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The Structure of the MultiverseAuthor(s): David DeutschSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 458, No. 2028 (Dec.8, 2002), pp. 2911-2923Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/3560091 .

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Hl THE ROYAL10.1098/rspa.2002.1015 Ul SOCIETY

The structure of the multiverseBY DAVID DEUTSCH

Centre for Quantum Computation, The Clarendon Laboratory,University of Oxford, Oxford OX1 3PU, UKReceived 18 January 2002; accepted 2 May 2002; published online 30 September 2002

The structure of the multiverse can be understood by analysing the ways in whichinformation can flow in it. We may distinguish between quantum and classical infor-mation processing. In any region where the latter occurs-which includes not onlyclassical computation but also all measurements and decoherent processes-the mul-tiverse contains an ensemble of causally autonomous systems, each of which resem-bles a classical physical system. However, even in those regions, the multiverse hasadditional structure.Keywords: multiverse; parallel universes; quantum information;information flow; quantum computation; Heisenberg picture

1. IntroductionThe idea that quantum theory is a true description of physical reality led Everett(1957) and many subsequent investigators (e.g. DeWitt & Graham 1973; Deutsch1985, 1997) to explain quantum-mechanical phenomena in terms of the simultaneousexistence of parallel universes or histories. Similarly, the power of quantum computa-tion has been explained in terms of many classical computations occurring in parallel('quantum parallelism'). However, if reality, which in this context is called the mul-tiverse, is indeed literally quantum-mechanical, then it must have a great deal morestructure than merely a collection of entities each resembling the universe of classicalphysics. For one thing, elements of such a collection would indeed be 'parallel': theywould have no effect on each other and would therefore not exhibit quantum interfer-ence. For another, a 'universe' is a global construct-say, the whole of space and itscontents at a given time-but, since quantum interactions are local in space-time,it must in the first instance be local physical systems, such as qubits, measuringinstruments and observers, that become differentiated into multiple copies, and thismultiplicity must propagate across the multiverse at subluminal speeds. And foranother, the Hilbert-space structure of quantum states provides an infinity of waysof slicing up the multiverse into 'universes', each way corresponding to a choice ofbasis. (This is the so-called 'preferred basis problem' (see Vaidman 2002).)This isreminiscent of the infinity of ways in which one can slice ('foliate') a space-time intospace-like hypersurfaces in the general theory of relativity. Given such a foliation,the theory partitions physical quantities into those 'within' each of the hypersur-faces and those that relate hypersurfaces to each other. In this paper I shall sketcha somewhat analogous theory for the multiverse.The quantum theory of computation is useful in this investigation because, aswe shall see, the structure of the multiverse is determined by information flow, andProc. R. Soc. Lond. A (2002) 458, 2911-2923 ? 2002 The Royal Society

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D. Deutschthe universality of computation ensures that by studying quantum computationalnetworks it is possible to obtain results about information flow that must also holdfor quantum systems in general. This approach was used by Deutsch & Hayden(2000) to analyse information flow in the presence of entanglement and to prove thatall information in quantum systems is, notwithstanding Bell's theorem, localized. Inthat analysis, as in this one, no quantitative definition of information is required; thefollowing two qualitative properties suffice.Property 1: a physical system S contains information about a parameter b if

(though not necessarily only if) the probability of some outcome of some mea-surement on S alone depends on b.Property 2: a physical system S contains no information about b if (and for presentpurposes we need not take a position about 'only if') there exists a completedescription of S that is independent of b.

I shall assume that an entity S qualifies as a 'physical system' if (but not necessarilyonly if) it is possible to store information in S and later to retrieve it. That isto say, it must be possible to cause S to satisfy the condition of property 1 forcontaining information about some parameter b. It is implicit in this assumption, andin properties 1 and 2, that b must be capable of taking more than one possible value,so there must exist some suitable sense in which if S contained different information itwould still be the same physical system. This condition raises interesting questionsabout the counterfactual nature of information which it will not be necessary toaddress here. It is also necessary that S be identifiable as the same system over time.This is particularly straightforward if S is causally autonomous, that is to say, if itsevolution depends on nothing outside itself that is deemed to be a variable.

2. Classical computersConsider a classical reversible computational network containing N bits B1,..., BN.A specification of the values bl(t),...,bN(t) of the bits just after the tth compu-tational step constitutes a complete description of the computational state of thenetwork at that instant. Given the structure of the network (its gates and how thecarriers of the bits move between them), this also determines the computational statejust after every other computational step. We are not interested in the network'sstate during computational steps, nor in its non-computational degrees of freedom,because we know that the computational degrees of freedom at integer values of tform a causally autonomous system, and it is that system which we shall regard asfaithfully modelling, with some finite but arbitrarily high degree of accuracy, the flowof information in a classical system or classical universe.Information flow in the network is local in the sense that if some information isconfined to a set of bits C at time t, then at time t +1 that information is confined tobits that have passed through the same gate as some member of C during the (t+l)thcomputational step. In particular, if a network contains two or more subnetworksthat are disconnected for a period, then information cannot flow from one of thosesubnetworks to another during that period. Where a system S has local dynamics-for instance, if it is a field governed by a differential equation of motion-and wewant to draw conclusions about information flow in S by studying networks thatProc. R. Soc. Lond. A (2002)

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The structure of the multiverseb

Ko1i - :; ;IN: Nii:i.-ii.i.f(i...-.. .. .........Ii

: :0 3

of the network.

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D. Deutschinitial state. One way of describing such a collection is as a single network consistingof M disconnected subnetworks. The network has NM bits B1,..., BNM, whereB1,... ,BN belong to the 'first' subnetwork, BN+1,.. ,B2N to the 'second', etc.But since the structure of the network is invariant under any permutation of thesubnetworks, we must regard any pair of bit sequences of length NM that are relatedby such a permutation as referring to physically identical states.When two or more of the subnetworks are themselves in identical states, theyare fungible. The term is borrowed from law, where it refers to objects, such asbanknotes, that are deemed identical for the purpose of meeting legal obligations. Inphysics we may define entities as fungible if they are not merely deemed identical butare identical, in the sense that, although they can be present in a physical systemin varying numbers or amounts, permuting them does not change the physical stateof that system. Fungibility is not new to physics. Many physical entities, such asamounts of energy, are fungible even in classical physics: one can add a joule ofenergy to a physical system, but one cannot later extract the same joule. In quantumphysics some material objects, bosons, are fungible too: it makes sense to ask howmany identical photons there are in a cavity, and it makes sense to add one more ofthe same kind and then to remove one, but it does not make sense to ask whetherthe photon that has been removed is or is not the photon that was previously added(unless there was exactly one photon of that kind present).Hence an alternative way of describing our NM-bit network is as a multiset ofM networks, each with N bits. A multiset is like a set except that some of itselements are fungible. Each element is associated with an integer, its multiplicity,which specifies how many instances of it appear in the multiset. In the present case,if ,Lb(t) is the number of subnetworks that are in the state b E 22N at time t, so that

b lb(t) = M, then the state of the network at time t is completely specified bythe 2N multiplicities {/-b(t)}. Their equation of motion isILb(t + 1) = Af- (b)(t). (3.1)

An ensemble is a limiting case of a multiset where the total number of elementsM goes to infinity and the proportions ,ub(t)/M converge to limits. Considered asa function of b, limM-+oo(pb(t)/M) is the distribution function for the ensemble attime t. Henceforth in this paper I shall use the term 'ensemble' for both ensemblesand multisets, and the term 'multiplicity' to denote both discrete multiplicities andreal-valued proportions.Figure 2 illustrates schematically a computation being performed by an ensem-ble of 12 structurally identical computers. Four of them are performing the samecomputation as the computer referred to in figure 1, with input P. Three other com-putations are being performed in parallel with that one. They have inputs a, y and6, and are being performed by two, one and five of the computers respectively.I shall refer to the sub-ensemble consisting of all the computers in the ensemblethat are in a given state, if that sub-ensemble is non-empty, as a branch of theensemble; so, for instance, the ensemble in figure 2 has four branches throughout the

computation. Note the following elementary properties of branches under reversibleclassical physics. First, the total number of branches is conserved: they cannot split,join, come into existence or be destroyed. Second, the multiplicity of each branchis conserved, and third, the branches are causally autonomous; that is to say, thebehaviour of each branch is determined by its own initial state and f, and it isProc. R. Soc. Lond. A (2002)

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The structure of the multiverse

Figure 2. History of an ensemble of reversible classical computations.therefore independent of how many other branches are present and what their statesand multiplicities are. These properties give each branch a well-defined identity overtime, even though the values of its bits change.There are such things as fungible processes as well as fungible objects. Becauseeach of the computations takes place within a particular branch over time, and thewhole ensemble over time is invariant under permutations of the computations withina branch, those computations are themselves fungible.In this representation of our network as an ensemble, the equation of motion (3.1)specifies how the multiplicity of a fixed bit sequence b changes with time. This isnot well suited to the analysis of information flow because, as illustrated in figure 2,the information in the ensemble flows entirely in branches that are characterizedby constant multiplicities and time-varying bit sequences. It is possible to makethis manifest by using an alternative representation, or 'picture', of an ensemble ofclassical reversible systems that bears the same relationship to the standard oneas the Heisenberg picture does to the Schr6dinger picture in quantum theory. Tomotivate this picture, consider first a list b(O) of the computational states of all thebranches in the ensemble at time zero, sorted by any convenient criterion such asthe value of b. For example, b(O) for the ensemble of figure 2 could be (a,/3, 7, 6).It will turn out to be convenient to consider such lists as vectors, with an algebrathat I shall define below, and I shall call such vectors e-numbers by analogy with theterms 'q-number' for quantum operators and 'c-number' for scalars, and I shall referto the kth bits of each of the networks in the ensemble collectively as the ensemble'skth e-bit. For each t > 0, define the e-number b(t) as a list, or vector, of the statesof all the branches at time t, with the branches appearing in the same order as theydo in b(O), so that the values of b need no longer appear in numerical order: in thefigure 2 case, b(1) = (fo(a), fo(/), fo(Y), foo()), and so on. Thus, each component ofb(t), as t varies, is the evolving state of one particular branch of the ensemble. Sincethe multiplicities of branches are constant, we can list them as a single, constant e-number f = (/,a(O), l,3(O), /(O), ,u6(0)). The quantities f and b(t) together containthe same information as the {/(b(t)}, and therefore amount to a complete descriptionof the state of the ensemble at time t.Proc. R. Soc. Lond. A (2002)

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D. DeutschIn general, e-numbers for an ensemble of N-bit reversible classical networks maybe defined as follows. They are elements of a 2N-dimensional vector space V. (Lowerdimensional representations are sometimes possible, as in the example above.) Wecan express b(t), or any other e-number, in terms of an orthonormal basis {Pb(t)} as

t) bPb(t). (3.2)b

A straightforward representation for Pb(0) would be a list of length 2N, whose(b + 1)th element was unity, with the rest all zero, in which case b(0) would simplybe a list of all the integers from zero to 2N-1 (i.e. all possible states of a constituentcomputer) in ascending order.The equation of motion for the ensemble in this picture is

Pb(t + 1) = Pf-l(b)(t). (3.3)I shall refer to Pb(t) as the 'projector for the bits to take the values b at time t'.The reason for this terminology is, as will become apparent below, that these are thee-number analogues of quantum projection operators or Boolean observables. Notethat they are not projection operators on V, but are elements of V.If {Ab} are any c-number coefficients and g is any function initially defined onc-numbers, we can define

g( AA(t)) E g(Ab)P(t). (34)b bFrom (3.2)-(3.4) it follows that

b(t + 1) = ft(b(t)). (3.5)Hence, the equation of motion (3.5) of the e-bits has the same form as its counterpart(2.1) for a single classical computer, with e-numbers replacing integers.Apart from multiplication by a c-number in the usual vector-space manner, thealgebra has three further forms of multiplication, which may all be defined by theiractions on the projectors. The scalar product .x y is defined by

Pa (t) Pb(t) = a (t)6ab. (3.6)The e-number product xy, which is the e-number analogue of the product of classicalor quantum observables, is defined by

Pa(t)Pb(t) = Pa (t)6ab. (3.7)We can also define a unit e-number I = Zb Pb(t) for any t, and a zero e-number6 = 0, which have the usual multiplicative properties with respect to e-number mul-tiplication. The tensor product x 0 y, which is used for combining ensembles, could,for instance, be defined by

P(t) t) ( = Na(t) (3.8)where the two projectors on the left of (3.8) refer to ensembles E and E', the secondhaving N e-bits, and the projector on the right refers to the combination E x E' ofthose two ensembles.Proc. R. Soc. Lond. A (2002)

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The structure of the multiverseThe connection with the conventional picture is

I-b(t) = 'ft P(t). (3.9)We can now regard figure 2 as showing the history of b(t) in the 'state' ,f, rather

than the history of the {,Ub(t)}. Note that, from a complete specification of thealgebra generated by the e-numbers ft, b(t) and i (i.e. a means of calculating thescalar products of all the expressions that can be constructed from those e-numbersby addition, scalar multiplication and e-number multiplication), one can obtain theprojectors Pb(t) for all states b that are present in the ensemble at any time t,Pb(t) = ((t) - bi), (3.10)

where a is an e-number version of the Kronecker delta function, defined by2n-_1

(2N - 1)! (bi- ). (3.11)b=1It then follows from (3.9) that a complete description of the ensemble is containedentirely in the algebra of its e-number descriptors f/, b(t) and 1, independently of anyparticular representation of these e-numbers as 2N-tuples.

4. Quantum computers performing classical computationsThe central question addressed in this paper (see, for example, Steane 2000) cannow be stated as follows: in what sense, and in what approximation, can a quantumcomputation be said to contain an ensemble of classical computations?Consider a quantum computational network containing N qubits Q1 ... QN. Fol-lowing Gottesman (1999) and Deutsch & Hayden (2000), let us represent each qubitQk at time t in the Heisenberg picture by a triple,

bk(t) = (bkx(t), bky(t), bkz(t)), (4.1)of 2N x 2N Hermitian matrices representing Boolean observables (projection opera-tors) of Qk, satisfying

[bk(t),bk,(t)] = 0 (k # k')(1- 2))- ()( - 2ky(t)) =i(1- 2bk (t)) and cyclic permu- (4.2)

bkx(t)2 = bk(t) ktations over (x, y,z) JThe Heisenberg state It) of the network is a constant, so we can adopt the abbre-viated notation (X) = (jIXjl) for the expectation value of any observable X of thenetwork.The effect of an n-qubit quantum gate during one computational step is to trans-

form the 3n matrices representing the n participating qubits into functions of eachother in such a way that the relations (4.2) are preserved.Rotations of each qubit's three-dimensional 'spin vector'(i - 2bkx(t), i - 2bky(t), i - 2bkz(t))

Proc. R. Soc. Lond. A (2002)

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D. Deutschare such functions, corresponding to single-qubit gates. This allows a large class ofpossible alternative representations of the qubits, corresponding to the freedom tomake such transformations for each qubit independently. By convention we use thisfreedom to choose a representation (if one exists) in which the z-components bkz(t)are stabilized by any decoherence or measurement that may occur or, more generally,in which those components are performing classical computations (see below). In anycase, we can define

b(t) = 2N- bNz(t) + ... 2b2z(t) + blz(t), (4.3)as for a classical computer, although we note that b(t) is not a complete specificationof the state of the quantum computer at time t: there are also the x- or y-componentsof the descriptors {bk(t)}, and the Heisenberg state 1f). In principle, one couldchange the representation at every computational step, but that adds no generality,being the same as studying a different network in a constant representation. It wouldalso be possible to construct alternative representations that were related to thisone by more general transformations that are not expressible as compositions ofsingle-qubit transformations. However, these would not be appropriate in the presentinvestigation because the 'qubits' in such representations would not be local in thenetwork, and in order to model information flow we are using local interactions(gates) of the network to model local interactions in general quantum systems.A quantum network (or subnetwork) is said to be 'performing a classical com-putation' during the (t + l)th computational step if its b(t + 1) = f(b(t)) for somefunction f (not necessarily invertible). This occurs if and only if all its gates that acton qubits during that step have classical analogues, including one-qubit gates withthe effect

bk(t+ 1) = (anything, anything, bkz(t)), (4.4)which may be a non-trivial quantum computation even though it corresponds to theclassical gate whose only computational effect is a one-step delay. It would thereforenot be true to say that a quantum network is a classical computer during such aperiod: it still has qubits rather than bits; the network is still undergoing coherentmotion; and its computational state is not specified by any sequence of N binarydigits.

The Toffoli gate, which is universal for reversible classical computations, is definedas having the following effect on the kth, Ith and mth bits of a classical network:(bk(t+ 1) bk(t)bl(t + 1) bl(t) . (4.5)

bm(t + 1)/ bm(t) + bk(t)bl(t) - 2bk(t)bl(t)bm(t)It follows from the results of ?3 that, in an ensemble of networks containing aToffoli gate, its effect on three e-bits has the same functional form as (4.5), withe-numbers replacing c-numbers:

/k(t+ 1)\ bk(t)bi(t +?) = b1 t). (4.6)

bm(t + 1)] bm(t) + bk(t)bl(t) - 2bk(t)bl(t)bm(t)Proc. R. Soc. Lond. A (2002)

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The structure of the multiverseTable 1. Correspondence between e-numbers and quantum observables

ensemble ++ quantumbk(t) + bkz(t)b(t) + b(t)ti i

Pb(t) X Pb(t) - Ib;t)(b;tl

X(g) , 5:Compare this with the effect of the quantum version of the Toffoli gate:

bk(t 1 ( +1) (bk blzbmx - 2bkxblzbmx, bky + blzbmx - 2bkyblzbmx, bkz)bl(t + 1) = (bx + bkzbmx 2bkzblxbmx, bly + bkzbmz - 2bkzblybmx, bz)

bm(t + 1) (bm, bmy + bkzblz - 2bkzblzbmy' bmz + bkzbiz - 2bzblzbmz) (4.7)For the sake of brevity, the parameter t has been suppressed from all the matriceson the right of (4.7). Notice that the z-components bkz(t + 1), blz(t + 1), bmz(t + 1)of the descriptors of the qubits emerging from the gate (the third component in eachrow on the right of (4.7)) depend only on the z-components bkz(t), blz(t), bmz(t) ofthe descriptors of the qubits entering the gate. Notice also that these z-componentscommute with each other and that their equation of motion has the same functionalform as that of the corresponding ensemble of classical computers (4.6). Given theuniversality of the Toffoli gate, all these properties must hold whenever a quantumnetwork, or any part of it, performs a classical computation. In other words, wheneverany quantum network (including a subnetwork of another network) is performing aclassical computation f, the matrices {bkz(t)} for that network evolve independentlyof all its other descriptors. Moreover, under the correspondence given in table 1 thecommuting algebra of these matrices forms a faithful representation of the algebraof e-numbers describing an ensemble of classical networks performing f. In table 1,Ib;) is the eigenvalue-b eigenstate of bkz(t) and X and Y are the same functions ofthe {bkz(t)} as X and Y, respectively, are of the {bk(t)}.We also have b(t + 1) = ft(b(t)), the analogue of (3.5). Thus, figure 2, showing thecourse of an ensemble of classical computations, could equally well be a graph of thequantities (Pb(t)) in a quantum computer that was performing the same classicalcomputation as that ensemble. Note also that while the quantities ft *Pb(t) form acomplete description of the ensemble of classical computations, the (Pb(t)) are not acomplete description of the state of the quantum computation.Thus, in any subnetwork R of a quantum computational network where a reversibleclassical computation is under way, half the parameters describing R are precisely thedescriptors of an ensemble of classical networks. It is half the parameters because,from (4.2), any two of the three components of {bk(t)} determine the third. This doesnot imply that such a subsystem constitutes half the region of the multiverse in whichR exists. Proportions in the latter sense, which play the role of probabilities undersome circumstances, as shown in Deutsch (1999), are determined by the HeisenbergProc. R. Soc. Lond. A (2002)

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D. Deutschstate as well as the observables and do not concern us here because the presentdiscussion is not quantitative.The other half of the parameters, say the {bkx(t)}, contains information that isphysically present in R (it can affect subsequent measurements performed on R alone)but cannot reach the ensemble (the descriptors of the ensemble being independent ofthat information). But the reverse is not true: as (4.7) shows, information can reachthe quantum degrees of freedom from the ensemble.The proposition that parts of the multiverse have the same description as anensemble with given properties is not quite the same as the proposition that such anensemble is actually present in those parts of the multiverse, for the description mightrefer to entities that are not present in addition to those that are. In particular, anensemble has an alternative interpretation as a notional collection, only one memberof which is physically real, with the multiplicity of each branch representing theprobability that the properties of that branch were the ones prepared in the realsystem at the outset, by some stochastic process. However, no such interpretation ispossible if the branches affect each other, as they do in general quantum computationsand hence in general quantum phenomena.

5. Quantum computationsWhen a quantum computational network is performing a general computation, itneed not be the case that the descriptors of any part of the network over two ormore computational steps constitute a representation of an evolving e-algebra. Thereneed exist no functions ft and no choice of the 'z-directions' for defining the bk,(t)and hence b(t), such that b(t + 1) = ft(b(t)), so the conditions (discussed in ?3) forbranches to have an identity over time need not hold. At each instant t, it is stillpossible to extract a set of numbers (Pb(T)) from the description of the network attime t, and these still constitute a partition of unity, and still indicate which of theeigenvalues b of the observable b(t) are present in the multiverse at time t (in the sensethat if b(t) were measured immediately after time t, the possible outcomes would beprecisely the values for which (Pb(t)) 7#0). But although the physical evolution is ofcourse always continuous, there is in general no way of 'connecting up the dots' ina graph of the quantities (Pb(t)) against b and t that would correctly represent theflow of information. Hence, there exists no entity (such as a 'branch' or 'universe'),associated with exactly one of the values b at each time, that can be identified as aphysical system over time.In a typical quantum algorithm, as illustrated schematically in figure 3, the qubitsfirst undergo a non-classical unitary transformation U, then a reversible classicalcomputation, and finally another unitary transformation, which is often the inverseU-1 of the first one. Despite the fact that the branches lose their separate identitiesduring the periods of the quantum transformations U and U-1, we can still trackthe flow of information reasonably well in terms of ensembles: for t < -1, there is ahomogeneous ensemble, in all elements of which the computer is prepared with theinput /. For -1 < t < 0, this region of the multiverse does not resemble an ensemble:it has a more complicated structure, but the quantum computer as a whole does stillcontain the information that the input was /. For 0 < t < 3 an ensemble is presentagain, this time with four branches. The information about 3 may no longer bewholly present in that ensemble; some or all of it may be in the other half of theProc. R. Soc. Lond. A (2002)

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*gtoFigure 3. History of a quantum computation.

computer's degrees of freedom. For t > 3 the story is similar to that for t < 0, butin reverse order, so that finally there is a homogeneous ensemble with all elementsholding the value g(3).Consider a quantum computation whose (t + l)th step has the effect 'if qubit Nis 1, evaluate the invertible function ft on qubits 1 to N - 1, and otherwise performthe unitary transformation Ut on those qubits'. In other words, during the (t + l)thstep the computer performs the transformation

2N-1_1Vt= bNz E Ift(b))(b + (i -bNz)Ut (5.1)

b=Oon all N qubits. (Since bNz does not change during this process, we can drop itsparameter t.) If the Ut do not represent classical computations then clearly thenetwork as a whole is not performing a classical computation (though in the spe-cial case (bNz) = 1 we could not detect the difference, so the point would per-haps be moot). Nevertheless, it is still the case that some of the descriptors ofthis network-only about a quarter of them this time, namely the {bN,bkz(t)}-are those of a causally autonomous ensemble of classical computers, which, by theargument above, means that such an ensemble is present. Half the descriptors, saythe {(1 - bNz)bkz(t)} U {1 - bNz)bkx(t)}, do form a causally autonomous system butdo not form a representation of an e-algebra, while the remaining quarter, say, the{bNzbkx(t)}, neither are causally autonomous nor (therefore) form a representationof an e-algebra. Thus, this system has the following information-flow structure: itconsists of two subsystems between which information does not flow. One of them isperforming a quantum computation and cannot be further analysed into autonomoussubsystems; the other contains both an ensemble of 2N-1 classical computations anda further system that cannot be analysed into autonomous subsystems; moreover,information can reach it from the ensemble but not vice versa.In this network, the individual branches of the ensemble, the e-number alge-bra of which is represented by the matrices {bNzbkz(t)}, qualify as physical sys-Proc. R. Soc. Lond. A (2002)

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tems according to the criteria of ?1 because, for instance, if X(t) is any observ-able on the network at time t and 0 < k < 2N-1, a measurement of the observablebNz(t)Pk()X(X(t)Pk(t)bNz(t) is a measurement on one such branch alone: the one inwhich the classical computation is taking place and all the classical computers in thebranch are in state k at time t.The 'controlled NOT' gate, or measurement gate, which has the effect

fbm(t + 1) (bnx + bmx-2bnnx bnx+ bmy- 2bnxbmy,bmz) (52)bn(t + 1) J (bnx, bny + bny + nz - 2bnybmz, bnz + bmz - 2bnzbmz)

where again the parameter t has been suppressed from all the matrices on the rightof the equation, can be used to model the effects of measurement and decoherence.Qm is known as the 'control' qubit and Qn as the 'target' qubit. Because this is thequantum analogue of a reversible classical computation, the last terms in each rowof (5.2) (the z-components of bm(t + 1) and bm(t - 1)) again depend only on thez-components bmz(t) and bnz(t) of the descriptors of the qubits entering the gate.Furthermore, the z-component of the descriptor of the control qubit is unaffectedby the action of the measurement gate (i.e. bmz(t + 1) = bmz(t)). Therefore, if somesubnetwork of a quantum network performs a classical computation for a period, andis then modified so that some or all of the observables {bkz} are repeatedly measuredbetween computational steps, the modified network will perform the same classi-cal computation and will contain an ensemble identical to that which the originalnetwork contained (though its other descriptors will be very different). Since deco-herence can be regarded as a process of measurement of a quantum system by itsenvironment, the same conclusion holds in the presence of decoherence. It also holds,by trivial extension, if the classical computation is irreversible, since an irreversibleclassical computation is simply a reversible classical computation in which some ofthe information leaves the subnetwork in question.Since a generic quantum computational network does not perform anything likea classical computation on a substantial proportion of its qubits for many computa-tional steps, it may seem that when we extend the above conclusions to the multi-verse at large, we should expect parallelism (ensemble-like systems) to be confinedto spatially and temporally small scattered pockets. The reason why such systemsin fact extend over the whole of space-time with the exception of many small regions(such as the interiors of atoms and quantum computers), and why they approxi-mately obey classical laws of physics, is studied in the theory of decoherence (seeZurek 1981; Hartle 1991). For our present purposes, note only that, although mostof the descriptors of physical systems throughout space-time do not obey anythinglike classical physics, the ones that do form a system that, to a good approximation,is not only causally autonomous but can store information for extended periods andcarry it over great distances. It is therefore that system which is most easily acces-sible to our senses; indeed, it includes all the information processing performed byour nervous systems (Tegmark 2000). It has the approximate structure of a classicalensemble comprising 'the Universe' that we directly perceive and participate in, andother 'parallel' universes.In ? 1 I mentioned that the theory presented here does roughly the same job forthe multiverse as the theory of foliation into space-like hypersurfaces does for space-time in general relativity. There are strong reasons to believe that this must be moreProc. R. Soc. Lond. A (2002)

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The structure of the multiversethan an analogy. It is implausible that the quantum theory of gravity will involveobservables that are functions of a c-number time. Instead, time must be associatedwith entanglement between clock-like systems and other quantum systems, as in themodel constructed by Page & Wootters (1983), in which different times are seenas special cases of different universes. Hence, the theory presented here and theclassical theory of foliation must in reality be two limiting cases of a single, yet-to-be-discovered theory: the theory of the structure of the multiverse under quantumgravity.I thank Dr Simon Benjamin for many conversations in which the ideas leading to this paperwere developed, and him and Patrick Hayden for suggesting significant improvements to previousdrafts.

ReferencesDeutsch, D. 1985 Int. J. Theor. Phys. 24, 1Deutsch, D. 1997 The fabric of reality, ch. 2. London: Allen Lane.Deutsch, D. 1999 Proc. R. Soc. Lond. A 455, 3129-3197.Deutsch, D. & Hayden, P. 2000 Proc. R. Soc. Lond. A 456, 1759-1774.DeWitt, B. S. & Graham, N. 1973 The many worlds interpretation of quantum mechanics.Princeton, NJ: Princeton University Press.Everett, H. 1957 Rev. Mod. Phys. 29, 454-462.Gottesman, D. 1999 In Group22: Proc. XXIIth Int. Colloquium on Group Theoretical Methodsin Physics (ed. S. P. Corney, R. Delbourgo & P. D. Jarvis), pp. 32-43. Cambridge, MA:International Press.Hartle, J. B. 1991 Phys. Rev. D 44, 3173.Page, D. N. & Wootters, W. 1983 Phys. Rev. D 27, 2885-2892.Steane, A. 2000 Preprint. (Available from http://xxx.lanl.gov/abs/quant-ph/0003084.)Tegmark, M. 2000 Phys. Rev. E61, 4194-4206.Vaidman, L. 2002 The many worlds interpretation of quantum mechanics. In Stanford encyclope-dia of philosopy, ?6.2. (Available from http://plato.stamford.edu/entries/qm-manyworlds/.)Zurek, W. H. 1981 Phys. Rev. D24, 1516-1525.

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