}}$ 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 Z07 ).

The collection of ideas discussed here allows one to understand how “the
classical” emerges from the quantum substrate staring from more basic
assumptions than decoherence. We have bypassed a related question of why is
our Universe quantum to the core. The nature of quantum 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, $|\psi\rangle$ 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 JAW .

In the end one might ask: “How Darwinian is Quantum 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 ${\mathcal{E}}$ – so that only pointer
states can multiply (at the expense of their complementary competition). There
is also another aspect of this competition: Huge memory available in the
Universe 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
questions seem far more interesting than deciding closeness of the analogy
with natural selection Darwin . 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 this question, but neither can I
resist raising it.

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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.