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Relative States and the Environment: Einselection, Envariance, QuantumDarwinism, and the Existential Interpretation

Wojciech Hubert Zurek

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

(Dated: February 3, 2008)

Starting with basic axioms of quantum theory we revisit Relative State Interpretation set out 50years ago by Hugh Everett III (1957a,b). His approach explains collapse of the wavepacket bypostulating that observer perceives the state of the rest of the Universe relativeto his own state,or to be more precise relative to the state of his records. This allows quantum theory to beuniversally valid. However, while Everett explains perception of collapse, relative state approachraises three questions absent in Bohrs Copenhagen Interpretation which relied on independentexistence of anab intioclassical domain. One is now forced one to seek sets of preferred, effectivelyclassical but ultimately quantum states that can define branches of the universal state vector, andallow observers to keep reliable records. Without such (i) preferred basis relative states are toorelative, and the approach suffers from basis ambiguity. Moreover, universal validity of quantumtheory raises the issue of the (ii) origin of probabilities, and of the Borns rulepk = |k|

2 which issimply postulated in textbook discussions. Last not least, even preferred quantum states (definede.g. by the einselection environment - induced superselection) are still quantum. Therefore theycannot be found out by initially ignorant observers through direct measurement without gettingdisrupted. Yet, states of macroscopic object exist objectively and can be found out by anyone. So,we need to identify the (iii) quantum origin of objective existence. Here we show how mathematical

structure of quantum theory supplemented by the only uncontroversial measurement axiom (thatdemands immediate repeatability and, hence, predictability of idealized measurements) leadsto preferred sets of states: Line of reasoning reminiscent of the no cloning theorem yields (i)pointer states which correspond to potential outcomes. Their stability is needed to establisheffectively classical domain within quantum Universe, and to define events such as measurementoutcomes. This leads one to enquire about their probabilities or more specifically about therelation between probabilities of measurement outcomes and the underlying quantum state. Weshow that symmetry of entangled states (ii) entanglement - assisted invarianceor envariance implies Borns rule. It also accounts for the loss of physical significance of local phases betweenSchmidt states. (in essence, for decoherence). Thus, loss of coherence between pointer states isa consequence of symmetries of entanglement (e.g., with the environment). It can be establishedwithout usual tools of decoherence (reduced density matrices and trace operation) that rely onBorns rule for physical motivation. Finally, we point out that monitoring of the system by theenvironment (process responsible for decoherence) will typically leave behind multiple copies ofits pointer states. Only states that can survive decoherence can produce information theoreticprogeny in this manner. This (iii) quantum Darwinism allows observers to use environment

as a witness to acquire information about pointer states indirectly, leaving system of interestuntouched and its state unperturbed. In conjunction with Everetts relative state account of theapparent collapse these advances illuminate relation of quantum theory to the classical domainof our experience. They complete existential interpretation based on the operational definitionof objective existence, and justify our confidence in quantum mechanics as ultimate theory thatneeds no modifications to account for the emergence of the classical.

I. INTRODUCTION

Quantum mechanics is often regarded as an essentiallyprobabilistic theory, in which random collapses of thewavepacket governed by the rule conjectured by MaxBorn (1926) play a fundamental role (Dirac, 1958). Yet,

unitary evolution dictated by Schrodingers equation isdeterministic. This clash of quantum determinism andquantum randomness is at the heart of interpretationalcontroversies reflected in the axioms that provide a text-book summary of quantum foundations:

(i)State of a quantum system is represented by a vectorin its Hilbert spaceHS.

(ii) Evolutions are unitary (i.e., generated bySchrodinger equation).

These two axioms imply, respectively, quantum princi-

ple of superposition and unitarity of quantum evolutions.They provide essentially complete summary of the formalstructure of the theory. They seem to contain no pre-monition of either collapse or probabilities. However, inorder to relate quantum theory to experiments one needsto establish correspondence between abstract state vec-

tors inHSand physical reality. The task of establishingthis correspondence starts with the next axiom:

(iii) Immediate repetition of a measurement yields thesame outcome.

Axiom (iii) is regarded as idealized (it is hard to de-vise in practice such non-demolition measurements, butin principle it can be done). Yet as a fundamental pos-tulate it is also uncontroversial. The very concept ofa state embodies predictability which requires axiom(iii). The role of the state is to allow for predictions, and

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the most basic prediction is confirmation that the stateis what it is known to be. However, in contrast to classi-cal physics (where an unknown preexisting state can befound out by an initially ignorant observer) the very nextquantum axiom limits predictive attributes of the state:

(iv) Measurement outcome is one of the orthonormalstates eigenstates of the measured observable.

This collapse postulate is the first truly controversial

axiom in the textbook list. It is inconsistent with thefirst two postulates: Starting from a general state |S ina Hilbert space of the system (axiom (i)), an initial state|A0 of the apparatus A, and assuming unitary evolution(axiom (ii)) one is led to a superposition of outcomes:

|S|A0 = (k

ak|sk)|A0 k

ak|sk|Ak , (1.1)

which is in apparent contradiction with axiom (iv).The impossibility to account starting with (i) and

(ii) for the collapse to a single outcome postulated by(iv) was appreciated since Bohr (1928) and von Neumann(1932). It was and often still is regarded as an indi-

cation of the ultimate insolubility of the measurementproblem. It is straightforward to extend such insolu-bility demonstrations to various more realistic situationsby allowing e.g. the state of the apparatus to be initiallymixed. As long as axioms (i) and (ii) hold, one is forcedto admit that the state of the Universe after the mea-surement contains a superposition of many alternativeoutcomes rather than just one of them as the collapsepostulate (and our immediate experience) would have it.

Given this clash between mathematical structure of thetheory and collapse (that captures the subjective impres-sions of what happens in real world measurements) onecan proceed in two directions: One can accept with

Bohr primacy of our immediate experience and blamethe inconsistency of (iv) with the core of quantum for-malism (i) and (ii) on the nature of the apparatus:According to Copenhagen Interpretation apparatus, ob-server, and, generally, any macroscopic object is classical:It does not abide by the quantum principle of superposi-tion (that follows from (i)), and its evolution need not beunitary. So, axiom (ii) does not apply, and the collapsecan happen on the border between quantum and classi-cal. Uneasy coexistence of the quantum and the classicalwas a challenge to the unification instinct of physicists.Yet, it has proved to be surprisingly durable.

The alternative to Bohrs Copenhagen Interpretationand a new approach to the measurement problem wasproposed by Hugh Everett III, student of John ArchibaldWheeler, half a century ago. The basic idea is to aban-don the literal view of collapse and recognize that a mea-surement (including appearance of the collapse) is al-ready described by Eq. (1.1). One just need to includeobserver in the wavefunction, and consistently interpretconsequences of this step. The obvious problem Whydont I, the observer, perceive such splitting? is thenanswered by asserting that while the right hand side ofEq. (1.1) contains all the possible outcomes, the observer

who recorded outcome #17 will (from then on) perceivethe Universe that is consistent with that random eventreflected in his records. In other words, when global stateof the Universe is|, and my state is|I17, for me thestate of the rest of the Universe collapses toI17|.

This view of the collapse is supported by axiom (iii);upon immediate re-measurement the same state will befound. Everetts (1957a) assertion: The discontinuous

jump into an eigenstate is thus only a relative propo-sition, dependent on the mode of decomposition of thetotal wave function into the superposition, and relativeto a particularly chosen apparatus-coordinate value... isconsistent with quantum formalism: In the superpositionof Eq. (1.1) record state |A17 can indeed imply detectionof the corresponding state of the system,|s17.

Two questions immediately arise. First one concernspreferred states of the apparatus (or of the observer, or,indeed, of any object that becomes entangled with an-other quantum system). By the principle of superposi-tion (axiom (i)) the state of the system or of the appa-ratus after the measurement can be written in infinitelymany ways, each corresponding to one of the unitarilyequivalent basis sets in the Hilbert spaces of the pointerof the apparatus (or memory cell of the observer). So;

k

ak|sk|Ak =k

ak|sk|Ak =k

ak |sk|Ak =...(1.2)

Thisbasis ambiguityis not limited to pointers of measur-ing devices (or cats, which in Schrodinger (1935) play arole of the apparatus). One can show that also very largesystems (such as satellites or planets (Zurek, 1998a)) canevolve into very non-classical superpositions. In reality,this does not seem to happen. So, there is somethingthat (in spite of the egalitarian superposition principle

enshrined in axiom (i)) picks out certain preferred quan-tum states, and makes them effectively classical. Axiom(iv) anticipates this. Before there is collapse, a set ofpreferred states one of which is selected by the collapsemust be somehow chosen. There is nothing in writingsof Everett that would hint at a criterion for such pre-ferred states, and nothing to hint that he was aware ofthis question. The second question concerns probabili-ties: How likely it is that after I measureS I willbecome |I17? Everett was very aware of its significance.

Preferred basis problemwas settled by environment -induced superselection (einselection), usually discussedalong with decoherence: As emphasized by Dieter Zeh(1970), apparatus, observers, and other macroscopic ob-

jects are immersed in their environments. This leads tomonitoring of the system by its environment, describedby analogy with Eq. (1.1). When this monitoring isfocused on a specific observable of the system, its eigen-states form a pointer basis (Zurek, 1981): They entan-gle least with environment (and, therefore, are least per-turbed by it). This resolves basis ambiguity.

Pointer basis and einselection were developed and arediscussed elsewhere (Zurek, 1982; 1991; 1993; 2003a; Pazand Zurek, 2001; Joos et al., 2003, Schlosshauer, 2004;

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2007). However, they come at a price that might havebeen unacceptable to Everett: Decoherence and einse-lection usually employ reduced density matrices. Theirphysical significance derives from averaging, and is thusbased on probabilities on Borns rule:

(v)Probabilitypk of finding an outcome|skin a mea-surement of a quantum system that was previously pre-pared in the state| is given by|sk||2.

Borns rule (1926) completes standard textbook discus-sions of the foundations of quantum theory. In contrastto collapse postulate (iv), axiom (v) is not in obviouscontradiction with (i) - (iii), so one can adopt the atti-tude that Borns rule can be used to complete axioms (i)- (iii) and thus to justify preferred basis and symptomsof collapse via decoherence and einselection. This is theusual practice of decoherence (Zeh, 1970; Zurek, 1991;1998b; Paz and Zurek, 2001; Joos et al., 2003).

Everett believed that axiom (v) was inconsistent withthe spirit of his approach. So, one might guess, he wouldnot have been satisfied with the usual discussion of deco-herence and its consequences. Indeed, he attempted to

derive Borns rule from other quantum postulates. Weshall follow his lead, although not his strategy which asis now known was flawed (DeWitt, 1971; Kent, 1990).

One more axiom should added to postulates (i) - (v):(o)The Universe consists of systems.

Axiom (o) is often omitted form textbooks as obvious.But, as pointed out by DeWitt (1970; 1971), it is usefulto make it explicit in relative state setting where in con-trast to Copenhagen Interpretation all of the Universeis quantum. As was noted before (Zurek, 1993; 2003a;Schlosshauer, 2007), in absence of systems measurementproblem disappears: Schrodinger equation provides ade-terministicdescription of evolution of such an indivisible

Universe, and questions about outcomes cannot be evenposed. The measurement problem arises only becausein quantum theory state of a collection of systems canevolve from a Cartesian product (where overall purityimplies definite state of each subsystem) into an entan-gled state represented with a tensor product: The wholeis definite and pure, but states of the parts are indefinite(and so there are no definite outcomes).

This transition illustrated by Eq. (1.1) does not accordwith our perception of what happens. We see a definitestate of the apparatus in a Cartesian product with thecorresponding state of the system. Relative state inter-pretation with observer included in the wavefunction restores the correspondence between equations and per-ceptions by what seems like a slight of hand: Everettdecomposes the global entangled tensor state into a su-perposition of branches Cartesian products labeledby observers records. But we need more: Our goal is tounderstand the emergence of stable classical states fromthe quantum substrate, and origin of the rules govern-ing randomness at the quantum-classical border. To thisend in the next two sections we shall derive postulates(iv) and (v) from the non-controversial axioms (o)-(iii).We shall then, in Section IV, account for the objective

existence of pointer states. This succession of resultsprovides a quantum account of the classical reality.

We start with the derivation of the preferred set ofpointer states the business end of the collapse pos-tulate (iv) from axioms (o) - (iii). We will show thatany set of states will do providing they are orthogonal.We will also see how these states are (ein)selected bythe dynamics of the process of information acquisition,

thus following the spirit of Bohrs approach which em-phasized the ability to communicate results of measure-ments. Orthogonality of outcomes implies that the mea-sured quantum observable must be Hermitean. We shallthen compare this approach (obtained without resort-ing to reduced density matrices or any other appeals toBorns rule) with decoherence - based approach to pointerstates and the usual view of einselection.

Pointer states are determined by the dynamics of in-formation transfer. They define outcomes independentlyof the instantaneous reduced density matrix of the sys-tem (and, hence, of its initial state). Fixed outcomesdefine in turn events, and are key in discussion of prob-

abilities in Section III. There we also take a fresh lookat decoherence: It arises along with Borns rule fromsymmetries of entangled quantum states.

Given Borns rule and preferred pointer states oneis still facing a problem: Quantum states are fragile.Initially ignorant observer cannot find out an unknownquantum state without endangering its existence: Col-lapse postulate means that selection of what to measureimplies a set of outcomes. So, only a lucky choice ofan observable could let observer find out a state withoutre-preparing it. The criterion for pointer states impliedby axioms (o) - (iii) turns out to be equivalent to theirstability under decoherence, and still leaves one with thesame difficulty: How to find out effectively classical butultimately quantum pointer state without re-preparingit? How to account for objective existence of classicalreality using only unreal quantum ingredients?

The answer turns out to be surprisingly simple: Con-tinuous monitoring ofSby its environment results in re-dundant records of inE. Thus, many observers can findout state of the system indirectly, from small fragmentsof the sameE that caused decoherence. Recent and stillongoing studies discussed in Section IV show how thisreplication selects fittest states that can survive moni-toring, and yield copious information-theoretic offspring:Quantum Darwinism favors pointer observables at theexpense of their complements. Objectivity of preferred

is quantified by their redundancy by the number ofcopies of the state of the system deposited inE. Stabil-ity in spite of the environment is clearly a prerequisitefor large redundancy. Hence, pointer states do best inthis information - theoretic survival of the fittest.

Several interdependent steps account for the collapse,preferred states, probabilities, and objectivity for theappearance of the classical. It is important to take themin the right order, so that each step is based only onwhat is already established. This is our aim. Neverthe-

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less, each section of this paper can be read separately:Other sections are important to set the context, but aregenerally not required as a background.

II. EINSELECTION: BREAKING OF QUANTUMUNITARY SYMMETRY BY CLONING

Unitary equivalence of all states in the Hilbert space the essence of axiom (i) is the basic symmetry of quan-tum theory. It is reaffirmed by axiom (ii) which admitsonly unitary evolutions. But unitarity is also a center-piece of various proofs of insolubility of the measure-ment problem. Unitary evolution of a general initial stateof a systemS interacting with the apparatusA leads as seen in Eq. (1.1) to an entagled state ofSA. Thus,in the end there is no single outcome no collapse andan apparent contradiction with our experience. So thestory goes measurement problem cannot be solved un-less unitarity is somehow circumvented (e.g., along theadhoc lines suggested by the Copenhagen Interpretation).

We shall start with the same initial assumptions and

follow similar steps, but arrive at a very different con-clusion. This is because instead of the demand of a sin-gle outcome we shall only require that the results of themeasurement can be confirmed (by a re-measurement),or communicated (by making a copy of the record). Ineither case one ends up with multiple copies of some state(of the system or of the apparatus). This amplification(implicit in axiom (iii)) calls for nonlinearity that wouldappear to be in conflict with the unitarity (and, hence,linearity) demanded by axiom (ii). Resolution of the ten-sion between linear and non-linear demands brings tomind spontaneous symmetry breaking. As we shall see,amplification is possible, but only for a single orthogonal

subset in the Hilbert space of the system. This conclusion(Zurek, 2007) extends the reach of no cloning theorem.It explains the need for quantum jumps.

This section shows on the basis of axioms (o) - (iii)and reasoning with a strong no cloning flavor thatquantum jumps (and hence at least Everettian collapse)are inevitable. We shall also see how preferred Hermitianobservable defined by the resulting orthogonal basis isrelated to the familiar pointer basis.

A. Quantum origin of quantum jumps

Consider a quantum systemS

interacting with anotherquantum system E(which can be for instance an appara-tus, or as the present notation suggests, an environment).Let us suppose (in accord with axiom (iii)) that there isa set of states which remain unperturbed by this interac-tion e.g., that this interaction implements a measure-ment - like information transfer fromS toE:

|sk|0 = |sk|k. (2.1)We now show that a set of unperturbed states {|sk}must be orthogonal providing that the evolution de-

scribed above can start from an arbitrary initial state|S inHS (axiom (i)) and that it is unitary (axiom(ii)). From linearity alone we get:

|S|0 = (k

ak|sk)|0 k

ak|sk|k = |SE.(2.2)

The total norm of the state must be preserved. After

elementary algebra this leads to:

Rej,k

jksj |sk =Rej,k

jksj |skj|k . (2.3)

This equality must hold for all states in HS, and, in par-ticular, for any phases of the complex coefficients{k}.Therefore, for any two states in the set{|sk}:

sj |sk(1 j |k) = 0 . (2.4)This equation immediately implies that{|sk} must beorthogonal if they are to leave any imprint deposit anyinformation in

Ewhile remaining unperturbed: It can

be satisfied only whensj |sk =jk , unlessj |k = 1 unless states ofEbear no imprint of the states ofS.

In the context of quantum measurements Eq. (2.4)establishes the essence of axiom (iv) the orthogonalityof outcome states. On other hand, when outcome statesare orthogonal, any value ofj|k is admitted, includingj |k = 0 which corresponds to a perfect record.

The necessity to choose between distinguishable (or-thogonal) outcome states is then a direct consequenceof the uncontroversial axioms (o) - (iii). It can be seenas a resolution of the tension between linearity of quan-tum theory (axioms (i) and (ii)) and nonlinearity of theprocess of proliferation of information of amplification.

This nonlinearity is especially obvious in cloning thatin effect demands two of the same. Our derivationabove parallels proofs of no-cloning theorem (Woottersand Zurek, 1982; Dieks, 1982; Yuen, 1986): The only dif-ference in cloning copies must be perfect. Hence, scalarproducts must be the same, j,k =j |k =sj |sk.Consequently, we have a special case of Eq. (2.4):

j,k(1 j,k) = 0 . (2.5)Clearly, there are only two possible solutions; j,k = 0(which implies orthogonality), or the trivial j,k= 1.

Indeed, we can deduce orthogonality of states that re-main unperturbed while leaving small but distinct im-prints inEdirectly from the no-cloning theorem: As thestates ofSremain unperturbed by assumption, arbitrar-ily many imperfect copies can be made. But each extraimperfect copy brings the collective state of all copiescorrelated with, say,|sj, closer to orthogonality withthe collective state of all of the copies correlated withany other state|sk. Therefore, one could distinguish|sj for|sk by a measurement on a collection of suffi-ciently many of their copies with arbitrary accuracy, and,consequently, produce their clones. So even imperfect

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copying (any value ofj |k except 1) that preserves theoriginal is prohibited by no-cloning theorem.

Similar argument based on unitarity was put forwardin discussion of security of quantum cryptographic pro-tocols. There, however, focus was on the ability to de-tect an eavesdropper through the perturbation she wouldhave to inflict on the transmitted states (Bennett, Bras-sard, and Mermin, 1992), which is rather different from

the quest for preferred basis considered here.Reader might be concerned that above discussion is

based on idealized assumptions, which include purity ofthe initial state ofE. Regardless of whether Edesignatesan environment or an apparatus, this is unlikely to bea good assumption. But this assumption is also easilybypassed: A mixed state ofEcan be always representedas a pure state of an enlarged system. This is the so-called purification strategy: Instead of a density matrixE =

kpk|kk| one can deal with a pure entangled

state ofE andE: |kk =

k

pk|k|k defined inHEHE . So, when the initial state of E is mixed, there

is always a pure state in an enlarged Hilbert space, andall of the steps of the reasoning that lead to Eqs. (2.3) -(2.5) can be repeated leading to:

sj |sk(1 jj |kk) = 0, (2.6)forcing one to the same conclusions as Eq. (2.3).

Purification uses connection between pure states anddensity matrices by treatingE=

kpk|kk| as a re-

sult of a trace over some pure state. This appears tocontradict our stated goal of deriving axiom (iii) withoutappeal to measurement axioms (iv) and (v): The con-nectionEwith pure state |kk =

k

pk|k|k does

involve tracing and, hence, Borns rule. But there is away to weaken this assumption: It suffices for to assume

only that somesuch pure states in the enlarged Hilbertspace exists. This does not rely on Borns rule, but itdoes assert that ignorance that is reflected in a mixedlocal state (here, ofE) can be regarded as a consequenceof entanglement, so that some pure global state ofEEexists. So Borns rule isnotneeded for this purpose. Butthe existence ofsomesuch rule is needed.

For a reader who is still suspicious of the proce-dure employed above we have an alternative: Unitaryevolution preserves scalar products of density opera-tors defined by Tr. So, T r|sjsj |E|sksk|E =T r|sjsj |E|j |sksk|E|k, where E|j and E|k are mixedstates ofEaffected by the two states ofS. This yields:

|sj |sk|2(T r2E T rE|jE|k) = 0 (2.7)which can be satisfied only in the same two cases as be-fore: Eithersj |sk = 0, or T r2E = T rE|jE|k whichimplies (by Schwartz inequality) that E|j = E|k (i.e.,there can be no record of non-orthogonal states ofS).This conclusion can be reached even more directly: It isclear that E|j and E|k have the same eigenvalues pmas E =

mpm|mm| from which they have unitarily

evolved. Consequently, they differ from each other only

in their eigenstates that contain record of the state ofS,e.g.: E|k =

mpm|m|km|k|. Hence, T rE|jE|k =

mpm2|m|j|m|k|2, which coincides with T rE2 iff

|m|j |m|k|2 = 1 whenever pm= 0. So, E|j = E|k,and unlesssj |sk = 0, they can leave no record inE.

Economy of our assumptions stands in contrast withthe uncompromising nature of our conclusions: Perfectpredictability the fact that the evolution leading to in-

formation transfer preserves initial state of the system was, along with the principle of superposition, linearityof quantum evolutions, and preservation of the norm keyto our derivation of inevitability of quantum jumps.

B. Predictability killed the (Schrodinger) cat

There are several equivalent ways to state our conclu-sions. To restate the obvious, we have established thatoutcome states of non-perturbing measurements must beorthogonal. This is the interpretation - independent partof axiom (iv) all of it except for the literal collapse.

This is of course enough for the Everettian relative stateaccount of quantum jumps. So, a cat suspended betweenlife and death by Schrodinger (1935) is forced to make achoice between these two options because these are thepredictable options they allow (axiom (iii)) for confir-mation (hence the above title).

Another way of stating our conclusion is to note thata set of orthogonal states defines a Hermitian observablewhen supplemented with real eigenvalues. This is thena derivation of the nature of observables. It justifies thetextbook focus on the Hermitean operators sometimesinvoked in an alternative statement of axiom (iv).

We note that strict repeatability (that is, assertion

that states {|sk} cannot change at all in courseof a mea-surement) is not needed: They can evolve providing thattheir scalar products remain unaffected. That is:

j,k

jksj |sk =j,k

jksj |skj |k. (2.8)

leads to the same conclusions as Eq. (2.2) providing thatsj |sk =sj|sk. So, when|sj and|sk are relatedwith their progenitors by a transformation that preservesscalar product (e.g., by evolution in a closed system) theproof of orthogonality goes through unimpeded. Bothunitary and antiunitary transformations are in this class.

We can also consider situations when this is not thecase sj |sk =sj |sk. Extreme example of this ariseswhen the state of the measured system retains no mem-ory of what it was beforehand (e.g. |sj |0, |sk |0). Then the apparatus can (and, indeed, by unitarity,has to) inherit information previously contained in thesystem. The need for orthogonality of |sj and |sk disap-pears. Of course such measurements do not fulfill axiom(iii) they are not repeatable. For example, in quantumoptics photons are usually absorbed by detectors, andcoherent states play the role of the outcomes.

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It is also interesting to consider sequences of informa-tion transfers involving many systems:

|v|A0|B0 . . . |E0 = |v|Av|B0 . . . |E0 = . . .. . .= |v|Av|Bv . . . |Ev . (2.9a)

|w|A0|B0 . . . |E0 = |w|Aw|B0 . . . |E0 = . . .

. . .= |w|

Aw|

Bw . . . |Ew. (2.9b)Above we focused on a pair of states and simplified no-tation (e.g.,|v=|sk,|w=|sk). Such von Nemannchains appear in discussion of quantum measurements(von Neumann, 1933), environment-induced decoherence(Zurek, 1982; 2003a), and as we shall see in Section IV in quantum Darwinism. As information about systemis passed along this chain, links can be perturbed (as in-dicated bytilde). Unitarity implies that at each stage products of overlaps must be the same. Thus;

v||w = v||wAv||AwBv||Bv . . . Ev||Ew , (2.10)or after a logarithm of squares of the absolute values ofboth side is taken;

ln |v||w|2 = ln |v||w|2 + ln |Av||Aw|2 ++ ln |Bv||Bw|2 + + ln |Ev||Ew|2.(2.11)

Therefore, whenv||w = 0, as the information aboutthe outcome is distributed along the two von Neumannchains, quality of the records must suffer: Sum of loga-rithms above must equal ln |v||w|2, and the overlap oftwo states is a measure of their distinguishability. Fororthogonal states there is no need for such deteriorationof the quality of information; ln |v||w|2 = , so arbi-trarily many orthogonal records can be made.

We emphasize that Borns rule was not used in theabove discussion. The only two values of the scalar prod-uct that played key role in the proofs are 0 and 1.Both of them correspond to certainty; e.g. when we haveasserted immediately below Eq. (2.4) thatj |k = 1implies that these two states ofEare certainly identical.We shall extend this strategy of relying on certainty inthe derivation of probabilities in Section III.

C. Pointer basis, information transfer, and decoherence

We are now equipped with a set of measurement out-comes or to put it in a way that ties in with the studyof probabilities we shall embark on in the Section III with a set of possible events. Our derivation above didnot appeal to decoherence. However, symmetry breakinginduced by decoherence yields einselection (which is, af-ter all, due to information transfer to the environment).We will conclude that the two symmetry breakings arein effect two views of the same phenomenon.

Popular accounts of decoherence and its role in theemergence of the classical often start from the observa-tion that when a quantum systemSinteracts with some

environmentEphase relations inS are lost. This isa caricature, at best incomplete if not misleading: Itbegs the question: Phases between what?. This inturn leads directly to the main issue addressed by ein-selection: What is the preferred basis?. This questionis often muddled in folklore accounts of decoherence.

The crux of the matter the reason why interactionwith the environment can impose classicality is pre-

cisely the emergence of the preferred states. Its role andthe basic criterion for singling out preferred pointer stateswas discovered when the analogy between the role of theenvironment in decoherence and the role of the appara-tus in measurement were understood: What matters isthat there are interactions that transfer information andyet leave selected states of the system unaffected. Thisleads one to einselection to the environment inducedsuperselection of preferred pointer states (Zurek, 1981).

Our discussion showed that simple idea of preservinga state while transferring the information about it alsothe central idea of einselection is very powerful indeed.It leads to breaking of the unitary symmetry and singlesout preferred pointer states without any need to involveusual tools of decoherence. This is significant, as reduceddensity matrices and partial trace employed in decoher-ence calculations invoke Borns rule, axiom (v), whichrelates states vectors and probabilities.

To consider probabilities it is essential to identify out-comes separately from these probabilities (and, therefore,independently from the amplitudes present in the initialstate of the measured system). Both einselection imple-mented via the predictability sieve (Zurek, 1993; Zurek,Habib, and Paz, 1993; Paz and Zurek, 2001) and the nocloning approach above accomplish this goal.

To compare derivation of preferred states in decoher-ence with their emergence from symmetry breaking im-

posed by axioms (o) - (iii) we return to Eq. (2.2). Wealso temporarily suspend prohibition on the use of partialtrace to compute reduced density matrix of the system:

S=j,k

jkj |k|sjsk| =T rE|ESES| . (2.12)

Above we wrote S in the pointer basis defined by itsresilience in spite of the monitoring byE. Resilience isthe essence of the original definition of pointer states andeinselection (Zurek, 1981; 1982). We note that pointerstates will be in general different from Schmidt states ofS eigenstates ofS. They will coincide with Schmidtstates of

|ES

=

k

ak|sk

|k

only when

{|k

} their

records inE are orthogonal. We did not need suchperfect orthogonality of{|k} to prove orthogonality ofpointer states earlier in this section.

Folklore often assigns classicality to eigenstates ofS. This identification is occasionally supported by someof decoherence proponents (Albrecht, 1992; Zeh, 1990;2007) and taken for granted by others (e.g. Deutsch,1986), but by and large it is no longer regarded as viable(see e.g. Schlosshauer, 2004; 2007): Eigenstates ofSarenotstable. They depend on time and on the initial state

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ofS, which disqualifies them as events in the sense ofprobability theory, as the elements of classical reality.A related problem arises in a very long time limit, whenequilibrium sets in, so energy eigenstates diagonalize S.

As is frequently the case with folk wisdom, a grainof truth is nevertheless reflected in such oversimplifiedproverbs: When environment acquires perfect knowl-edge of the states it monitors without perturbing and

j |k =jk , pointer states become Schmidt, and endup on the diagonal ofS. Effective decoherence assuressuch alignment of Schmidt states with the pointer states.Given that decoherence is at least in the macroscopicdomain very efficient, this can happen essentially in-stantaneously. Still, this coincidence should not be usedto attempt a redefinition of pointer states as instanta-neous eigenstates ofS instantaneous Schmidt states.As we have already seen, and as will become even clearerin the rest of this paper, it is important to distinguishprocess that fixes preferred pointer states (that is, dy-namics of information transfer that results in measure-ment as well as decoherence, but does not depend on the

initial state of the system) from the probabilities of theseoutcomes (that are determined by this initial state).

D. Summary: The origin of outcome states

Preferred states of quantum systems emerge from dy-namics. Interaction between the system and the envi-ronment plays crucial role: States that are immune tomonitoring by the environment are predictable, and atleast in that sense the most classical.

Selection of pointer states is determined by the evo-lution i.e., in practice by the completely positive map

(CPM) that represents open system dynamics. There-fore, instantaneous eigenstates of the reduced densitymatrix of the open systems will in general not coincidewith the preferred pointer states singled out by einselec-tion. However, as pointer states are capable of evolvingpredictably under the CPM in question, they will be eventually, and after decoherence has done its job, butbefore the system has equilibrated found on or near di-agonal ofS. It is nevertheless important to emphasizethat definition of pointer states, e.g. via predictabilitysieve is based on stability, and noton this coincidence.

The lesson that derives from this section as well asfrom earlier studies, including the original definition ofpointer states (Zurek, 1981) is that the preferred, ef-fectively classical basis has nothing to do with the initialstate of the system. This is brings to mind Bohrs insis-tence that the measured observable of the system is de-fined by the classical apparatus, so that it arises outsideof the quantum domain. Here preferred observables aredefined not by the classical apparatus, but by the opensystem dynamics. Completely positive map describing itwill preserve (at least approximately) certain subset ofall the possible states. Extremal points of this set areprojection operators corresponding to pointer states. In

Bohrs view as well as here they are defined by some-thing else than just the preexisting quantum states. Wehave defended the idea that they are classical by pointingout to their predictability. In Section IV we shall showthat they can be often found out by observers withoutgetting disrupted (a telltale sign of objective existence,which will reinforce case for their classicality).

In the next section we derive Borns rule. We build on

einselection, but do it in a way that does not rely on ax-iom (v). In particular, use of reduced density matrices orcompletely positive maps we allowed in the latter part ofthis section shall be prohibited. We shall use them againin section IV, only after Borns rule has been derived.

III. PROBABILITIES AND BORNS RULE FROM THESYMMETRIES OF ENTANGLEMENT

The first widely accepted definition of probability wascodified by Laplace (1820): When there areN possibledistinct outcomes and observer is ignorant of what will

happen, all alternatives appear equally likely. Probabil-ity one should assign to any one outcome is then 1/N.Laplace justified this principle of equal likelihoodusinginvarianceencapsulated in his principle of indifference:Player ignorant of the face value of cards in front of him(Fig. 1a) will be indifferentwhen they are swapped be-fore he gets the card on top, even when one and only oneof the cards is favorable (e.g., a spade he needs to win).

Laplaces invariance under swaps captures subjectivesymmetry: Equal likelihood is a statement about ob-servers state of mind (or, at best, his records), and nota measurable property of the real physical state of thesystem which is altered by swaps (Fig. 1b). In classical

setting probabilities defined in this manner are thereforeultimately unphysical. Moreove, indifference, likelihood,and probability are all ill-defined attempts to quantifythe same ignorance. Expressing one undefined conceptin terms of another undefined concept is not a definition.

It is therefore no surprise that equal likelihood is nolonger regarded as a sufficient foundation for classicalprobability, and several other attempts are vying for pri-macy (Fine, 1973). Among them, relative frequency ap-proach has perhaps the largest following, but attemptsto use it in the relative states context (Everett, 1957a,b;DeWitt, 1970; 1971; Graham, 1973; Geroch, 1984) havebeen found lacking. This is because counting many worldbranches does not suffice. Maverick branches that havefrequencies of events very different from these predictedby Borns rule are also a part of the universal state vec-tor, and on the basis of relative frequencies alone thereis no good reason to claim that observer is unlikely to befound on such a branch. To get rid of them one wouldhave to assign to them without physical justification vanishing probabilities related to their small amplitudes.This goes beyond relative frequency approach, in effectrequiring in addition to frequencies another measureof probability. Papers mentioned above introduce it ad

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a)

b)

c)

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

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

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

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

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

~~

=

FIG. 1 Probabilities and symmetry: (a) Laplace (1820) appealed to subjective invariance associated with indifference of theobserver based on his ignorance of the real physical state to define probability through his principle of equal likelihood. Whenobservers ignorance means he is indifferent to swapping (e.g., of cards), alternative events should be considered equiprobable.So, for the cards above, subjective probability p = 12 would be inferred by an observer who does not know their face value,but knows that one (and only one) of the two cards is a spade. (b) The real physical state of the system is however alteredby the swap it is not indifferent, illustrating subjective nature of Laplaces approach. The subjectivity of equal likelihoodprobabilities posed foundational problems in statistical physics, and led to the introduction of imaginary ensembles. They werefictitious and subjective, but their state could be associated with probabilities defined through relative frequency an objectiveproperty (albeit of a fictitious infinite ensemble). (c)Quantum theory allows for an objective definition of probabilities basedona perfectly known stateof a composite system and symmetries of entanglement: When two systems (Sand E) are maximallyentangled (i.e, Schmidt coefficients differ only by phases, as in the Bell state above), a swap ||+ || in Scan be undoneby counterswap|| + || inE. So, as can be established more carefully (see text), p = p = 12 follows from anobjective symmetry of entanglement. This entanglement - assisted invariance (envariance) also causes decoherence of Schmidtstates, allowing for additivity of probabilities of the effectively classical pointer states. Probabilities derived through envariancequantify indeterminacy of the state ofS alone given the global entangled state ofSE. Complementarity between globalobservables with entangled eigenstates (such as|SE , Eq. (3.5)) and local observables (such as S = Pkk|sksk| 1E)is reflected in the commutator [|SE SE |, S 1E ] =

Pkk(|SE sk|k| h.c.. It does not vanish, implying quantum

indeterminacy responsible for the uncertainty about the outcome of a future measurement ofS =P

kk|sksk| onSalone.

hoc. This is consistent with Borns rule, but circular(Stein, 1984; Kent, 1990; Joos, 2000). Indeed, formal at-tempts based on the frequency operator lead to math-ematical inconsistencies (Squires, 1990).

The problem can be made to disappear coeffi-cients of maverick branches become small (along withcoefficients of all the branches) in a limit of infinite

and fictitious (and, hence, subjectively assigned) ensem-bles is introduced (Hartle, 1968; Farhi, Goldstone, andGuttman, 1989). Such infinite ensembles one mightargue are always required by the frequentist approachalso in the classical setting, but this is a weak excuse(see Kent, 1990). Moreover, in quantum mechanics theymay pose problems that have to do with the structure of

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infinite Hilbert spaces (Poulin, 2005; Caves and Shack,2005). It is debatable whether these mathematical prob-lems are fatal, but it is also difficult to disagree withKent (1990) and Squires (1990) that the need to go to alimit of infinite ensembles to define probability in a finiteUniverse disqualifies them in the relative states setting.

The other way of dealing with this issue is to modifyphysics so that branches with small enough amplitude

simply do not count (Buniy, Hsu, and Zee, 2006). We be-lieve it is appropriate to regard such attempts primarilyas illustration of seriousness of the problem at least untilexperimental evidence for the required modifications ofquantum theory is found.

Kolmogorovs approach probability as a measure (seee.g. Gnedenko, 1968) bypasses the question we aimto address: How to relate probability to states. It onlyshows that any sensible assignment (non-negative num-bers that for a mutually exclusive and exhaustive set ofevents sum up to 1) will do. And it assumes additiv-ity of probabilities while quantum theory provides onlyadditivity of complex amplitudes (and, typically, thesetwo additivity requirements are mutually inconsistent, asdouble slit experiment famously demonstrates).

Gleasons theorem (Gleason, 1957) implements ax-iomatic approach to probability by looking for an ad-ditive measure on Hilbert spaces. It leads to Bornsrule, but provides no physical insight into why the resultshould be regarded as probability. Clearly, it has notsettled the issue: Rather, it is often cited (Hartle, 1968;Graham, 1973; Farhi, Goldstone, and Guttman, 1989;Kent, 1990) as a motivation to seek a physically justifiedderivation of Borns rule.

We shall now demonstrate how quantum entanglementleads to probabilities based on a symmetry, but in con-trast to subjective equal likelihood on objective sym-

metry ofknownquantum states.

A. Envariance

A pure entangled state of a systemSand of anothersystem (which we call an environmentE, anticipatingconnections with decoherence) can be always written as:

|SE =Nk=1

ak|sk|k. (3.1)

Here ak are complex amplitudes while{|sk}and{|k}are orthonormal bases in the Hilbert spaces

HSand

HE.

This Schmidt decompositionof pure entangled|SEis aconsequence of a theorem of linear algebra that predatesquantum theory.

Schmidt decomposition demonstrates that any pureentangled bipartite state is a superposition of perfectlycorrelated outcomesof judiciously chosen measurementson each subsystem: Detecting|sk onS implies, withcertainity,|kforE, and vice versa.

Even readers unfamiliar with Eq. (3.1) have likely re-lied on its consequences: Schmidt basis{|sk} appears

on the diagonal of the reduced density matrix S =T rE|SESE|. But tracing presumes Borns rule we aretrying to derive (see e. g. Nielsen and Chuang, 2000, forhow Borns rule is used to justify physical significance ofreduced density matrices). Therefore, we shall not useSas this could introduce circularity. Instead, we shallderive Borns rule from symmetries of|SE.

Symmetries reflect invariance. Rotation of a circle by

an arbitrary angle, or of a square by multiples of/2 arefamiliar examples. Entangled quantum states exhibit anew kind of symmetry entanglement - assisted invari-anceor envariance: When a state|SE of a pairS,Ecan be transformed by US=uS 1Eacting solely onS,

US|SE = (uS 1E)|SE = |SE , (3.2)but the effect ofUScan be undone by acting solely onEwith an appropriately chosen UE= 1S uE:

UE|SE = (1S uE)|SE = |SE, (3.3)then |SE is called envariant underUS(Zurek, 2003a;b).

Envariance can be seen on any entangled|SE. Anyunitary operation diagonal in Schmidt basis{|sk}:

uS=Nk=1

exp(ik)|sksk| , (3.4a)

is envariant: It can be undone by a countertransforma-tion:

uE=Nk=1

exp(ik)|kk| , (3.4b)

acting solely on environment.

In contrast to familiar symmetries (when a transforma-tion has no effect on a state or an object) envariance is anassisted symmetry: The global state ofSEistransformedbyUS, but it can be restored by acting onE, physicallydistinct (e.g., spatially separated) fromS. When thestate ofSE is envariant under some US, the state ofSalone must be obviously invariant under it.

Entangled state might seem an unusual starting pointfor the discussion of probabilities. After all, textbook for-mulation of Borns rule begins with a pure state. How-ever, in Everetts approach entangled quantum state is amodel of a measurement, with the outcomes correspond-ing to different states of the apparatus pointer (or of thememory of the observer). A similar sort of entanglementhappens in course of decoherence. So it is natural to en-quire about the symmetries of such states. And as weshall see below, envariance allows one to reassess the roleof the environment and the origin of decoherence.

B. Decoherence as a result of envariance

Envariance of entangled states leads to our first conclu-sion: Phases of Schmidt coefficients are envariant under

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local (Schmidt) unitaries, Eqs. (3.4). Therefore, whena composite system is in an entangled state|SE, thestate ofS (orE) alone is invariant under the change ofphasesofak. In other words, the stateofS (understoodas a set of all measurable properties ofS alone) cannotdepend on phases of Schmidt coefficients: It can dependonly on their absolute values and on the outcome states on the set of pairs{|ak|, |sk}. In particular (as we shallsee below) probabilities cannot depend on these phases.

So loss of phase coherence between Schmidt states decoherence is a consequence of envariance: Decoher-ence is, after all, selective loss of relevance of phases forthe state ofS. We stumbled here on its essence while ex-ploring an unfamiliar territory, without the usual back-drop of dynamics and without employing trace and re-duced density matrices. This encounter is a good omen:Borns rule, the key link between the quantum formalismand experiments, is also involved in the transition fromquantum to classical. But decoherence viewed from thevantage point of envariance may look unfamiliar.

What other landmarks of decoherence can we get to

without using trace and reduced density matrices (whichrely on Borns rule something we do not yet have)? Theanswer is all the essential ones (Zurek, 2005; 2007). Wehave seen in Section II that pointer states (states thatretain correlations, and, hence, are predictable and goodcandidates for classical domain) are singled out directlyby the nature of information transfers. So we alreadyhave a set of preferred pointer states and we have seenthat when they are aligned with Schmidt basis, phasesbetween them lose relevance for S alone. Indeed, modelsof decoherence (Zurek, 1981, 1982, 1991, 2003a; Paz andZurek, 2001; Joos et al, 2003, Schlosshauer, 2007) predictthat after a brief (decoherence time) interlude Schmidtbasis will settle down to coincide with pointer states de-termined through other criteria (such as predictability inspite of the coupling to the environment).

One more lesson can be drawn from this encounterwith decoherence on the way to Borns rule: Quantumphases must be rendered irrelevant for additivity of prob-abilities to replace additivity of complex amplitudes. Ofcourse, one could postulate additivity of probabilities byfiat. This was done by Gleason (1957), but such an as-sumption is at odds with the overarching additivity prin-ciple of quantum mechanics with the quantum prin-ciple of superposition. So, if we set out with Everetton a quest to understand emergence of the classical do-main from the quantum domain defined by axioms (o)

(iii), additivity of probabilities should be derived (as it isdone in Laplaces approach, see Gnedenko, 1968) ratherthan imposed as an axiom (as it happens in Kolmogorovsmeasure - theoretic approach, and in Gleasons theorem).

Assuming decoherence to derive pk =|k|2 (Zurek,1998b; Deutsch, 1999; Wallace, 2003) means at beststarting half way, and courts circularity (Zeh, 1997;Zurek, 2003a,b; 2005; Schlosshauer, 2007) as physicalsignificance of reduced density matrix standard tool ofdecoherence is justified using Borns rule. By contrast,

envariant derivation, if successful, can be fundamental,independent of the usual tools of decoherence: It will

justify use of trace and reduced density matrices in thestudy of the quantum - classical transition.

There is clearly much more to say about preferredstates, einselection, and decoherence, and we shall comeback to these subjects later in this section. But now wereturn to the derivation of Borns rule to a problem

Everett did appreciate and attempted to solve.

C. Swaps and equal probabilities

Envariance of pure states is purely quantum: Classi-cal state of a composite system is given by a Cartesian(rather than tensor) product of its constituents. So, tocompletely know a state of a composite classical systemone must know a state of each subsystem. It follows thatwhen one part of a classical composite system is affectedby a transformation a classical analogue ofUS stateof the whole cannot be restored by acting on some otherpart. Hence, pure classical states are never envariant.

However, a mixed state (of, say, two coins) can mimicenvariance: When we only know that a dime and a nickelare same side up, we can undo the effect of the flip of adime by flipping a nickel. This classical analogue dependson a partial ignorance: To emulate envariance, we cannotknow individual states of coins, just the fact that theyare same side up just the correlation.

In quantum physics tensor structure of states for com-posite systems means that pure correlation is possible.We shall see that a maximally entangled state with equalabsolute values of Schmidt coefficients:

|SE

N

k=1

eik

|sk

|k

(3.5)

implies equal probabilities for any measurementS andE. Such an evenstate is envariant under a swap

uS(k l) = |sksl| + |slsk| . (3.6a)A swap exchanges two cards (Fig.1c). It permutes states|sk and|sl of the system. A swap|HeadsTails| +|TailsHeads| would flip a coin.

A swap onS is envariant when|ak| =|al| becauseuS(k l) can be undone by a counterswaponE;

uE(k l) = ei(kl)|lk| +ei(kl)|kl| .

(3.6b)Envariance under swaps is illustrated in Fig. 1c. Wewant toprovethat probabilities of envariantly swappableoutcome states must be equal. But let us proceed withcaution: Invariance under a swap is not enough proba-bility could depend on some other intrinsic property ofthe outcome. For instance, in a superposition|g + |e,the ground and excited state can be invariantly swapped,but theirenergiesare different. Why shouldnt probabil-ity like energy depend on some intrinsic property ofthe outcome state?

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Envariance can be used to prove that it cannot thatprobabilities of envariantly swappable states are indeedequal. To prove this, we first define what is meant bythe state and the system more carefully. Quantumrole of these concepts is elucidated by three Facts three assumptions that are tacitly made anyway, but areworth stating explicitly:

Fact 1: Unitary transformations must act on the systemto alter its state. That is, when an operator doesnot act on the Hilbert spaceHS ofS, i.e., whenit has a form ... 1S ... the state ofS does notchange.

Fact 2: Given the measured observable, the state of thesystemS is all that is needed (and all that isavailable) to predict measurement results, includ-ing probabilities of outcomes.

Fact 3: The state of a larger composite system that in-cludesSas a subsystem is all that is needed (andall that is available) to determine the state ofS.

Note that resulting states need not be pure. Also notethat Facts while naturally quantum are not in anyobvious conflict with the role of states in classical physics.

We can now prove that for an even|SE, Eq. (3.5),state ofSalone is invariant under swaps: Swap changespartners in the Schmidt decomposition (and, therefore,alters the global state). But, when the coefficients ofthe swapped outcomes differ only by phase, swap can beundone without acting on S by a counterswap in E. Asthe global state ofSEis restored, it follows (from fact 3)that the state ofS must have been also restored. But,(by fact 1) state ofScould not have been affected by acounterswap acting only onE. So, (by fact 2) the stateofSmust be left intact by a swap on Scorresponding toan even|SE. QED.

We conclude that envariance of a pure global state ofSEunder swaps implies invariance of a corresponding lo-cal state ofS. We could now follow Laplace, appeal toindifference, apply equal likelihood, and declare victory claim that subjective probabilities must be equal. How-ever, as we have seen with the example of the eigenstatesof energy, invariance of a local state under a swap impliesonly that probabilities get swapped when outcomes areswaped. This does not yet prove they are equal!

Envariance will allow us to get rid of subjectivity alto-gether. The simplest way to establish this desired equal-ity is based on perfect correlation between Schmidt statesofSandE. These are relative states in the sense of Ev-erett, and they are orthonormal, so they are correlatedone-to-one. This implies the same probability for eachmember of a pair. Moreover (and for the same reason)after a swap onSprobabilities of swapped states mustbe the same as probabilities of their two new partners inE. But (by Fact 1) the state ofE(and, by Fact 2, proba-bilities it implies) are not affected by the swap inS. So,swapping Schmidt states ofS exchanges their probabili-ties, and when the entangled state is even it also keeps

them the same!This can be true only if probabilities ofenvariantly swappable states are equal. QED.

We can now state our conclusion: When all N coeffi-cients in Schmidt decomposition have the same absolutevalue (as in Eq. (3.5)), probability of each Schmidt stateis the same, and, by normalization, it is pk = 1/N.

1

Reader may regard this as obvious, but (as noted bySchlosshauer and Fine (2005) and Barnum (2003)), this

is actually the hard part of the derivation, as it requiresestablishing a connection between quantum physics andmathematics with only minimal set of assumptions athand. Still, this may seem like a lot of work to establishsomething obvious: The case of unequal coefficients isour real goal. But as we now show it can be reducedto the equal coefficient case we have just settled.

It is important to emphasize that in contrast to manyother approaches to both classical and quantum prob-ability, our envariant derivation is based not on a sub-

jective assessment of an observer, but on an objective,experimentally verifiable symmetry of entangled states.Observer is forced infer equal probabilities not because

of his ignorance, but because his certainty about some-thing else about a global state of the composite system implies that local states are completely unknown.

D. Borns rule from envariance

To illustrate general strategy we start with an exam-ple involving a two-dimensional Hilbert space of the sys-tem spanned by states{|0, |2} and (at least) a three-dimensional Hilbert space of the environment:

|SE

2

3|0S|+E +

1

3|2S|2E . (3.7a)

System is represented by the leftmost kets, and|+E =(|0E+ |1E)/

2 exists in (at least two-dimensional) sub-

space ofE that is orthogonal to the state|2E, so that0|1=0|2=1|2=+|2 = 0. We already know wecan ignore phases in view of their irrelevance for statesof subsystems, so we omitted them above.

To reduce this case to an even state we extend|SEabove to a state |SEC with equal coefficients by lettingE act on an ancillaC . (By Fact 1, sinceS is not actedupon, so probabilities we shall infer for is cannot change.)This can be done by a generalization of controlled-not

1 There is an amusing corollary to this theorem: One can nowprovethat states which appear in a Schmidt decomposition withcoefficients al = 0 have 0 probability: To this end, consider adecomposition that has n such states. One can combine two ofthese states to form a new state, which still has the same co-efficient of 0. This purely mathematical step that cannot haveany physical implications for probabilities of states that were notinvolved. Yet, there are now only n1 states with equal coef-ficients. So the probability pl of any state with zero amplitudehas to satisfy npl = (n1)pl, which holds only when pl = 0.

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acting betweenE (control) andC (target), so that (inthe obvious notation)|k|0 |k|k, leading to;

2|0|+|0 + |2|2|0 =

=

2|0 |0|0 + |1|1

2+ |2|2|0 (3.8a)

Above, and from now on we skip subscripts: State of

Swill be always listed first, and state ofC will be primed.The cancellation of

2 yields equal coefficient state:

|SCE |0, 0|0 + |0, 1|1 + |2, 2|2 . (3.9a)Note that we have now combined state ofS andC and(in the next step) we shall swap states ofSC as if it wasa single system.

Clearly, for joint states|0, 0,|0, 1, and|2, 2 ofSCthis is a Schmidt decomposition of (SC )E. The three or-thonormal product states have coefficients with the sameabsolute value. So, they can be envariantly swapped. Itfollows that the probabilities of these Schmidt states |0|0,|0|1, and|2|2 are all equal, so by normal-ization they are

13 . Moreover, probability of state|2 of

the system is 13 . As |0 and |2 are the only two outcomestates forS, it also follows that probability of|0 mustbe 23 . Consequently:

p0 =2

3; p2 =

1

3 . (3.10a)

This is Borns rule!Note that above we have avoided assuming additiv-

ity of probabilities: p0 = 23 not because it is a sum of

two fine-grained alternatives each with probability 13 , butrather because there are only two (mutually exclusive andexhaustive) alternatives forS;|0 and|2, and p2 = 13 .So, by normalization, p0 = 1

1

3 .Bypassing appeals to additivity of probabilities is agood idea in interpreting a theory with another principleof additivity quantum superposition principle whichtrumps additivity of probabilities or at least classical in-tuitive ideas about what should be additive (e.g., in thedouble slit experiment). Here this conflict is averted:Probabilities of Schmidt states can be added because ofthe loss of phase coherence that follows directly from en-variance as we have established earlier (Zurek, 2005).

Consider now a general case. For simplicity we focuson entangled state with only two non-zero coefficients:

|SE

=

|0

|0

+

|1

|1

, (3.7b)

and assume =

mM; =

MmM , with integerm,M.

As before, the strategy is to convert a general entan-gled state into an even state, and then to apply envari-ance under swaps. To implement it, we assumeE hassufficient dimensionality to allow decomposition of|0and|1 in a different orthonormal basis{|ek}:

|0 =mk=1

|ek/

m; |1 =M

k=m+1

|ek/

M m

Envariance we need is associated with counterswaps ofE that undo swaps of the joint state of the compositesystem SC . To exhibit it, we let ancilla C interact with Eas before, e.g. by employingEas a control to carry out|ek|c0 |ek|ck, where|c0 is the initial state ofC insome suitable orthonormal basis{|ck}. Thus;

|SCE

m |0

m

k=1 |

ck

|ek

m +

M m |1

M

k=m+1 |

ck

|ek

M m(3.8b)

obtains. ThisCEinteraction can happen far fromS, soby Fact 1 it cannot influence probabilities inS. |SCEis envariant under swaps of states |s, ck of the compositeSC system (where s stands for 0 or 1, as needed). Thisis even more apparent after the obvious cancellations;

|SCE mk=1

|0, ck|ek +M

k=m+1

|1, ck|ek. (3.9b)

Hence,p0,k = p1,k = 1M. So, probabilities of|0and|1:

p0 = m

M = ||2; p1 = M m

M = ||2 (3.10b)

are given by Borns rule. It arises from the most quantumaspects of the theory entanglement and envariance.

In contrast with other approaches, probabilities in ourenvariant derivation are a consequence of complementar-ity, of the incompatibility of purity of entangled state ofthe whole with purity of the states of parts. Borns rulearises in a completely quantum setting, without any aprioriimposition of symptoms of classicality that violatespirit of quantum theory. In particular, envariant deriva-tion (in contrast to Gleasons successful but unphysical

proof and to Everetts unsuccessful attempt) does notrequire additivity as assumption: The strategy that by-passes appeal to additivity used in the simple case of Eq.(3.10a) can be generalized (Zurek, 2005). In quantumsetting this is an important advance. It can be madeonly because relative phases between Schmidt states areenvariant because of decoherence. The case of morethan two outcomes is straightforward, as is extension bycontinuity to incommensurate probabilities.

E. Relative frequencies from relative states

We can now use envariance to deduce relative frequen-cies. ConsiderNdistinguishableSCEtriplets, all in thestate of Eq. (3.9). The state of the ensemble is then;

|NSCE = N=1|()SCE (3.11)We now repeat steps that led to Eqs. (3.10) for theSCEtriplet, and think ofC as a counter, a detector in whichstates|c1 . . . |cm record 0 inS, while|cm+1 . . . |cMrecord 1. Carrying out tensor product and countingterms with n detections of 0 yields the total N(n) =

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Nn

mn(M m)Nn of fine-grained records or of the

corresponding envariantly swappable Everett brancheswith histories of detections that differ by a sequence of0s and 1s, and by the labels assigned to them by thecounter, but that have the same total of 0s. Normaliz-ing leads to probability of a record with n 0s:

pN(n) =Nn||2n||2(Nn)

e 1

2n||NN||

2

2N || (3.12)

Gaussian approximation of a binomial is accurate forlargeN: We shall assumeNis large not because envari-ant derivation requires this (we have already obtainedBorns rule forN = 1), but because relative frequencyapproach needs it (von Mises, 1939; Gnedenko, 1968).

The average number of 0s is, according to Eq. (3.12)n = ||2N, as expected, establishing a link between rel-ative frequency of events in a large number of trials andBorns rule. This connection between quantum statesand relative frequencies does not rest on either circu-lar and ad hocassumptions that relate size of the coef-

ficients in the global state vector to probabilities (e.g.,by asserting that probability corresponding to a smallenough amplitude is 0 (Geroch 1984); Buniy, Hsu, &Zee, 2006)), modifications of quantum theory (Weissman,1999), or on the unphysical infinite limit (Hartle, 1968;Farhi, Goldstone, and Guttmann, 1989). Such steps haveleft past frequentist approaches to Borns rule (includingalso these of Everett, DeWitt, and Graham) open to avariety of criticisms (Stein, 1984; Kent, 1990; Squires,1990; Joos, 2000; Auletta, 2000).

In particular, we avoid problem of two independentmeasures of probability (number of branches and sizeof the coefficients) that derailed previous relative state

attempts. We simply count the number ofenvariantlyswappable (and, hence provably equivalent) sequences ofpotential events. This settles the issue of maverick uni-verses atypical branches with numbers of e.g. 0s quitedifferent from the averagen. They are there (as theyshould be) but are very improbable. This is establishedthrough a physically motivated envariance under swaps.So, maverick branches did not have to be removed eitherby assumption (DeWitt, 1970; 1971; Graham, 1973;Geroch, 1984) or by an equally unphysicalN = .

F. Summary

Envariance settles major outstanding problem of rela-tive state interpretation: The origin of Borns rule. It canbe now established without assumption of the additivityof probabilities (as in Gleason, 1957). We have also de-rived pk = |k|2 without relying on tools of decoherence.

Recently there were other attempts to apply Laplaceanstrategy of invariance under permutations to prove in-difference. This author (Zurek, 1998b) noted that allof the possibilities listed on a diagonal of a unit densitymatrix (e.g., |00| + |11|) must be equiprobable, as

it is invariant under swaps. This approach can be thenextended to the case of unequal coefficients, and leads toBorns rule. However, a density matrix is not the rightstarting point for the derivation: A pure state, preparedby the observer in the preceding measurement, is. And toget from such a pure state to a mixed (reduced) densitymatrix one must trace average over e.g., the envi-ronment. Borns rule is involved in the averaging, which

courts circularity (Zeh, 1997; Zurek, 2003a,b; 2005).One could of course try to start with a pure state in-

stead. Deutsch (1999) and his followers (Wallace, 2002,2003; Saunders, 2004) pursue this strategy, couched interms of decision theory. The key is again invarianceunder permutations. It is indeed there for certain purestates (e.g.,|0 + |1) but not when relative phase is in-volved. That is,|0 + |1 equals the post-swap|1 + |0,but |0+e|1 = |1+e|0, and the difference is not theoverall phase. Indeed, |0+ i|1 isorthogonaltoi|0+ |1,so there is no invariance under swaps. In an isolatedsystem this problem cannot be avoided. (Envariance ofcourse deals with it very naturally.) The other prob-

lem is selection of events one of which will happen uponmeasurement the choice of preferred states. These twoproblems must be settled, either through appeal to deco-herence (as in Zurek, 1998b, and in Wallace, 2002; 2003),or by ignoring phases essentially ad hoc (Deutsch, 1999),which then makes readers suspect that some Copen-hagen assumptions are involved. Indeed, Barnum et al.(2000) criticized Deutsch (1999) paper interpreting hisapproach in the Copenhagen spirit. And decoherence invoked by Wallace (2002) employs reduced densitymatrices, and, hence, Borns rule (Zurek, 2003b, 2005;Baker, 2007; Forrester, 2007; Schlosshauer, 2007).

Envariant derivation of Borns rule we have presentedis an extension extension of the swap strategy in (Zurek,1998b): However, instead of tracing the environment, weincorporated it in the discussion. This leads to Bornsrule, but also to new appreciation of decoherence. Pointerstates can be inferred directly from dynamics of informa-tion transfers as was shown in Section II and, indeed, inthe original definition of pointer states (Zurek, 1981).Not everyone is comfortable with envariance (see e.g.Herbut, 2007, for a selection of views). But this is un-derstandable interpretation of quantum theory is a fieldrife with controversies.

Envariance is firmly rooted in physics. It is basedon symmetries of entanglement. One may be neverthe-less concerned about the scope of envariant approach:

pk =|k|2 for Schmidt states, but how about measure-ments with other outcomes? The obvious starting pointfor derivation of probabilities is not an entangled stateofS andE, but a pure state ofS. And such a statecan be expressed in any basis that spansHS. So whyentanglement? And why Schmidt states?

Envariance of phases of Schmidt coefficients is closelytied to einselection of pointer states: After decoherencehas set in, pointer states nearly coincide with Schmidtstates. Residual misalignment is not going to be a major

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problem. At most, it might cause minor violations oflaws obeyed by the classical probability for events definedby pointer states. Such violations are intriguing, andperhaps even detectable, but unlikely to matter in themacroscopic setting we usually deal with. To see why,we revisit pointer states Schmidt states (or einselection- envariance) link in the setting of measurements.

Observer

Ouses an (ultimately quantum) apparatus

A, initially in a known state|A0, to entangle withS,which then decoheres asA is immersed inE (Zurek,1991, 2003a; Joos et al., 2003; Schlosshauer, 2004; 2007).This sequence of interactions leads to: |S|A0|0

kak|sk|Ak|0 kak|sk|Ak|k. In a prop-

erly constructed apparatus pointer states|Ak are un-perturbed byE while|k become orthonormal on a de-coherence timescale. So in the end we have Schmidt de-composition ofSA (treated as a single entity) andE.

Apparatus is built to measure a specific observableS =

kk|sksk|, and O knows thatS starts in|S =

kak|sk. The choice ofA (of Hamiltonians,

etc.) commits observer to a definite set of potential out-

comes: Probabilities will refer to {|Ak}, or, equivalently,to {|Aksk} in the Schmidt decomposition. So, to answerquestions we started with, entanglement is inevitable,and only pointer states (e.g., nearly Schmidt states af-ter decoherence)of the apparatuscan be outcomes. Thisemphasis on the role of apparatus in deciding what hap-pens parallels Bohrs view captured by No phenomenonis a phenomenon untill it is a recorded phenomenon(Wheeler, 1983). However, in our caseA is quantumand symptoms of classicality e.g., einselection as wellas loss of phase coherence between pointer states ariseas a result of entanglement withE.

Envariant approach applies even when |sk arentorthogonal: Orthogonality of

|Aksk

is assured by

Ak|Al =kl by distinguishability of records in a goodapparatus. This is because events that we have directaccess to are records inA(rather than states ofS).

Other simplifying assumptions we invoked can be alsorelaxed (Zurek, 2005). For example, whenE is initiallymixed (as will be generally the case), one can purify

it by adding extra E in the usual manner (see SectionII). Given that we already have a derivation of Bornsrule, its use (when justified by the physical context) doesnot require apologies, and does not introduce circular-ity. Indeed, it is interesting to enquire what instances ofprobabilities in physicscannotbe interpreted envariantly.

Purifications, use of ancillae, fine - graining, and othersteps in the derivation need not be carried out in thelaboratory each time probabilities are extracted from astate vector: Once established, Borns rule is a law. Itfollows from the geometry of Hilbert spaces for compos-ite quantum systems. We used assumptions aboutC, E,etc., to demonstrate pk =|k|2, but this rule must beobeyed even when no one is there to immediately verifycompliance. So, even when there is no ancilla at hand, orwhenE is initially mixed or too small for fine-graining,one could (at some later time, using purification, extra

environments and ancillae) verify that bookkeeping im-plicit in assigning probabilities to|S or pre-entangled|SE abides by the symmetries of entanglement.

The obvious next question is how to verify envariancedirectly. One way is to carry out experimentally steps weoutlined earlier, Eqs. (3.4)-(3.10). To this end, one cane.g. attach ancillaC , carry out swaps, and make inter-mediate measurements that verify equal probabilities in

the fine-grained states ofSC through swaps. This leadsto probability of potential outcomes it can be deducedfrom sequences of reversible operations that involve justa single copy of|S (rather than a whole ensemble) bydetermining fraction of equiprobable alternatives|skckcorresponding to specific|sk. Such sequences of swaps,confirmatory measurements, and counterswaps can beeasily devised on paper, but harder to implement: Forinstance, it would be best if measurements probed an en-tangled global state ofSCE, as global outcomes can bepredicted with certainty (so that confirming envarianceat the root of Borns rule does not rely on Borns rule).However, even measurements that destroy global state

and require statistics to verify envariance (e.g., quantumtomography) would be valuable: What is at stake is a ba-sic symmetry of nature that provides a key link betweenthe unitary bare quantum theory and experiments.

Probabilities described by Borns rule quantify igno-rance ofO beforehe measures. So they admit ignoranceinterpretation O is ignorant of the future outcome,(rather than of an unknown pre-existing real state, aswas the case classically). Of course, onceOs memorybecomes correlated withA, its state registers whatOhas perceived (say,|o7 that registers |A7). Re-checkingof the apparatus will confirm it. And, when many sys-tems are prepared in the same initial state, frequenciesof outcomes will be in accord with Borns rule.

Envariant approach uses incompatibility between ob-servables of the whole and its parts. In retrospect itseems surprising that envariace was not noticed and usedbefore to derive probability or to provide insights intodecoherence and environment - induced superselection:Entangling interactions are key to measurements and de-coherence, so entanglement symmetries would seem rele-vant. However, entanglement is often viewed as paradox,as something that needs to be explained, and not used inan explanation. This attitude is, fortunately, changing.

IV. QUANTUM DARWINISM

Objective existence in quantum theory is a conse-quence of a relationshipbetween the system and the ob-server, and not just (as was the case classically) soleresponsibility of the system. This relational view of ex-istence is very much in concert with Everett: Relativestates can exist objectively, providing that observer willonly measure observables that commute with the pre-existing mixed state of the system (e.g., in the wake ofdecoherence, its pointer observable). But why should the

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observers measure only pointer observables?Quantum Darwinism provides a simple and natural ex-

planation of this restriction, and, hence, of the objectiveexistence bulwark of classicality for the einselectedstates: Information we acquire about the rest of theUniverse comes to us indirectly, through the evidencesystems of interest deposit in their environments. Ob-servers access directly only the record made in the envi-

ronment, an imprint of the original state ofSon the stateof a fragment ofE. And there are multiple copies of thatoriginal (e.g., of this text) that are disseminated by thephoton environment, by the light that is either scattered(or emitted) by the printed page (or by the computerscreen). We can find out the state of various systemsindirectly, because their correlations withE (which weshall quantify below using mutual information) allowEto be a witness to the state of the system.

The plan of this section is to define mutual infor-mation, and to use it to quantify information that canbe gained aboutS fromE. Objectivity arises becausethe same information can be obtained independently bymany observers from many fragments of

E. So, to quan-

tify objectivity of an observable we compute its redun-dancy inE count the number of copies of its record.

Results of two previous sections will be useful: Deriva-tion of the pointer observable we have reported in SectionII allows us to anticipate what is the preferred observablecapable of leaving multiple records in E: Only states thatcan be monitored without getting perturbed can survivelong enough to deposit multiple copies of their informa-tion - theoretic progeny in the environment. In otherwords, the no-cloning theorem that is behind the deriva-tion of Section II is not an obstacle when cloning in-volves not an unknown quantum state, but rather aneinselected observable that is fittest that has already

survived evolutionary pressures of its environment andcan produce copious information - theoretic offspring.

In order to make these arguments rigorous we shallcompute entropies ofS,E, and various fragmentsF ofE(see Fig. 2). This means we will need reduced densitymatrices and probabilities. It is therefore fortunate that,in Section III, we have arrived at a fundamental deriva-tion of Borns rule from symmetry of entanglement. Thisgives us a right to use the usual tools of decoherence reduced density matrix and trace to compute entropy.

A. Mutual Information, redundancy, and discord

Quantities that play key role in quantum Darwinismare often expressed in terms of von Neumann entropy:

H() = T r lg , (4.1)The density matrix describes the state of a system, or ofa collection of several quantum systems. As is the caseclassically, von Neumann entropy in the quantum case isa measure of ignorance. But density matrix providesmore that just its eigenvalues that determineH() more

FIG. 2 Quantum Darwinism and the structure of the envi-ronment. The decoherence paradigm distinguishes betweena system (S) and its environment (E) as in (a), but makesno further recognition of the structure ofE; it could be aswell monolithic. In the environment-as-a-witness paradigm,we recognize subdivision ofEinto subenvironments its nat-ural subsystems, as in (b). The only real requirement for asubsystem is that it should be individually accessible to mea-surements; observables corresponding to different subenviron-ments commute. To obtain information about the systemSfrom its environment Eone can then carry out measurementsof fragmentsF of the environment non-overlapping collec-tions of the subsystems ofE. Sufficiently large fragments ofEthat has monitored (and, therefore, decohered) Scan oftenprovide enough information to infer the state ofS, by combin-ing subenvironments as in (c). There are then many copiesof the information aboutS inE, which implies that some in-formation about the fittest observable that survived mon-itoring byE has proliferated throughoutE, leaving multipleinformational offspring. This proliferation of the informationabout the fittest states defines quantum Darwinism. Multiplecopies allow many observers to find out the state ofS: Envi-ronment becomes a reliable witness with redundant copies ofinformation about preferred observables, which leads to ob-jective existence of preferred pointer states.

than just the set of probabilities. It is an operator ithas eigenstates. So, one is tempted to add that alsodetermines what one is ignorant of. This is not neces-sarily the case: As Section II has demonstrated, pointerstates do not in general coincide with the states on thediagonal of the reduced density matrix. Therefore, theset of states that one should be curious about (because oftheir stability) may not coincide with the instantaneouseigenstates of the reduced density matrix (although afterdecoherence time they should be approximately aligned).

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Nevertheless, one might be interested in the informa-tion about some other observable that has eigenstates{|k}which differ from the eigenstates of. The corre-sponding entropy is then Shannon entropy given by:

H(pk) = k

pklgpk , (4.2)

where:

pk = T rk||k (4.3)are the associated probabilities. For example, {|k} maybe the pointer states. They will coincide with the eigen-states of the reduced density matrix S only after de-coherence has done its job. In this case, the differencebetween the two entropies disappears.

Strictly speaking, it is only then that one can associatethe usual interpretation of probabilities to the pointerstates. The problem one can encounter when this is notso is apparent in composite quantum systems. We shallexhibit it by computing mutual information which char-acterizes correlations between systems. Mutual informa-

tion will also be our measure of how much (and what)fragment of the environment knows about the system.

1. Mutual information

Mutual information is the difference between the en-tropy of two systems treated separately and jointly:

I(S: A) = H(S) +H(A) H(S, A) . (4.4)For classical systems this definition is equivalent to thedefinition that employs conditional information (e.g.,H(

S|A)). It is defined by separating out of the joint

entropy H(S, A) the information about one of the twosystems. For example:

H(S, A) = H(A) +H(S|A) = H(S) +H(A|S). (4.5)Conditional entropy quantifies information aboutS (orA) that is still missing even after the state of A (or S) isalready known.

In quantum physics knowing is not as innocent asin the classical setting: It involves performing a mea-surement, which in turn alters the joint density matrixinto the outcome - dependent conditional density matrixthat describes state of the system given the measurementoutcome e.g., state

|Ak

of the apparatus

A:

S|Ak= Ak|SA |Ak/pk , (4.6)where, in accord with Eq. (4.3), pk = T rAk|SA |Ak.Conditional entropy given an outcome|Ak is then:

H(S|Ak) = T rS|Aklg S|Ak , (4.7)which leads to the average:

H(S|{|Ak}) =k

pkH(S|Ak) . (4.8)

So this is how much information aboutS one can ex-pect will be still missing after the measurement of theobservable with the eigenstates{|Ak}onA.

One can pose a more specific question e.g., how muchinformation about a specific observable ofS (character-ized by its eigenstates{|sj}) will be still missing afterthe observable with the eigenstates{|Ak}onAis mea-sured. This can be answered by usingSA to compute

the joint probability distribution:p(sj , Ak) = sj , Ak|SA |sj , Ak. (4.9)

These joint probabilities are in effect classical. They canbe used to compute classical joint entropy for any twoobservables (one in S, the other in A), as well as entropyof each of these observables separately, and to obtain thecorresponding mutual information:

I({|sj} : {|Ak}) = H({|sj})+H({|Ak})H({|sj}, {|Ak(4.10)

We shall find uses for both of these definitions of mutualinformation. In effect, the von Neumann entropy basedI(

S :

A), Eq. (4.4), answers the question how much the

systems know about each other, while the Shannon ver-sion immediately above quantifies the mutual informa-tion between two specific observables. Shannon versionis (by definition) basis dependent. It is straightforwardto see (extending arguments of Ollivier and Zurek (2002))that, for the same underlying joint density matrix:

I(S: A) I({|sj} : {|Ak}). (4.11)Equality can be achieved only for a special choice of themeasured observables, and only when the eigenstates ofSA are not entangled.

One can also define half way (Shannon - von Neu-nann) mutual informations which presume a specific mea-surement on one of the two systems (e.g.,A), but makeno such commitment for the other one. For instance,

J(S : {|Ak}) = H(S) H(S|{|Ak}) (4.12)would be one way to express the mutual information de-fined asymmetrically in this manner. There are somesubtleties involved in such definition (Zurek, 2003c), andwe shall not pursue this asymmetric subject in muchgreater detail because we can discuss quantum Darwin-ism without making extensive use ofJ(S :{|Ak}) andsimilar quantities. Nevertheless, J(S :{|Ak}) can beused to quantify quantumness of correlations.

Quantum discord is a difference between mutual en-tropy defined using the symmetric von Neumann formula,Eq. (4.4), and one of the Shannon versions. For example:

I(S: {|Ak}) = I(S: A) J(S: {|Ak}). (4.13)Discord defined in this manner is a measure of how muchinformation about the two systems is accessible througha measurement onA with outcomes{|Ak}. It clearlybasis-dependent, and the minimum discord disappears:

I(S, A) = min{|Ak}{I(S: {|Ak})} = 0 (4.14)

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iffSA commutes with A =

kk|AkAk| (Ollivier andZurek, 2002). When that happens, quantum correlationis classically accessible from A. It is of course possible tohave correlations that are accessible only from one end(Zurek, 2003c). For instance:

SA =1

2(|||AA| + |||AA|) (4.15)

will be classically accessible through a measurement withorthogonal records {|A, |A} on A, but classically in-accessible to any measurement onSwhen | = 0.

Questions we shall analyze using mutual informationwill concern both; (i) redundancy of the information(e.g., how many copies of the record does the environ-ment have aboutS), and; (ii) what is this informationabout (that is, what observable of the system is recordedin the environment with largest redundancy).

One might be concerned that having