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The Structure of the Multiverse
David Deutsch
Centre for Quantum Computation
The Clarendon Laboratory
University of Oxford, Oxford OX1 3PU, UK
April 2001
Keywords: multiverse, parallel universes, quantum information, quantum
computation, Heisenberg picture.
The structure of the multiverse is determined by information flow.
1. Introduction
The idea that quantum theory is a true description of physical reality led Everett
(1957) and many subsequent investigators (e.g. DeWitt and Graham 1973, Deutsch
1985, 1997) to explain quantum-mechanical phenomena in terms of the simultaneous
existence of parallel universes or histories. Similarly I and others have explained the
power of quantum computation in terms of ‘quantum parallelism’ (many classical
computations occurring in parallel). However, if reality – which in this context is
called the multiverse – is indeed literally quantum-mechanical, then it must have a
great deal more structure than merely a collection of entities each resembling the
universe of classical physics. For one thing, elements of such a collection would
indeed be ‘parallel’: they would have no effect on each other, and would therefore
not exhibit quantum interference. For another, a ‘universe’ is a global construct –
say, the whole of space and its contents at a given time – but since quantum
interactions are local, it must in the first instance be local physical systems, such as
qubits, measuring instruments and observers, that are split into multiple copies, and
this multiplicity must propagate across the multiverse at subluminal speeds. And for
another, the Hilbert space structure of quantum states provides an infinity of ways
David Deutsch
The Structure of the Multiverse
of slicing up the multiverse into ‘universes’, each way corresponding to a choice of
basis. This is reminiscent of the infinity of ways in which one can slice (‘foliate’) a
spacetime into spacelike hypersurfaces in the general theory of relativity. Given such
a foliation, the theory partitions physical quantities into those ‘within’ each of the
hypersurfaces and those that relate hypersurfaces to each other. In this paper I shall
sketch a somewhat analogous theory for a model of the multiverse.
The quantum theory of computation is useful in this investigation because, as we
shall see, the structure of the multiverse is determined by information flow, and the
universality of computation ensures that by studying quantum computational
networks it is possible to obtain results about information flow that must also hold
for quantum systems in general. This approach was used by Deutsch and Hayden
(2000) to analyse information flow in the presence of entanglement. In that analysis,
as in this one, no quantitative definition of information is required; the following
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
measurement on S alone depends on b .
·
Property 2 : A physical system S contains no information about b if (and for present
purposes we need not take a position about ‘only if’) there exists a complete
description of S that is independent of b .
I shall assume that an entity S qualifies as a ‘physical system’ if (but not necessarily
only if) it is possible to store information in S and later to retrieve it. That is to say, it
must be possible to cause S to satisfy the condition of Property 1 for containing
information about some parameter b . It is implicit in this, and in 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 it would still be the
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David Deutsch
The Structure of the Multiverse
same physical system. This condition raises interesting questions about the counter-
factual nature of information which it will not be necessary to address 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 its evolution depends
on nothing outside itself.
2. Classical computers
Consider a classical reversible computational network containing N bits
B 1 2 B N . A
specification of the values
b 1 t
( )
, 2 , b N t
( )
of the bits just after the t ’th computational
step constitutes a complete description of the computational state of the network at
that instant. Given the structure of the network (its gates, and how the carriers of the
bits move between them), this also determines the computational state just after
every other computational step. We are not interested in the network’s state during
computational steps, nor in its non-computational degrees of freedom, because we
know that the computational degrees of freedom at integer values of t form a
causally autonomous system, and it is that system which we shall regard as
faithfully modelling, with some finite but arbitrarily high degree of accuracy, the
flow of information in a classical system or classical universe.
Information flow in the network is local in the sense that if some information is
confined to a set of bits C at time t, then at time t
1 that information is confined to
+
bits that have passed through the same gate as some member of C during the
t
(
+
1
)
‘th computational step. In particular, if a network consists of two or more sub-
networks that are disconnected for a period, then information cannot flow from one
of those sub-networks 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 we want to draw conclusions about information flow in S by studying networks
that model S to some degree of approximation, we must consider only models with
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The Structure of the Multiverse
the property that local regions of S correspond to local (in the above sense) regions
of the network.
If we were to construct such a network in the laboratory, then each of the 2 N
possible bit-sequences
b 1 , 2 , b N would specify a physically and computationally
different state of the network. But if reality consisted of such a network, that would
not necessarily be so, because there would then be no external labels, such as spatial
location, to distinguish one bit from another. So, for instance, if the network
consisted of two disjoint sub-networks with identical structures, containing bits
B 1 , 2 , B N 2 and B N 2+1 , 2 , B N respectively, then any two bit-sequences of the form
b N and b N 2+1 , 2 ,
b N 2 would refer to the same physical state. The same
b 1 , 2 ,
b N ,
b 1 , 2 ,
applies when we are considering a hypothetical network that models information
flow in reality as a whole: if the structure of such network is invariant under some
permutation
of its bits, then any two bit-sequences that are related by
refer to
P
P
the same state of reality.
Let us refer to a bit-sequence
b 1 t
( )
, 2 , b N t
( )
collectively as b t
( )
(which can be thought
2 N -1 b N t
( ) +
( ) + b 1 t
( ) Î Z 2 N ). During each
of as the binary number
2
+
2 b 2 t
computational step, the values of the bits in the network change according to
(
) =
(
( )
)
b t
+
1
f t b t
,
(1)
where each f t is some invertible function from Z 2 N to itself, which characterises the
action of all the gates through which the bits pass during the ( t +1)’th computational
step.
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David Deutsch
The Structure of the Multiverse
Fig. 1: History of a classical computation
The course of such a computation with initial state b 0
( ) = b
is shown schematically
in Fig. 1. The parts of the graph in the shaded regions ( i.e. during computational
steps), and the non-integer values of b , have no significance except to indicate that
the motion of a real computer would interpolate smoothly between computational
states.
3. Ensembles of classical computers
Consider a collection of M classical networks of the kind described in Section 2, all
with the same structure in terms of gates, but not necessarily all starting in the same
initial state. One way of describing such a collection is as a single network consisting
of M disconnected sub-networks. The network has NM bits
B 1 2 B NM , where
B 1 2 B N
belong to the ‘first’ sub-network,
B N + 1 2 B 2 N to the ‘second’, and so on. But since the
structure of the network is invariant under any permutation of the sub-networks, we
must regard any pair of bit-sequences of length NM that are related by such a
permutation as referring to physically identical states.
In other words, when such sub-networks are in identical states, they are fungible . The
term is borrowed from law, where it refers to objects, such as banknotes, that are
deemed identical for the purpose of meeting legal obligations. In physics we may
define entities as fungible if they are not merely deemed identical but are identical, in
the sense that although they can be present in a physical system in varying numbers
or amounts, permuting them does not change the physical state of that system.
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