Antigravity Agent
388 KiB
388 KiB
Quantum Darwinism
Wojciech Hubert Zurek
Theory Division, MS B213, LANL Los Alamos, NM, 87545, U.S.A.
QuantumDarwinismdescribestheproliferation, intheenvironment, ofmultiplerecordsofselected
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 I. DECOHERENCEANDEINSELECTION
any combination of quantum states is also a legal state.
This seems to be in conflict with everyday reality: States Decoherence turns one of the two problems we noted
we encounter are localized. Classical objects can be ei- above–fragility of quantum states – into a solution of the
ther here or there, but never both here and there. Yet, the other. Environment-induced decoherence recognizes that
principle of superposition says that localization should be if a measurement can put a state at risk and re-prepare
a rare exception and not a rule for quantum systems. it, so can accidental information transfers that happen
Fragility of states is the second problem with quantum- whenever a system interacts with its environment.
classical correspondence: Upon measurement, a general Decoherence is by now well understood [3, 4, 5]:
preexisting quantum state is erased – it “collapses” into Fragility of states makes quantum systems very difficult
an eigenstate of the measured observable. How is it then to isolate. Transfer of information (which has no effect on
possible that objects we deal with can be safely observed, classical states) has dramatic consequences in the quan-
even though their basic building blocks are quantum? tum realm. So, while fundamental problems of classical
To bypass these obstacles Bohr [1] followed Alexander physics were always solved in isolation (it sufficed to pre-
the Great’s example: Rather than try disentangling the vent energy loss) this is not so in quantum physics (leaks
Gordian Knot at the beginning of his conquest, he cut of information are much harder to plug).
it. The cut separates the quantum from the classical. Whenaquantumsystemgivesupinformation, its own
Bohr’s Universe consists of two realms, each governed by state becomes consistent with the information that was
its own laws. Fragile superpositions were banished from disseminated. “Collapse” in measurements is an extreme
the classical realm deemed more fundamental and indis- example, but any interaction that leads to a correlation
pensable to interpret or even practice quantum theory. can contribute to such re-preparation: Interactions that
Thus, instead of trying to understand Universe (includ- depend on a certain observable correlate it with the en-
ing “the classical”) in quantum terms one “quantized” vironment, so its eigenstates are singled out, and phase
this and that, always starting from the classical base. relations between such pointer states are lost [6].
Negative selection due to decoherence is the essence of
This was a brilliant tactical move: Physicists could environment-induced superselection, or einselection [7]:
conquerthequantumrealmwithoutgettingdistractedby Under scrutiny of the environment, only pointer states
interpretational worries. In those days only gedankenex- remain unchanged. Other states decohere into mixtures
periments like the famous Schr¨odinger cat [2] were truly of stable pointer states that can persist, and, in this sense,
disturbing: Real experiments dealt with electrons, pho- exist: They are einselected.
tons, atoms, or other microscopic systems. Bohr’s rule of These ideas can be made precise. The basic tool is the
arXiv:0903.5082v1 [quant-ph] 29 Mar 2009thumb – that the macroscopic is classical – was enough.reduced density matrix ρS. It represents the state of the
Moreover, many (including Einstein) believed that quan- system that obtains from the composite state ΨSE of S
tum physics is just a step on a way to a deeper theory and E by tracing out the environment E:
that will solve or bypass interpretational conundrums. ρ =Tr |Ψ ihΨ | . (1)
That did not happen. Instead, old gedankenexperi- S E SE SE
ments were carried out. They confirmed validity of quan- Evolution of ρS reveals preferred states: It is most pre-
tumlawsonscales that have, of recent, begun to infringe dictable when the system starts in a pointer state. To
on “the macroscopic”. Quantum theory is here to stay. quantify this one can use (von Neumann) entropy, H =
S
It is also increasingly clear that its weirdest predictions H(ρ ) = −Trρ lgρ , as a function of time. Pointer
S S S
– superpositions and entanglement – are experimental states result in smallest entropy increase. By contrast,
facts, in principle relevant also for macroscopic objects. their superpositions produce entropy rapidly, at decoher-
Therefore, questions about the origin of “the classical”, ence rates, especially when S is macroscopic.
with its restriction to localized states that are robust, un- When pure states of the system are sorted by pre-
perturbed by measurements, can no longer be dismissed. dictability, according to entropy of the evolved ρS,
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 |deadi and |alivei
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. ENVIRONMENTASAWITNESS
Monitoring by the environment means that informa- � - + , & # " . / 0 1 2
tion about S is deposited in E. What role does it play, 3 4 5 # " 6 . . 0 6 6 .
and what is its fate? Decoherence theory ignores it. En- . / 7 8 9 0 1 : 7 ; / 1 . <
vironment is “traced out”. Information it contains is FIG. 1: Quantum Darwinism and the structure of the envi-
treated as inaccessible and irrelevant: E is a “rug to sweep ronment. Decoherence theory distinguishes between a system
under” the data that might endanger classicality. (S) and its environment (E) as in (a), but makes no further
Quantum Darwinism recognizes that “tracing out” is recognition of the structure of E; it could as well be mono-
not what we do: Observers eavesdrop on the environ- lithic. In Quantum Darwinism the focus is on redundancy.
ment. Vast majority of our data comes from fragments Werecognize the subdivision of E into subsystems, as in (b).
of E. Environment is a witness to the state of the system. The only requirement for a subsystem is that it should be
For example, this very moment you intercept a fraction individually accessible to measurements; observables of dif-
of the photon environment emitted by a screen or scat- ferent subsystems commute. To obtain information about S
tered by a page. We never access all of E. Tiny fractions from E one can then measure fragments F of the environ-
sufficetoreveal the state of various “systems of interest”. ment – non-overlapping collections of subsystems of E, (c).
This insight captures the essence of Quantum Darwin- ically, there are many copies of the information about S in E
ism: Only states that produce multiple informational off- – “progeny” of the “fittest observable” that survived monitor-
spring – multiple imprints on the environment – can be ing by E proliferates throughout E. This proliferation of the
multiple informational offspring defines Quantum Darwinism.
found out from small fragments of E. The origin of the Theenvironment becomes a witness with redundant copies of
emergentclassicality is then not just survival of the fittest information about the preferred observable. This leads to the
states (the idea already captured by einselection), but objective existence of pointer states: Many can find out the
their ability to “procreate”, to deposit multiple records state of the system independently, without prior information,
– copies of themselves – throughout E. and they can do it indirectly, without perturbing S.
Proliferation of records allows information about S to
be extracted from many fragments of E (in the example of the system was the basic tool of decoherence. To study
above, photon E). Thus, E acquires redundant records of Quantum Darwinism we focus on correlations between
S. Now, many observers can find out the state of S in- fragments of the environment and the system. The rele-
dependently, and without perturbing it. This is how pre- vant reduced density matrix ρ is given by:
ferred states of S become objective. Objective existence SF
– hallmark of classicality – emerges from the quantum ρ =Tr |Ψ ihΨ | . (2)
substrate as a consequence of redundancy. SF E/F SE SE
Decoherencetheorywasfocusedonthesystem. Itsaim Above, trace is over “E less F”, or E/F – all of E except
was to determine what states survive information leaks for the fragment F. How much F knows about S can be
to E. Now we ask: What information about the system quantified using mutual information:
can be found out from fragments of E? This change of
focus calls for a more realistic model of the environment I(S : F) = H +H −H , (3)
(Fig. 1): Instead of a monolithic E we recognize that envi- S F S,F
ronments consist of subsystems that comprise fragments defined as the difference between entropies of two sys-
independently accessible to observers. tems (here S and F) treated separately and jointly. For
The reduced density matrix ρ representing the state example, the mutual information between an original and
S
3
and indirectly – without perturbing S.
Rapid rise and gradual leveling of I(S : Ff), Fig. 2,
implies redundancy. The information in F allows one
f
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 H is the classical
S
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-
FIG. 2: Information about S stored in E and its redundancy. lowedbyredundancyisalsoimportant: Fragilityofquan-
Mutual information is monotonic in f. When global state of tumstatesmeansthatcopiesinF’saredamagedbymea-
SE is pure, I(S : F ) in a typical fraction f of the environ- surements (we destroy photons!), and may be measured
f in a “wrong” basis. One cannot access records in E with-
ment is antisymmetric around f = 0.5 [13]. For pure states
picked out at random from the combined Hilbert space H , out endangering their existence. But with many (R )
SE δ
there is little mutual information between S and a typical F copies, state of S can be found out by ∼ Rδ observers
smaller than half of E. However, once a threshold f = 1 is who can get their information independently, and with-
2
attained, nearly all information is in principle at hand. Thus, out prior knowledge about S. Consensus between copies
such random states (green line) exhibit no redundancy. By suggests objective existence of the state of S.
contrast, states of SE created by decoherence (where the en- The mutual information I(S : Ff) computed in mod-
vironment monitors preferred observable of S) contain almost els of decoherence exhibits behavior illustrated by the red
all (all but δ) of the information about S in small fractions plot of Fig. 2. In the family of models representing spin
f of E. The corresponding I(S : F ) (red line) quickly rises
δ f S surrounded by environments of many spins [12, 13, 14]
to HS (entropy of S due to decoherence), which is all of the the same number of spins suffices to reach the plateau:
information about S available from either E or S. (More, up AddingmorespinstoE onlyextendslengthoftheplateau
to 2H , can be obtained only through global measurements
S
on S and nearly all E). HS is therefore the classically acces- measured in “absolute units” – in the number of the en-
sible information. As (1 − δ)H of information is contained
S vironment spins. In this model (that can be viewed as
in f = 1/R of E, there are R such fragments in E: R
δ δ δ δ a simplified model of a photon environment) redundancy
is the redundancy of the information about S. Large redun- is then proportional to the number of the environment
dancy implies objectivity: The state of the system can be subsystems that interact with the system of interest.
found out indirectly and independently by many observers, Quantum Brownian motion – harmonic oscillator sur-
who will agree about their conclusions. Thus, Quantum Dar- rounded by many environmental oscillators – is the other
winism accounts for the emergence of objective existence. well known model of decoherence. It is exactly solvable,
and the case of an underdamped oscillator yields sur-
a perfect copy (of, say, a book) is equal to the entropy of prisingly simple results [15, 16]: (i) Mutual information
the original, as either contains the same text. So, every is approximately given by I(S : F) ≈ H + 1 ln f ,
bit of information in the first copy reveals a bit of infor- S 2 (1−f)
and; (ii) Redundancy for an initially squeezed state of S
mationintheoriginal. However, having extra copies does 2δ
reaches R ≈ s , where s, the squeeze factor, quantifies
not increase the information about the original. Yet, it δ
determines how many can independently access this in- delocalization of the state. Similar equation should hold
formation. The number of copies defines redundancy. for more general “Schr¨odinger cat” states, with s quan-
Similar ideas apply to the quantum case. Initially, ev- tifying the separation of the two localized alternatives.
ery bit of information gained from a fraction f 1 of These results confirm intuitions that originally moti-
E that was pure before it monitored (and decohered) the vated Quantum Darwinism [4, 17]: Monitoring of the
system is a bit about S. The red plot in Fig. 2 starts with system by the environment can deposit multiple records
this steep “bit for bit” slope, but moderates as I(S : Ff) of preferred states of S in E. States of SE that arise from
approaches redundancy plateau at H , where additional decoherence are special [13, 14], as I(S : Ff) for a typ-
S ical pure state selected with Haar measure in the whole
bits only confirm what is already known. Hilbert space of SE (green plot in Fig. 2) shows. In
Redundancy is the number of independent fragments such random states small fragments reveal almost noth-
of the environment that supply almost all classical infor- ing about the rest of the state. Only when half of E is
mation about S, i.e., (1 − δ)H . In other words;
S found out the whole state is suddenly revealed.
R =1/f . (4) States that arise from decoherence are then far from
δ δ random. Roughlyspeaking, theyhaveabranch structure.
Rδ is the number of times one can acquire (1−δ) of the This is why the rest of such a branch including the state
information about S independently (from distinct F’s) of the system – the “bud” from which this branch has
4
originated – can be deduced from its fragment. We shall not interact with each other. This is why light deliv-
see how such branches grow in the next section. ers most of our information. Moreover, photons emitted
Plots of I(S : Ff) for pure SE are antisymmetric by the usual sources (e.g., sun) are far from equilibrium
around the point {H ,f = 1} for typical fragments of with our surroundings. Thus, even when decoherence is
S 2 dominated by other environments (e.g., air) photons are
E [13]. Thus, rapid rise for small f must be matched at muchbetter in passing on information they acquire while
the other end, for f ∼ 1. This is a signature of entan- “monitoring the system of interest”: Air molecules scat-
glement that allows state to be known “as the whole”,
while states of subsystems are unknown. The joint state ter from one another, so that whatever record they may
of SE is then pure, so that H =0, and I(S : F ) have gathered becomes effectively undecipherable.
S,F=E f Stability of the level of the redundancy plateau at H ,
must rise to H +H =2H when f approaches 1. S
S E S even for mixed E’s, is a compelling reason to think of it as
This is a very quantum aspect of information. In clas-
sical physics knowing a composite object implies knowing “classical”. The question we shall now address concerns
each of its subsystems. This is not so in quantum physics, the nature of that information – what does the environ-
where composite states are given by tensor (rather than ment know about the system, and why?
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 III. FROMCOPYINGTOQUANTUMJUMPS
how this feature can be used to derive Born’s rule [18]
that relates probabilities with wavefunctions. Quantum Darwinism leads to appearance, in the en-
To reveal this latent quantumness one would have to vironment, of multiple copies of the state of the system.
measure the right global observable on all of SE. For However, the no-cloning theorem [20, 21] prohibits copy-
example, when mutual information, Eq. (3), is defined ing of unknown quantum states. If cloning is outlawed,
using Shannon entropy with probabilities corresponding how can redundancy seen in Fig. 2 be possible?
to optimal observables in S and in E, the resulting Shan- Quickansweristhatcloningrefers to (unknown) quan-
non I(S : Ff) graph for small f would look very similar tum states. So, copying of observables evades the theo-
to Fig. 2. However, using Shannon entropy involves lo- rem. Nevertheless, the tension between the prohibition
cal probabilities (precluding global observables), so such on cloning and the need for copying is revealing: It leads
Shannon I(S : F ) never exceeds H , antisymmetry is
f S to breaking of unitary symmetry implied by the super-
lost, and the plateau continues until the end at f ∼ 1. position principle, accounts for quantum jumps, and sug-
Effective unattainability of the f ∼ 1 part of the plot gests origin of the “wavepacket collapse”, setting stage for
also shows why decoherence is so hard to undo: Correla- the study of quantum origins of probability in Section IV.
tions that reveal coherence can be usually detected only Quantum physics is based on several “textbook” pos-
by such global measurements of whole SE. We intercept tulates [22]. The first two; (i) States are represented by
small fractions of E, and never have the luxury of perfect vectors in Hilbert space, and; (ii) Evolutions are unitary –
global measurements needed to undo decoherence. Yet, give complete account of mathematics of quantum theory,
because of redundancy, we get ∼ HS information with but make no connection with physics. For that one needs
“sloppy” measurements of f 1. to relate calculations made possible by the superposition
Quantum Darwinism does not require pure E. Mixed principle of (i) and unitarity of (ii) to experiments.
environmentisanoisycommunicationchannel: Itsinitial Postulate (iii) Immediate repetition of a measurement
entropy of h per bit can still increase after interaction yields the same outcome starts this task. This is the only
with S, reflecting mutual information buildup. However, uncontroversial measurement postulate (even if it is diffi-
nowabitgainedfromE yieldsonly1−hofabitaboutS. cult to approximate in the laboratory): Such repeatability
So, a completely mixed E (h = 1) is useless (even though or predictability is behind the very idea of “a state”.
it can still induce decoherence!). For a partly mixed E In contrast to (i)-(iii), collapse postulate (iv) Outcomes
mutual information will increase more slowly, pure case correspond to eigenstates of the measured observable, and
“bit per bit” rate tempered to ∼ 1 − h. Yet, it can still only one of them is detected in any given run of the ex-
climb the same redundancy plateau at H [19]. periment, is inconsistent with (i) and (ii). Conflict arises
S
These conclusions apply when E is initially mixed, but for two reasons: Restriction to a preferred set of outcome
are also relevant when this channel is noisy for other rea- states seems at odds with with the egalitarian principle
sons (e.g., imperfect measurements). In all such cases one of superposition, embodied in (i). This restriction pre-
can still reach the same redundancy plateau, although vents one from finding out unknown quantum states, so
now a proportionally larger fragment of the environment it is responsible for their fragility. And a single outcome
is needed to get the same information about S. per run is at odds with unitarity (and, hence, linearity)
Suitability of the environment as a channel depends of quantum dynamics that preserves superpositions.
on whether it provides a direct and easy access to the The last axiom; (v) Probability of an outcome is given
records of the system. This depends on the structure by the square of the associated amplitude, p = |ψ |2,
k k
and evolution of E. Photons are ideal in this respect: is known as Born’s rule [18]. It completes the relation
They interact with various systems, but, in effect, do between mathematics of (i) and (ii) and the experiments.
5
a) b) c)
1.0 50 1.0
0.8 ) 40 )0.8
) σ e
σ0.6 ( 30 :0.6
1
( .
0 σ
ˆN0.4 R (0.4
I 20 I
0.2 µ=0.23 0.2
10
0 0 0
0 0 0
π/4 π/4 π/4 40 50
µ µ π/4 π/4 µ 30
π/8 π/8 π/2 10 20 m
π/2 0 a a 0
π/2 0
FIG. 3: Quantum Darwinism in a simple model of decoherence [12]. The spin-1 S interacts with N = 50 spin-1 subsystems of E
P 2 2
N S E 1 E E
withanIsingHamiltonianHSE = g σ ⊗σ k. TheinitialstateofS⊗E is √ (|0i+|1i)⊗|0i 1⊗...⊗|0i N. Couplingsg are
k=1 k z y 2 k
distributed randomly in the interval (0,1]. All the plotted quantities are a function of the observable σ(µ) = cos(µ)σ +sin(µ)σ ,
z x
where µ is the angle between its eigenstates and the pointer states of S – eigenstates of σS. a) Information acquired by the
z
ˆ
optimal measurement on the whole environment, IN(σ), as a function of the inferred observable σ(µ) and the average interaction
action hgkti = 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 hg ti = a. R (σ) counts the number of times 90% of the total information can be “read off” independently
k δ=0.1
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].
Bohrbypassedconflictof(i)and(ii)with(iv)byinsist- demand”: As in cloning, one asks for “two (or more) of
ing that apparatus is classical, so unitarity and the prin- the same”. Its conflict with linearity of quantum the-
ciple of superposition need not apply to measurements. ory can be resolved only by restricting states that can
Butthis is an excuse, not an explanation. We are dealing be copied. Such pointer states then act as “buds” of
with a quantum environment, and redundancy of previ- branches that grow by reproducing, in E, multiple copies
ous section strengthened motivation for postulate (iii) – of the original in S. Interaction Hamiltonians do not per-
repeatability. Let us see where this demand takes us in turb observables that commute with them. So, buds of
a purely quantum setting of postulates (i), (ii), and (iii). branches coincide with the einselected pointer states.
Suppose there are states of S (say, |ui and |vi) that Evidence of such symmetry breaking is seen in Fig.
produce an imprint in a subsystem of E (which plays a 3. Mutual information and redundancy shown there are
role of an apparatus), but remain unperturbed (so they obtained using Eq. (3), but with Shannon (rather than
can produce more imprints). This repeatability implies: von Neumann) entropies of specific observables of S and
|ui|e i ⇒ |ui|e i, |vi|e i ⇒ |vi|e i in obvious notation.
0 u 0 v F,i.e., using probabilities of their eigenstates. While von
In a unitary process scalar product is preserved. Thus; Neumann-based I(S : F ) and R characterized total
f δ
hu|vi = hu|vihe |e i , (5) information, Shannon-based counterparts are well suited
u v to enquire: What observable is this information about?
where we have set he |e i = 1. This simple equation
0 0 It turns out that the environment as a whole “knows”
can be satisfied only when; (a) either he |e i = 1 (which
u v manyobservables of S, as is seen in Fig. 3a. By contrast,
meansthat copying was completely unsuccessful), or; (b) in Fig. 3b symmetry breaking is evident: The ridge of
hu|vi = 0, i.e., they are orthogonal. In that case he |e i
u v redundancy appears abruptly only when test observable
is arbitrary – perfect record he |e i = 0 is also possible.
u v σ(µ) and the preferred pointer observable σz (that re-
It follows that multiple (perfect or imperfect) copies mains unperturbed by the environment) nearly coincide.
of |ui and |vi can be imprinted in disjoint F’s. As a Why are pointer states favored? Commonsense says
consequence of unitarity, only sets of orthogonal states that, to be reproduced, state must survive copying. This
(that define Hermitean observables [22]) can be so copied, leads to a theorem [12, 24] that only pointer states can be
explaining selection of a set of outcomes – terminal points discovered from fractions of E. Other observables (such
of quantumjumps[23]. Before, they had to be postulated as σ(µ) in Fig. 3) can be deduced only to the extent they
by the first part of axiom (iv). We emphasize that this are correlated with the pointer observable. So, fragments
result relies on just two values of the scalar product – 0 of the environment offer a very narrow, projective point
and 1 – and, thus, does not appeal to Born’s rule. of view. Redundant imprinting of some observables hap-
This breaking of unitary symmetry (choice of preferred pens at the expense of their complements.
states in an egalitarian Hilbert space) is induced by re-
peatability of the information transfer. It is a “nonlinear Structure of branching state betrays its origin and fore-
6
P
shadows “collapse”. Starting from |ψ i = nψ |s i, Selection of the set of outcomes by the proliferation of
S k k k information essential for Quantum Darwinism parallels
n n
|Ψ i = Xψ |s i|e(1)i...|e(N)i = Xψ |s i|ε i (6) Bohr’s insistence [1] that a “classical apparatus” should
SE k k k k k k k determine the outcomes. However, it follows from purely
k k quantum Eq. (5), and is caused by a unitary evolution
branches grow to include N subsystems of E. Branch responsible for the information transfer. Nevertheless, as
J (j) (j) classical apparatus would, preferred pointer states desig-
fragments can be nearly orthogonal; Π he |e 0 i '
j=1 k k nate possible future outcomes, precluding measurements
δkk0 for large enough J. This means that a pointer state of complementary observables or determining preexist-
|ski of S can be determined (along with the rest of the ing state of the system. Thus, information acquisition –
branch) from a sufficiently long fragment (which may still a copying process – results in preferred states.
be short compared to the length of the branch, J N). Consensus between records deposited in fragments of
In the huge Hilbert space H branching state is a
SE E looks like “collapse”. In this sense we have accounted
very atypical minimally entangled superposition of only for postulate (iv) using only very quantum postulates (i)-
n product “branches” labelled by the pointer states of (iii). In particular, in deriving and analyzing Eq. (5) we
the system. This is tiny compared to the dimension of have not employed Born’s rule, axiom (v). We shall be
HSE that exceeds n by a factor exponential in N. This therefore able to use our results as a starting point for
is why the two plots in Fig. 2 are so different: Branch- such a derivation in the next section.
ing state is, to a good approximation, a multi-system There was nothing nonunitary above – unitarity was
Schmidt decomposition, with long branch fragments con- the crux of our argument, and orthogonality of branch
stituting “systems”. In a Schmidt decomposition, states seeds our main result. Relative states of Everett [26, 27,
of partners are in one-to-one correspondence. Thus, in 28] come to mind. One could speculate about reality of
Eq. (6), |s i implies |ε i (and, vice versa), and measur-
k k branches with other outcomes. We abstain from this –
ing a branch fragment F can reveal the whole branch. our discussion is interpretation-free, and this is a virtue.
Initial part of I(S : Ff), Fig. 2, represent buildup of Indeed, “reality” or “existence” of universal state vector
this correlation: When f = 0, observer is ignorant of seems problematic. Quantum states acquire objective
what branch he will find out, but the structure of the existence when reproduced in many copies. Individual
correlations within |ΨSEi leaves no doubt of what these states – one might say with Bohr – are mostly informa-
branches are. Using Born’s rule one could assign to them tion, too fragile for objective existence. And there is only
probabilities p = |ψ |2 and the corresponding entropy
k k one copy of the Universe. Treating its state as if it really
H . Next section shows how one can deduce these prob-
S existed [26, 27, 28] seems unwarranted and “classical”.
abilities without axiom (v) – how symmetries of entan-
glement imply Born’s rule.
When observer measures enough of E, he finds out IV. PROBABILITIES FROM ENTANGLEMENT
the branch (and what the state of S is). Additional
data are redundant. They only confirm what is already
known. Probabilities associated with |Ψ i are replaced Observer prepared S in a state |ψ i, but wants to mea-
SE S
with certainty of a branch. This transition from uncer- sure observable with eigenstates {|ski}. This will lead to
tainty (initial presence of many branches – potential for entangled |ΨSEi with branch structure, Eq. (6). Pointer
multiple outcomes) to certainty (once a sufficiently long states {|s i} define the outcomes, but, as yet, observer
branch fragment becomes known) accounts for percep- k
tion of “collapse”. The initial, steeply rising, part of has not measured E, and does not know the result. Given
I(S : F ) “resolves” it: Collapse is brief compared to |ΨSEi, what is the probability of, say, |s17i?
f To derive it we cannot use reduced density matrices,
the ensuing period of certainty about the outcome, as Eqs. (1,2). Tracing out is averaging [25, 29, 30] – it relies
fδ 1, but, nevertheless, not instantaneous. on p = |ψ |2, Born’s rule we want to derive. We have
Assumptionsthatleadfromcopyingtopreferredstates k k
imposed that ban while deriving and analyzing Eq. (5),
can be relaxed. Thus, E need not be initially pure [23]. but relaxed it to plot Fig. 3. Now we reimpose it again.
Moreover, it suffices that the records (e.g., in the appara- So, Born’s rule and standard tools of decoherence are
tus A) are “repeatably accessible”. Transfer of responsi- off limits – using them courts circularity. Our derivation
bility for repeatability from a quantum S to a (still quan- will rest instead on certainty and symmetry, cornerstones
tum)Aallowsonetomodelnon-orthogonalmeasurement that mark two extremal cases of probability.
outcomes (POVM’s): A entangles with the system, and The case of certainty was just settled without Born’s
then acts as ancilla. Its orthogonal pointer states |Aki rule using Eq. (5). When one re-measures an observable,
P ˜
correlate with non-orthogonal |ς i of S, ψ |ς i|A i. the same outcome will be seen again. Thus, when {|s i}
k k k k k k
Interaction of A with the environment results in multiple includes |ψ i (e.g., |ψ i = |s i), newly added copies
S S 17
copies of |A i. The usual projective measurement imple- just extend the branch already correlated with observer’s
k
mentation of POVM’s (see e.g. [25]) is now straightfor- state, and the outcome is certain; p =1. Certainty of
17
ward. Branches are labelled by |Aki. Indeed, we usually correlations between partners in Schmidt decomposition,
experience “quantum jumps” via an apparatus pointer. Eq. (6) is another important example.
7
a) ~
~
b)
=
| | | |
+
S E S E
> > > >
| | | | | | | |
+ +
S E S E S E S E
c) > > > > > > > >
| | | | | | | |
+ +
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 for the top card. (b) The real physical
♠ 2
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
0 0♥ ♦
p =p . After a swap u = |♠ih♥|+|♥ih♠| in S, the resulting state |♠i|♦i+|♥i|♣i must have p = p , and p = p . (We
♠ ♣ S ♠ ♦ ♥ ♣
‘primed’ probabilities in S, as it was acted upon by a swap, so they might have changed.) A counterswap u = |♦ih♣|+|♣ih♦|
0 0 E
in E restores the original entangled state, proving that p =p andp =p ,afterall(ascounterswap u leaves S untouched).
♥ ♥ ♠1 ♠ E
This sequence of equalities implies p =p =p ,sothat p =p = ,as probabilities in S must add up to 1.
♠ ♦ ♥ ♠ ♥ 2
Certainty seems trivial but is important. Confirmation Figure 4 illustrates how this classical intuition yields –
that a state “is what it is” – postulate (iii) – is a part of far more convincingly — quantum probabilities.
standard quantum lore [22]. We re-affirmed it, but with Symmetry is probed by invariance. Transformations
a key insight: Redundancy allows observers to discover that respect it take system between states that exhibit
(and not just confirm) that S is in a certain pointer state. no measurable differences. For example, change of phase
in the coefficients in the Schmidt decomposition |ΨSEi =
We now turn to the opposite case of complete inde- P
nψ |s i|ε i cannot influence the state of S: It is in-
terminacy. Its connection with symmetry was noted by k k k k
duced by u = eiφk|s ihs |, local unitary on S, that can
Laplace. He wrote: “The theory of chance consists in re- S k k
be “undone” by u = e−iφk|ε ihε | on E, or;
ducing all the events ... to a certain number of cases that E k k
are equally possible... The ratio of this number to that of
all the cases possible is the measure of probability” [31]. u ⊗1 |Ψ i=|Φ i; 1 ⊗u |Φ i=|Ψ i (7)
S E SE SE S E SE SE
8
So, phases of ψk cannot matter for a local state or influ- of S. However, this is done by a unitary “countertrans-
ence probabilities in S. This symmetry, Eq. (7), is the formation” acting solely on E. Hence, by fact (1), state
entanglement-assisted invariance or envariance [32, 33]. of S must have been unaffected by u in the first place.
S
Such loss of phase significance for S entangled with E So, by fact (2), phases of ψ cannot change outcomes of
k
implies decoherence [33]. We arrived at its essence using any measurement on S. Equiprobability follows.2
envariance, without reduced density matrices, trace, etc. One can now derive Born’s rule, p = |ψ | , with
k k
We now use phase envariance to show that equal ab- straightforward algebra from the above two simple cases
of complete certainty (p = 1) and equiprobability (p =
solute values of the coefficients ψ imply equal prob- k k
k 1): The general case can be always reduced to the case
abilities. For equal |ψk| any orthogonal basis of S n
is “Schmidt” (i.e., has an orthogonal partner in E). case of equal coefficients by “finegraining” (see Box).
|0i |0i +|1i |1i |+i |+i +|−i |−i The origin of probability is a fascinating problem that
S E S E S E S E
Thus, |ϕ¯ i = √ = √ ,
SE 2 2 is older than quantum measurement problem, and is for-
|0i±|1i iπ
where |±i = √2 . Sign change induced by e |−ih−| gotten primarily because it is so old. We have seen how
|+i |+i −|−i |−i quantum physics sheds a new, very fundamental, light
S E S E
acting on S produces |η¯ i = √ =
SE 2 on probability. We cannot do justice to the history of
|1i |0i +|0i |1i
S E S E
√2 . In other words, one can swap |0iS with this subject here, but Ref. [34] provides a basic overview
|1i by rotating phase in a |±i basis by π. Yet, we just and exhaustive set of references. In particular, envariant
S
saw that phases of Schmidt coefficients do not matter for derivation is very different from the classic proof of Glea-
the state of S, so probabilities of 0 and 1 in S must have son [35] in that it sheds light on the physical significance
remained the same. Moreover, probabilities of paired up of the resulting measure. Moreover, it does not assume
Schmidt states are equal, so that p (0) = p (0) in |ϕ¯ i probabilities are additive (except to posit that probabil-
S E SE
and p (1) = p (0) in |η¯ i. Hence, p (0) = p (1) = 1, ity of an event and its complement are certain, i.e., to
S E SE S S 2
where we assumed that probabilities add up to 1. establish normalization; see Box and Ref. [33, 38]). By-
In contrast to Laplace’s subjective “ignorance-based” passing additivity of probabilities is essential when deal-
approach, we obtained objective probabilities for a com- ing with a theory with another principle of additivity
pletely known entangled state. Phase envariance implied – the quantum superposition principle – which trumps
equiprobability in S. To paraphrase Beatles, “All you additivity of probabilities or at least classical intuitiions
need is phase...”. We rotated phases of the coefficients to about it (e.g., in the double-slit experiment). Discus-
induce a swap in a complementary basis. Another proof sion of the implications of envariance has already started,
(that implements swap more directly) is given in Fig. 4. with [36, 37], and [5] providing insightful commentary.
This equiprobability case is the difficult part of the BOX
proof. Instead of subjectivity (that undermined appli- We show here how “finegraining” reduces the case of
cability of Laplace’s approach to physics) we relied on arbitrary ψ to equiprobability. To illustrate general
objective symmetries of entangled quantum states. This k
was made possible by the nature of quantum states of strategy consider state in a 2D Hilbert space HS of S
spanned by orthonormal {|0i,|2i} and (at least) 3D HE:
composite systems. Classically, pure states have struc- |ψ i ∝ q2 |0i |+i + q1 |2i |2i .
ture of a Cartesian product – knowing the whole implies SE 3 S E 3 S E
knowledge of each subsystem. In quantum theory they |0i +|1i
E E
The state |+iE = √ exists in (at least 2D) sub-
are tensor products – one can know state of the whole, 2
space of E orthogonal to |2i , i.e., h0|1i = h0|2i = h1|2i =
and thus know nothing about parts, as envariance shows. E
This was the basis of our proof of equiprobability. We h+|2i = 0. We know we can ignore phases.
To reduce |ψ i to equal coefficients case we “extend
assumed unitarity. Moreover, we assumed; (1) When a SE
¯
it” to a state |Ψ i by letting E act on an ancilla C.
system is not acted upon by a unitary transformation, its SEC
state remains unaffected. This state is a property of (S is not acted upon, so, by fact (1), probabilities for S
S alone, so; (2) Predictions regarding measurement out- cannot change.) This can be done by a generalization of
comes on S (including their probabilities) can be inferred controlled-not acting between E (control) and C (target),
from the state of S. Last not least; (3) When S is entan- so that (in obvious notation) |ki|00i ⇒ |ki|k0i, leading to
gled with other systems (e.g., the environment) the state √ √ 0 0
0 0 |0i|0 i+|1i|1 i 0
of S alone is determined by the state of the whole SE. 2|0i|+i|0 i+|2i|2i|0 i ⇒ 2|0i √ +|2i|2i|2 i.
These “facts of life” are accepted properties of systems 2
and states, but given the fundamental nature of our dis- Above, and from now on we skip subscripts: The state of
cussion it seems a good idea to make them explicit [33]. S will be listed first, and the state of C will be primed.
For instance, to establish independence from phases of Thecancellation of √2 yields an equal coefficient state:
the coefficients ψ we noted that the state of S is un-
k
affected by the unitaries u diagonal in Schmidt basis ¯ 0 0 0
S |ΨSCEi ∝ |0,0 i|0i + |0,1 i|1i + |2,2 i|2i .
acting on S (like changes of Schmidt coefficient phases)
that would normally affect isolated S: The global state We have combined S and C in a single ket and (below)
Ψ isrestored by u . Thus, by fact (3), so is local state we shall swap states of SC as if it was a single system.
SE E
9
Clearly, this is a Schmidt decomposition of (SC)E. “single idea” category. Several ideas, applied in the right
Three orthonormal product states have coefficients with order, led to advances described here. Logically, we may
the same absolute value. Therefore, they can be en- well have started with the derivation of Eq. (5) and the
variantly swapped. Thus, the probabilities of states analysis of quantum jumps. Their randomness leads to
0 0 0
|0i|0 i, |0i|1 i, and |2i|2 i are all equal. By normalization probabilities. And symmetries of entangled states (that
they are 1. So, probability of detecting state |2i of S is arise in decoherence and Quantum Darwinism) allow one
3
1. Moreover, |0i and |2i are the only two outcome states to derive Born’s rule. As we have seen, phase envariance
3
for S. It follows that probability of |0i must be 2; is (nearly) “all you need”. With probabilities at hand
3 one has then every right to use reduced density matrices
p = 2; p = 1 .
0 3 2 3 to analyze Quantum Darwinism and decoherence.
ThisisBorn’srule. Wehavejustseenwhytheamplitudes
in the initial |ψSEi “get squared” to yield probabilities. Ourpresentation was “historical”. We started with de-
Notethatwehaveavoidedassumingadditivityofprob- coherence, and used it to introduce Quantum Darwinism.
abilities: p = 2 not because it is a sum of two fine- Analysis of copying essential to information flows in both
0 3
grained alternatives for SE, each with probability of 1, of these phenomena led to quantum jumps. This in turn
3 motivated entangelment-based derivation of Born’s rule.
but rather because there are only two (mutually exclu- QuantumDarwinism – upgrade of E to a communication
sive and exhaustive) alternatives for S; |0i and |2i, and
p = 1. Therefore, by normalization, p = 1− 1. Prob- channel from a mundane role it played in decoherence –
2 3 0 3 tied together all of the other developments. This order
abilities of Schmidt states can be added because of the had the advantage of making motivations clear, but it is
loss of phase coherence that follows directly from phase different from more logical presentation where postulates
envariance established earlier (see also Ref. [32, 33]). (i)-(iii) are the starting point (strategy followed in [38]).
Extension of this proof to the case where proba-
bilities are commensurate is conceptually straightfor- Thecollection of ideas discussed here allows one to un-
ward but notationally cumbersome. The case of non- derstand how “the classical” emerges from the quantum
commensurate probabilities is settled with an appeal to substrate staring from more basic assumptions than de-
continuity. Frequency of the outcomes can be also de- coherence. We have bypassed a related question of why is
duced, allowing one to establish connection with the fa- our Universe quantum to the core. The nature of quan-
miliar relative frequency approach to probabilities [32, tum state vectors is a part of this larger mystery. Our
33, 38], but in a quantum setting probability arises as a focus was not on what quantum states are, but on what
consequence of symmetries of a single entangled state. they do. Our results encourage a view one might describe
Weendbynotingthatthefinegrainingdiscussedabove (with apologies to Bohr) as “complementary”. Thus, |ψi
does not need to be carried out experimentally each time is in part information (as, indeed, Bohr thought), but
probabilities are discussed: Rather, it is a way to de- also the obvious quantum object to explain “existence”.
duce a measure that is consistent with the geometry of Wehave seen how Quantum Darwinism accounts for the
the Hilbert spaces using entanglement as a tool. Still, transition from quantum fragility (of information) to the
given fundamental implications of envariance experimen- effectively classical robustness. One can think of this
tal tests would be most useful. transition as “It from bit” of John Wheeler [39].
In the end one might ask: “How Darwinian is Quan-
tumDarwinism?”. Clearly, there is survival of the fittest,
V. DISCUSSION and fitness is defined as in natural selection – through
the ability to procreate. The no-cloning theorem implies
We derived the two controversial quantum postulates competition for resources – space in E – so that only
from the first three. We have thus seen how classical do- pointer states can multiply (at the expense of their com-
mainoftheUniversearisesfromthesuperpositionprinci- plementary competition). There is also another aspect
ple (postulate (i)) and unitarity (postulate (ii)) as well as of this competition: Huge memory available in the Uni-
rudimentary assumptions about information flows (pos- verse as a whole is nevertheless limited. So the question
tulate (iii)), and a few basic facts about states of com- arises: What systems get to be “of interest”, and imprint
posite quantum systems (including their tensor nature, their state on their obliging environments, and what are
often cited as additional “axiom (0)”). the environments? Moreover, as the Universe has a finite
The essence of the measurement problem – accounting memory, old events will be eventually “overwritten” by
for axioms (iv) and (v) – has been largely settled. It is of new ones, so that some of the past will gradually cease
course likely one may be able to clarify assumptions and to be reflected in the present record. And if there is no
simplify proofs. Much work remains to be done on Quan- record of an event, has it really happened? These ques-
tum Darwinism and envariance. Nevertheless, nature of tions seem far more interesting than deciding closeness
the quantum-classical correspondence has been clarified. of the analogy with natural selection [40]. They suggest
Physicists take it for granted that even hard problems one more question: Is Quantum Darwinism (a process of
are solved by a single good idea. Therefore, when a single multiplication of information about certain favored states
idea does not do the whole job, often our first instinct is to that seems to be a “fact of quantum life”) in some way
dismiss it. Measurement problem does not fall into this behind the familiar natural selection? I cannot answer
10
this question, but neither can I resist raising it.
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