grammar.dvi


The Grammar of Teleportation

Christopher G. Timpson∗

Division of History and Philosophy of Science,

School of Philosophy, University of Leeds,

Leeds, LS2 9JT, UK

September 7, 2005

Abstract

Whilst a straightforward consequence of the formalism of non-relativistic quantum
mechanics, the phenomenon of quantum teleportation has given rise to considerable
puzzlement. In this paper, the teleportation protocol is reviewed and these puzzles
dispelled. It is suggested that they arise from two primary sources: 1) the familiar
error of hypostatizing an abstract noun (in this case, ‘information’) and 2) failure
to differentiate interpretation dependent from interpretation independent features
of quantum mechanics. A subsidiary source of error, the simulation fallacy, is also
identified. The resolution presented of the puzzles of teleportation illustrates the
benefits of paying due attention to the logical status of ‘information’ as an abstract
noun.

1 Introduction

‘The questions “What is length?”, “What is meaning?”, “What is the number
one?” etc. produce in us a mental cramp. We feel that we can’t point to
anything in reply to them and yet ought to point to something. (We are up
against one of the great sources of philosophical bewilderment: a substantive
makes us look for a thing that corresponds to it.)’ Wittgenstein (1958)

Quantum teleportation (Bennett et al. (1993)) is one of the singular fruits of the
burgeoning field of quantum information theory. In this theory, one seeks to describe
and make use of the distinctive possibilities for information processing and commu-
nication that quantum systems allow: quantum features like entanglement and non-
commutativity are put to work.

Teleportation is perhaps the most striking example of the use of entanglement in
assisting communication and it illustrates vividly several of the general features associ-
ated with quantum information protocols, most notably the fact that entanglement (a
characteristically quantum property) serves as an important resource, and that unknown
quantum states cannot be cloned (Dieks (1982); Wootters and Zurek (1982)).

Although a straightforward consequence of the formalism of non-relativistic quan-
tum mechanics, teleportation has given rise to some confusion and to a good deal of

∗c.g.timpson@leeds.ac.uk

1



|φ+〉 = 1/
√

2(|↑〉|↑〉 + |↓〉|↓〉),
|φ−〉 = 1/

√
2(|↑〉|↑〉 − |↓〉|↓〉),

|ψ+〉 = 1/
√

2(|↑〉|↓〉 + |↓〉|↑〉),
|ψ−〉 = 1/

√
2(|↑〉|↓〉 − |↓〉|↑〉).

Table 1: The four Bell states, a maximally entangled basis for 2⊗2 dimensional systems.

controversy. In this paper I review the main lines of controversy (Sections 2 and 3) and
seek to dispell the confusion that has surrounded the interpretation of the protocol.

I will suggest (Section 4) that puzzlement has generally arisen as a consequence of a
familiar philosophical error—in fact the one that Wittgenstein famously warns us of in
the Blue Book —that is, the error of assuming that every grammatical substantive is a
referring term. Here the culprit is the word ‘information’. ‘Information’ is an abstract
(mass) noun and hence does not refer to a spatio-temporal particular, to an entity or
a substance1. It follows that one should not be seeking in an information theoretic
protocol—quantum or otherwise—for some particular, denoted by ‘the information’,
whose path one is to follow, but rather concentrating on the physical processes by
which the information is transmitted, that is, by which the end result of the protocol is
brought about. Once this is recognised, I suggest, much of our confusion is dispelled.
(A subsidiary source of difficulty—what I term the simulation fallacy—is also remarked
upon.)

With this clarification in place, the other major source of controversy is thrown into
relief: just what are the physical processes by which teleportation is effected? This is, in
fact, a relatively straightforward question; but it is a question that will find a different
answer depending on what interpretation of quantum mechanics one wishes to adopt
(Section 5) a point which has not been sufficiently recognised to date.

The central theme of this paper is that the conceptual puzzles surrounding telepor-
tation arise from thinking about information in the wrong way. The converse point
holds too: the clarification of these puzzles clearly illustrates the value of recognising
the logico-grammatical status of ‘information’ as an abstract noun.

Let us begin by briefly reviewing the teleportation protocol2.

2 The quantum teleportation protocol

In the teleportation protocol we consider two parties, Alice and Bob, who are widely
separated, but each of whom possess one member of a pair of particles in a maximally
entangled state. Alice is presented with a system in some unknown quantum state, and
her aim is to transmit this state to Bob. In the standard example, Alice and Bob share
one of the four Bell states (Table 1) and she is presented with a spin-1/2 system in the
unknown state |χ〉 = α|↑〉 + β|↓〉.

By performing a suitable joint measurement on her half of the entangled pair and
the system whose state she is trying to transmit (in this example, a measurement in the

1For elaboration of this claim, see Timpson (2004, Chpt.1). The point is also made in Timpson
(2005).

2Helpful discussions of further conceptual aspects of teleporation, in particular concerning the re-
lation of teleportation to nonlocality, may be found in Hardy (1999), Barrett (2001) and Clifton and
Pope (2001). Mermin (2001) also provides an interesting perspective.

2



χχ

χχ

Bob

Alice

Figure 1: Teleportation. A pair of systems is first prepared in an entangled state and shared
between Alice and Bob, who are widely spatially separated. Alice also possesses a system in an
unknown state |χ〉. Once Alice performs her Bell-basis measurement, two classical bits recording
the outcome are sent to Bob, who may then perform the required conditional operation to
obtain a system in the unknown state |χ〉. (Continuous black lines represent qubits, dotted
lines represent classical bits. Time runs along the horizontal axis.)

Bell basis), Alice can flip the state of Bob’s half of the entangled pair into a state that
differs from |χ〉 by one of four unitary transformations, depending on what the outcome
of her measurement was. If a record of the outcome of Alice’s measurement is then sent
to Bob, he may perform the required operation to obtain a system in the state Alice
was trying to send (Fig. 1).

The end result of the protocol is that Bob obtains a system in the state |χ〉, with
nothing that bears any relation to the identity of this state having traversed the space
between him and Alice. Only two classical bits recording the outcome of Alice’s mea-
surement were sent between them; and the values of these bits are completely random,
with no dependence on the parameters α and β. Meanwhile, no trace of the identity of
the unknown state remains in Alice’s region, as is required in accordance with the no-
cloning theorem (the state of her original system will usually now be maximally mixed).
The state has ‘disappeared’ from Alice’s region and ‘reappeared’ in Bob’s, hence the use
of the term teleportation for this phenomenon.

To fix the process in our minds, let’s review how the standard example goes. We
begin with system 1 in the unknown state |χ〉 and with Alice and Bob sharing a pair of
systems (2 and 3) in, say, the singlet state |ψ−〉. The total state of the three systems at
the beginning of the protocol is therefore simply

|χ〉1|ψ−〉23 =
1√
2

(

α|↑〉1 + β|↓〉1
)(

|↑〉2|↓〉3 − |↓〉2|↑〉3
)

. (1)

Notice that at this stage, the state of system 1 factorises from that of systems 2 and
3; and so in particular, the state of Bob’s system is independent of α and β. We may
re-write this initial state in a suggestive manner, though:

3



|χ〉1|ψ−〉23 =
1√
2

(

α|↑〉1|↑〉2|↓〉3 + β|↓〉1|↑〉2|↓〉3 − α|↑〉1|↓〉2|↑〉3 − β|↓〉1|↓〉2|↑〉3
)

(2)

=
1

2

(

|φ+〉12
(

α|↓〉3 − β|↑〉3
)

+ |φ−〉12
(

α|↓〉3 + β|↑〉3
)

+ |ψ+〉12
(

−α|↑〉3 + β|↓〉3
)

+ |ψ−〉12
(

−α|↑〉3 − β|↓〉3
)

)

.

(3)

The basis used is the set

{|φ±〉12|↑〉3, |φ±〉12|↓〉3, |ψ±〉12|↑〉3, |ψ±〉12|↓〉3},

that is, we have chosen (as we may) to express the total state of systems 1,2 and 3 using
an entangled basis for systems 1 and 2, even though these systems are quite independent.
But so far, of course, all we have done is re-written the state in a particular way; nothing
has changed physically and it is still the case that it is really systems 2 and 3 that are
entangled and wholly independent of system 1, in its unknown state.

Looking closely at (3) we notice that the relative states of system 3 with respect to
particular Bell basis states for 1 and 2 have a very simple relation to the initial unknown
state |χ〉; they differ from |χ〉 by one of four local unitary operations:

|χ〉1|ψ−〉23 =
1

2

(

|φ+〉12
(

−iσ3y|χ〉3
)

+ |φ−〉12
(

σ3x|χ〉3
)

+ |ψ+〉12
(

−σ3z|χ〉3
)

+ |ψ−〉12
(

−13|χ〉3
)

)

, (4)

where the σ3i are the Pauli operators acting on system 3 and 1 is the identity. To re-
iterate, though, only system 1 actually depends on α and β; the state of system 3 at
this stage of the protocol (its reduced state, as it is a member of an entangled pair) is
simply the maximally mixed 1/2 1.

Alice is now going to perform a measurement. If she were simply to measure system 1
then nothing of interest would happen—she would obtain some result and affect the state
of system 1, but systems 2 and 3 would remain in the same old state |ψ−〉. However, as
she has access to both systems 1 and 2, she may instead perform a joint measurement,
and now things get interesting. In particular, if she measures 1 and 2 in the Bell basis,
then after the measurement we will be left with only one of the terms on the right-
hand side of eqn. (4), at random; and this means that Bob’s system will have jumped
instantaneously into one of the states −iσ3y|χ〉3, σ3x|χ〉3, −σ3z |χ〉3 or −|χ〉3, with equal
probability.

But how do things look to Bob? As he neither knows whether Alice has performed her
measurement, nor, if she has, what the outcome turned out to be, he will still ascribe
the same, original, density operator to his system—the maximally mixed state3. No
measurement on his system could yet reveal any dependence on α and β. To complete
the protocol therefore, Alice needs to send Bob a message instructing him which of four

3Notice that an equal mixture of the four possible post-measurement states of his system results in
the density operator 1/2 1.

4



unitary operators to apply (iσy, σx, −σz, −1) in order to make his system acquire the
state |χ〉 with certainty; for this she will need to send two bits4. With these bits in
hand, Bob applies the needed transformation and obtains a system in the state |χ〉.

Now of course, this quantum mechanical process differs from science fiction versions
of teleportation in at least two ways. First, it is not matter that is transported, but
simply the quantum state |χ〉; and second, the protocol is not instantaneous, but must
attend for its completion on the arrival of the classical bits sent from Alice to Bob.
Whether or not the quantum protocol approximates to the science fiction ideal, however,
it remains a very remarkable phenomenon from the information-theoretic point of view5.
For consider what has been achieved. An unknown quantum state has been sent to Bob;
and how else could this have been done? Only by Alice sending a quantum system
in the state |χ〉 to Bob6, for she cannot determine the state of the system and send a
description of it instead. (Recall, it is impossible to determine an unknown state of an
individual quantum system. See Busch (1997) for a nice review.)

If, however, Alice did per impossibile somehow learn the state and send a description
to Bob, then systems encoding that description would have to be sent between them. In
this case something that does bear a relation to the identity of the state is transmitted
from Alice to Bob, unlike in teleportation. Moreover, sending such a description would
require a very great deal of classical information, as in order to specify a general state
of a two dimensional quantum system, two continuous parameters need to be specified.

The picture we are left with, then, is that in teleportation there has been a transmis-
sion of something that is inaccessible at the classical level (often loosely described as a
transmission of quantum information); in the transmission this information has been in
some sense disembodied; and finally, the transmission has been very efficient—requiring,
apart from prior shared entanglement, the transfer of only two classical bits.

2.1 Some information-theoretic aspects of teleportation

2.1.1 Preamble

The notion of information that is central to quantum information theory is that deriving
from the seminal work of Shannon (1948) in communication theory. He introduced a
measure of information H(X) to characterise a source X of messages which are produced
from a fixed alphabet {x1, . . . ,xn} whose elements occur with probability p(xi). The
Shannon information H(X) measures in bits (classical two-state systems) the resources
required to transmit all the messages that the source produces (Shannon’s noiseless
coding theorem see Appendix A). That is, it measures how much the messages from the
source can be compressed. Shannon also introduced the mutual information, H(X : Y ),
which indicates how much information it is possible to transmit over a noisy channel
(intuitively: how much can be inferred about the input to a channel, given the output
obtained).

The Shannon information measure has many important applications in the quantum

4Two bits are clearly sufficient, for the argument that they are strictly necessary, see Bennett et al.
(1993) Fig.2.

5Interestingly, it can be argued that quantum teleporation is perhaps not so far from the sci-fi ideal as
one might initially think. Vaidman (1994) suggests that if all physical objects are made from elementary
particles, then what is distinctive about them is their form (i.e. their particular state) rather than the
matter from which they are made. Thus it seems one could argue that objects really are teleported in
the protocol.

6Or by her sending Bob a system in a state explicitly related to |χ〉 (Park (cf. 1970)).

5



context7, for example, when considering the transmission of classical (Shannon) informa-
tion over a channel consisting of quantum devices; but it is also possible to introduce an
important and closely related concept—that of quantum information properly so-called.
This was done by Schumacher in the early 1990s8.

Schumacher followed Shannon’s lead: consider a device—a quantum source—which,
rather than outputting systems corresponding to elements of a classical alphabet, pro-
duces systems in particular quantum states ρxi with probabilities p(xi). By reasoning
analogous to Shannon’s, Schumacher showed that the output of this source could be
compressed by an amount measured by the von Neumann entropy of the source (the
quantum noiseless coding theorem, Appendix A). We therefore have, analogously to the
classical case, a notion of the amount of quantum information that the source produces:
a measure of the minimum number of two-state quantum systems (qubits) required to
encode the output of the source.

While it is important to distinguish these technical notions of information from
the everyday notion of information linked to knowledge, language and meaning (pace
Dretske (1981), cf. Timpson (2004) Chpt.1) there is at least one interesting property
held in common: in both the technical and everyday settings ‘information’ functions as
an abstract noun, that is, as a term which does not serve to denote a kind of entity
having a location in space and time. Briefly: in the latter case, because ‘information’
is a nominalization of the verb ‘to inform’; in the former, because an answer to the
question ‘what is transmitted?’ will refer to an abstract type rather than a concrete
thing; and because what is measured is a property of a source (compressibility) or a
channel (capacity), not an amount of some stuff present in a message9.

2.1.2 Application to Teleportation

There are two information-theoretic aspects of the teleporation protocol it is helpful to
go into in somewhat more detail. The first concerns our reason for saying that a very
large amount of information is required to specify the state that is teleported.

As we have noted, in order to describe an arbitrary (pure) state of a two dimensional
quantum system, it is necessary to specify two continuous parameters. A useful means
of picturing this is via the Bloch sphere representation. The pure states of a two-state
quantum system are in one-to-one correspondence with the points on the surface of the
unit 3-sphere, and we may specify two real numbers (angles) to determine a point on the
sphere. But why should doing this have associated with it an amount of information?
If it is to do so we will need to imagine a classical information source that is selecting
these pairs of angles with various probabilities; then a certain Shannon information
may be ascribed to the process. Given a particular output of this information source,
a quantum system is prepared in the state corresponding to the two angles selected.
The quantum states prepared in this manner will then have associated with them a
specification information10 given by the information of the source. Once a system has
been prepared in some state in this way, it is presented to Alice, who may proceed to
teleport the state to Bob.

7Surprisingly, perhaps, this point has occasioned some controversy (Brukner and Zeilinger (2001)).
See Timpson (2003) for a detailed discussion of the applicability of the Shannon information in the
quantum context.

8Published as Schumacher (1995), Jozsa and Schumacher (1994).
9I will not seek to defend these views in more detail here. See Timpson (2004) Part 1 passim and

Section 1.2.3 esp. for further discussion.
10Cf. Schumacher (1995), Timpson (2003) Section 4.1.

6



Rather than the pairs of angles being selected from their full, continuous, range
of possible values, the surface of the sphere might be coarse-grained evenly to give a
finite number of choices. One might pick the angles specifying the mid-point, say, of
each small element of surface area to provide the finite set of pairs of angles to choose
between. Loosely speaking this coarse-graining corresponds to considering angles only to
a certain degree of accuracy. As this accuracy is increased (the choices made more finely
grained), the number of bits required to specify the choice increases without bound.
If our information source is selecting states to an arbitrarily high accuracy then, the
specification information is unboundedly large. (On the other hand, if the information
source is only selecting between a small number of distinct states, then the specification
information is correspondingly small. From now on we will assume that unless otherwise
stated, the unknown states to be teleported are selected from a suitable coarse-graining
of the whole range of possible angles.) It is essential to note, however, that even if the
specification information associated with the state that has been teleported to Bob is
exceedingly large, the majority of this information is not accessible to him. This leads
on to the second point.

When one considers encoding classical information in quantum systems, it is neces-
sary to distinguish between specification information and accessible information11. The
specification information refers to the information of the classical source that selects
sequences of quantum states, the accessible information to the maximum amount of
information that is available following measurements on the systems prepared in these
states. Because of the existence of non-orthogonal states, specification and accessible
information may differ; and as it is impossible to distinguish non-orthogonal states per-
fectly, the specification information associated with a string of systems selected from
a source may be much greater than the accessible information. In teleportation, the
systems are prepared near Alice before teleportation of their states to Bob. He may
then perform various measurements to try and learn something. Call the information of
the source selecting the states to be teleported by Alice H(A); the mutual information
H(A : B) will determine the amount of classical information per system that Bob is able
to extract by performing some measurement, B, following successful teleportation of the
unknown state. The accessible information is given by the maximum over all decoding
measurements of H(A : B). A well known result due to Holevo (1973)—the Holevo
bound—restricts the amount of information that Bob may acquire to a maximum of
one bit of information per qubit, that is, to a maximum of one bit of information per
successful run of the teleportation protocol.

So this gives us the sense in which the very large amount of information that may be
associated with the unknown state being teleported to Bob is largely inaccessible to him.
Note that the amount of information that Bob may acquire from the teleported state
is less than the amount of classical information—two bits—that Alice had to send to
him during the protocol. This fact is of the utmost importance, for if the Holevo bound
did not guarantee this, and Bob were able to extract more than two bits of information
from his system, then teleportation would give rise to paradox (when embedded in a
relativistic theory) as superluminal signalling would be possible12.

So the Holevo bound ensures that teleportation is not paradoxical, but it also means
that teleportation, when considered as a mode of ordinary classical information transfer,

11Cf. Schumacher (1995).
12The argument parallels the one given by Bennett et al. (1993) to the effect that two full classical

bits are required in teleportation. In essence, if Bob were able to gain more than two bits of information
in the protocol, then even if he were not to wait for Alice to send him the pair of bits each time and
simply guessed their values instead, then some information would still get across.

7



is pretty inefficient, requiring two classical bits to be sent for every bit of information
that Bob can extract at his end.

3 The puzzles of teleportation

Let us return to the picture of teleportation that was sketched earlier. An unknown
quantum state is teleported from Alice to Bob with nothing that bears any relation
to the identity of the state having travelled between them. The two classical bits sent
are quite insufficient to specify the state teleported; and in any case, their values are
independent of the parameters describing the unknown state. The unboundedly large
specification information characterizing the state—information that is inaccessible at the
classical level—has somehow been disembodied, and then reincarnated at Bob’s location,
as the quantum state first ‘disappears’ from Alice’s system and then ‘reappears’ with
Bob.

The conceptual puzzles that this process presents seem to cluster around two essential
questions. First, how is so much information transported? And second, most pressingly,
just how does the information get from Alice to Bob?

Perhaps the prevailing view on how these questions are to be answered is the one
that has been expressed by Jozsa (1998, 2004) and Penrose (1998). In their view, the
classical bits used in the protocol evidently can’t be carrying the information, for the
reasons we have just rehearsed; therefore the entanglement shared between Alice and
Bob must be providing the channel down which the information travels. They conclude
that in teleportation, an indefinitely large, or even infinite amount of information travels
backwards in time from Alice’s measurement to the time at which the entangled pair was
created, before propagating forward in time from that event to Bob’s performance of his
unitary operation and the attaining by his system of the correct state. Teleportation
seems to reveal that entanglement has a remarkable capacity to provide a hitherto
unsuspected type of information channel, which allows information to travel backwards
in time; and a very great deal of it at that. Further, since it is a purely quantum link
that is providing the channel, it must be purely quantum information that flows down
it. It seems that we have made the discovery that quantum information is a new type
of information with the striking, and non-classical, property that it may flow backwards
in time.

The position is summarized succinctly by Penrose:

‘How is it that the continuous “information” of the spin direction of the
state that she [Alice] wishes to transmit...can be transmitted to Bob when
she actually sends him only two bits of discrete information? The only other
link between Alice and Bob is the quantum link that the entangled pair
provides. In spacetime terms this link extends back into the past from Alice
to the event at which the entangled pair was produced, and then it extends
forward into the future to the event where Bob performs his [operation].

Only discrete classical information passes from Alice to Bob, so the complex
number ratio which determines the specific state being “teleported” must be
transmitted by the quantum link. This link has a channel which “proceeds
into the past” from Alice to the source of the EPR pair, in addition to
the remaining channel which we regard as “proceeding into the future” in

8



the normal way from the EPR source to Bob.There is no other physical
connection.’ (Penrose (1998, p.1928))

But one might feel, with good reason, that this explanation of the nature of infor-
mation flow in teleportation is simply too outlandish. This is the view of Deutsch and
Hayden (2000), who conclude instead that with suitable analysis, the message sent from
Alice to Bob can, after all, be seen to carry the information characterizing the unknown
state. The information flows from Alice to Bob hidden away, unexpectedly, in Alice’s
message. This approach, and the question of what light it may shed on the notion of
quantum information, has been considered in detail elsewhere (Timpson (2005)). Suffice
it to say for present purposes that Deutsch and Hayden disagree with Jozsa and Penrose
over the nature of quantum information and how it may flow in teleportation.

One might adopt yet a third, and perhaps more prosaic response to the puzzles that
teleportation poses. This is to adopt the attitude of the conservative classical quantity
surveyor 13. According to this view, an amount of information cannot be said to have
been transmitted to Bob unless it is accessible to him. But of course, as we noted
above, the specification information associated with the state teleported to Bob is not
accessible to him: he cannot determine the identity of the unknown state. On this view,
then, the information associated with selecting some unknown state |χ〉 will not have
been transmitted to Bob until an entire ensemble of systems in the state |χ〉 has been
teleported to him, for it is only then that he may determine the identity of the state14.
To teleport a whole ensemble of systems, though, Alice will need to send Bob an infinite
number of classical bits; and now there isn’t a significant disparity between the amount
of information that has been explicitly sent by Alice and the amount that Bob ends
up with. One needs to send a very large number of classical bits to have transmitted
by teleportation the very large amount of information associated with selecting the
unknown state.

This approach does not seem to solve all our problems, however. Someone sympa-
thetic to the line of thought espoused by Jozsa and Penrose can point out in reply that
there still remains a mystery about how the information characterizing the unknown
state got from Alice to Bob—the bits sent between them, recall, have no dependence on
the identity of the unknown state. So while the approach of the conservative classical
quantity surveyor may mitigate our worry to some extent over the first question, it does
not seem to help with the second.

4 Resolving (dissolving) the problem

Dwelling on the question of how the information characterizing the unknown state is
transmitted from Alice to Bob has given rise to some conundrums. Should we side with
Jozsa and Penrose and admit that quantum information may flow backwards in time
down a channel constituted by shared entanglement? Or perhaps with Deutsch and

13A resolution along these lines, tied also to an ensemble view of the quantum state (vide infra) has
been suggested by Barrett (2001) and Morgan (2001).

14Note that we will need to adjust our scenario slightly to incorporate this view. In our initial set-up,
the source A selected a sequence of states which were then teleported one by one to Bob. Now we
imagine instead that following some particular output of A, an entire ensemble of systems is prepared
in the pure state associated with that output; then this ensemble of systems — all in the same unknown
pure state — is teleported. This adjustment is required because in our initial set-up for the teleportation
procedure, the only way in which an ensemble of systems all in the same state could be teleported to
Bob would be by setting the information of the source A to zero, with the tiresomely paradoxical result
that Bob could now determine the state all right, but would gain no information by doing so.

9



Hayden, and agree that information should flow in a less outlandish fashion, but that
quantum information may be squirrelled away in seemingly classical bits? Counting
conservatively the amounts of information available after teleportation may make us
less anxious about the load carried in a single run of the protocol, but the question still
remains: how did the information, in the end, get to Bob? Should we just conclude that
it is transported nonlocally in some way? But what might that mean?

If the question ‘How does the information get from Alice to Bob?’ is causing us
these difficulties, however, perhaps it might pay to look at the question itself rather
more closely. In particular, let’s focus on the crucial phrase ‘the information’.

Our troubles arise when we take this phrase to be referring to a particular, to some
sort of substance or entity whose behaviour in teleportation it is our task to describe.
The assumption common to the approaches of Deutsch and Hayden on the one hand,
and Jozsa and Penrose on the other, is that we need to provide a story about how some
thing denoted by ‘the information’ travels from Alice to Bob. Moreover, it is assumed
that this supposed thing should be shown to take a spatio-temporally continuous path.

But notice that ‘information’ is an abstract noun. This means that ‘the information’
certainly does not refer to a substance or to an entity. The shared assumption is thus
a mistaken one, and is based on the error of hypostatizing an abstract noun. If ‘the
information’ doesn’t introduce a particular, then the question ‘How does the information
get from Alice to Bob?’ cannot be a request for a description of how some thing travels.
It follows that the locus of our confusion is dissolved.

But if it is a mistake to take ‘How does the information get from Alice to Bob?’ as
a question about how some thing is transmitted, then what is its legitimate meaning, if
any? It seems that the only legitimate use that can remain for this question is as a flowery
way of asking: what are the physical processes involved in the transmission? Now this
question is a perfectly straightforward one, even if, as we shall see (Section 5), the answer
one actually gives will depend on the interpretation of quantum mechanics one adopts.
But there is no longer a conceptual puzzle over teleportation. Once it is recognised that
‘information’ is an abstract noun, then it is clear that there is no further question to be
answered regarding how information is transmitted in teleportation that goes beyond
providing a description of the physical processes involved in achieving the aim of the
protocol. That is all that ‘How is the information transmitted?’ can intelligibly mean;
for there is not a question of information being a substance or entity that is transported,
nor of ‘the information’ being a referring term. Thus, one does not face a double task
consisting of a) describing the physical processes by which information is transmitted,
followed by b) tracing the path of a ghostly particular, information. There is only task
(a).

The point should not be misunderstood: I am not claiming that there is no such
thing as the transmission of information, but simply that one should not understand
the transmission of information on the model of transporting potatoes, or butter, say,
or piping water15.

15Note that we do sometimes talk of a flow of information; and we do say of many physical quantities
that are not entities or substances — for example, energy, heat — that they flow. But there is no
analogy between the two cases, for what this latter description means is that the quantities in question
obey a local conservation equation. It is not clear that it is at all intelligible to suggest that information
should obey a local conservation equation. Certainly, the concept of quantity of information that is
provided by the Shannon theory does not give us a concept of a quantity it makes sense to suggest
might obey such an equation. (On this, see Section 6 below.)

10



4.1 The simulation fallacy

Whilst paying due attention to the status of ‘information’ as an abstract noun provides
the primary resolution of the problems that teleportation can sometimes seem to present
us with, there is a secondary possible source of confusion that should be noted. This is
what may be termed the simulation fallacy.

Imagine that there is some physical process P (for example, some quantum-mechanical
process) that would require a certain amount of communication or computational re-
sources to be simulated classically. Call the classical simulation using these resources
S. The simulation fallacy is to assume that because it requires these classical resources
to simulate P using S, there are processes going on when P occurs that are physically
equivalent to (are instantiations of) the processes that are involved in the simulation
S itself (although these processes may be being instantiated using different properties
in P). In particular, when P is going on, the thought is that there must be, at some
level, physical processes involved in P which correspond concretely to the evolution of
the classical resources in the simulation S. The fallacy is to read off features of the
simulation as real features of the thing simulated16.

A familiar example of the simulation fallacy is provided by Deutsch’s argument that
Shor’s factoring algorithm supports an Everettian view of quantum mechanics (Deutsch
(1997, p.217)). The argument is that if factoring very large numbers would require
greater computational resources than are contained in the visible universe, then how
could such a process be possible unless one admits the existence of a very large number
of (superposed) computations in Everettian parallel universes? A computation that
would require a very large amount of resources if it were to be performed classically is
explained as a process that consists of a very large number of classical computations.
But of course, considered as an argument, this is fallacious. The fact that a very large
amount of classical computation might be required to produce the same result as a
quantum computation does not entail that the quantum computation consists of a large
number of parallel classical computations17.

The simulation fallacy is also evident in the common claim that Bell’s theorem shows
us that quantum mechanics is nonlocal, or the claim that the experimental violation of
Bell inequalities means that the world must be nonlocal. Of course, what is in fact shown
by these well-known results is that no local hidden variable model can simulate the
predictions of quantum mechanics, nor provide a model for the experimentally observed
correlations. But these facts about simulation don’t lead directly to facts about the
simulated: the fact that any adequate hidden variable model must be nonlocal does not
show that quantum mechanics is nonlocal (this, of course, is an interpretation dependent
property), nor show the world to be nonlocal.

While the question of what classical resources would be required to simulate a given
quantum process is an indispensible guide in the search for interesting quantum informa-
tion protocols and is vitally important for that reason, the simulation fallacy indicates
that it is by no means a sure guide to ontology.

With regard to teleportation, it is important to recognise the simulation fallacy
in order to assuage any worries that might remain over the question ‘How does so

16Note that it will not always be fallacious to take features of a simulation to correspond to features of
the simulated — if the features in question are explicitly analogues of features of the system or process
being simulated. One should thus distinguish between i) simulations that involve analogues and ii)
functional ‘black-box’, or input-output simulations.

17For further discussion of Deutsch’s conception of quantum ‘parallel processing’, see Steane (2003)
and Hewitt-Horsman (2002).

11



much information get from Alice to Bob?’, and to undermine further the thought that
teleportation must be understood as a flow of information.

For the fact that it would take a very large number of classical bits to transmit the
identity of an unknown state from Alice to Bob does not entail that in teleportation
there is a real corresponding transmission of information, some physical process going
on that instantiates, albeit in a different medium, the transport of this large amount
of information18. (Note that the flow of the hypostatized ‘quantum information’ of
Jozsa and Penrose plays precisely this rôle: the analogue, in a different medium, of the
transport of the large amount of classical information.) Equivalence from the point of
view of information processing does not imply physical equivalence.

Awareness of the simulation fallacy is particularly relevant when we consider the
approach of the conservative classical quantity surveyor. Recall that the point of this
approach is to deny that a large amount of information can be said to have been trans-
ported to Bob in teleportation until that information is actually available to him. How-
ever, it might be objected to this that after a single run of the teleportation protocol,
the information characterizing the state is certainly present at Bob’s location, even if
inaccessible to him, as a system in the unknown state is present19.

This contention would seem to rest on an argument of the following form: The only
way the unknown state can appear at Bob’s location is if the information characterizing
the state has actually been transported to Bob, hence on appearance of the state, the
specification information associated with the state has indeed been transported to Bob’s
location. (Crudely, if a system in the given state is present, then the information is
present, as it takes this information to specify the state.) But such an argument needs
to be treated with care, for the main premise appears to rest on the simulation fallacy.
Just because it would take a large amount of information to specify a state doesn’t mean
that we should conclude that this amount of information has been physically transported
in teleportation when Bob’s system acquires the state.

In any event, the simplest way to remain clear on whether or not, or in what way,
information can be said to be present at Bob’s location following a single run of the
teleportation protocol is to respect the distinction between the specification information
associated with a system and the amount of information that may be said to be encoded
or contained in the system. Once Bob’s system has acquired the state |χ〉 teleported
by Alice, then his system has associated with it the same specification information,
H(A): if one were now asked to specify the state of Bob’s system, then this number
of bits would be required, on average. This quantity of information is not encoded or
contained in the system however. The mutual information H(A : B) and the accessible
information provide the relevant measures of how much information Bob’s system can
be said to contain, for they govern the amount that may be decoded. But of course, as
‘information’ is an abstract noun, containing information is not containing some thing,
however aethereal.

18Nor, for example, does the fact that there are protocols in which the state of a qubit can be
substituted for an arbitrarily large amount of classical information (Galvão and Hardy (2003)) imply
that this large amount of information is really there in the qubit.

19It is for this reason that it is natural to marry conservative classical quantity surveying with an
ensemble view of the quantum state (see footnote 13), for then this objection would not go through —
when the two positions are conjoined, not only is the information characterizing the state not available
until the whole ensemble is teleported, but neither has the state been teleported until the whole ensemble
has been teleported to Bob.

12



5 The teleportation process under different interpre-

tations

By reflecting on the logico-grammatical status of the term ‘information’ we have been
able to replace the (needlessly) conceptually puzzling question of how the information
gets from Alice to Bob in teleportation, with the simple, genuine, question of what
the physical processes involved in teleportation are. While this may not, perhaps, be
quite enough to still all the controversy that trying to understand teleportation has
evoked, the controversy is now of a very familiar kind: it concerns what interpretation
of quantum mechanics one adopts. For the detailed story one tells about the physical
processes involved in teleporation will of course depend upon one’s interpretive stance.
Two questions in particular will find different answers under different interpretations:
first, is nonlocality involved in teleportation? and second, has anything interesting
happened before Alice’s classical bits are sent to Bob and he performs the correct unitary
operation?

We will now see how some of these differences play out in the following familiar
interpretations (the list of approaches considered is by no means exhaustive).

5.1 Collapse interpretations: Dirac/von Neumann, GRW

The natural place to begin is with the orthodox approach of Dirac (1947) and von
Neumann (1955) in which there is a genuine process of collapse on measurement20. (The
vagueness over where, when, why and how this collapse takes place might be alleviated
along lines suggested by Ghirardi et al. (1986), perhaps.) If one has a genuine process
of collapse then as noted long ago by Einstein, Podolsky and Rosen (1935)21, one has
action-at-a-distance. In the presence of entanglement, a measurement on one system
can result in a real change to the possessed properties of another system, even when the
two systems are widely separated. (Although, as is well known, these changes do not
allow one to send signals superluminally—this is known as the no-signalling theorem22.)

In teleportation, then, under a collapse interpretation, the effect of Alice’s Bell-basis
measurement will be to prepare Bob’s system, at a distance, in one of four pure states
which depend on the unknown state |χ〉, by using the nonlocal effect of collapse. It then
only remains for Alice to send her two bits to Bob to tell him which (type of) state he
now has in his possession. Under this interpretation, teleportation explicitly involves
nonlocality, or action-at-a-distance; and it is precisely because of the nonlocal effect of
collapse, preparing Bob’s system in a state that differs in one of only four ways from
|χ〉, that a mere two classical bits need be sent by Alice in order for Bob’s system to
acquire a state parameterised by two continuous values.

It is enlightening to compare the effect of collapse in this scenario to that of a rigid
rod held by two parties. Imagine that Alice wanted to let Bob know the value of a
parameter that could take on values in the interval [0, 1]. If they were each holding
one end of a long rigid rod, then Alice could let Bob know the value she has in mind
simply by moving her end of the rod along in Bob’s direction by a suitable distance.
Bob, seeing how far his end of the rod moves, may infer the value Alice is thinking

20One of the defining features of what I here term ‘orthodoxy’ is the adoption of the standard
eigenstate-eigenvalue link for the ascription of definite values to quantum systems. See e.g. Bub (1997).

21See Timpson and Brown (2002) for a recent discussion.
22An early version of the no-signalling theorem, specialised to the case of spin 1/2 EPR-type exper-

iments appears in Bohm (1951). Later, more general versions are given by Tausk (1967); Eberhard
(1978); Ghiradi et al. (1980). See also Shimony (1984), Redhead (1987).

13



of23. There is no mystery here about how the value of the continuous parameter is
transmitted from Alice to Bob. Alice, by moving her end of the rod, moves Bob’s by a
corresponding amount. In teleportation, the effect of collapse is somewhat analogous:
Bob’s system is prepared, by the nonlocal effect of collapse, in a state that depends on
the two continuous parameters characterizing |χ〉. As we have said, collapse allows a
real change in the physical properties that a distant system possesses, if there was prior
entanglement. Compare: pushing one end of a rigid rod axially leads to a change in the
position of the far end. The nonlocal effect of collapse, which is here understood as a
real physical process, is providing the main physical mechanism behind teleportation;
and recall that once the physical mechanisms have been described (I have argued) there
is no further question to be asked about how information is transmitted in the protocol.

In a collapse interpretation, teleportation thus involves nonlocality, in the sense of
action-at-a-distance, crucially. Also, something interesting certainly has happened once
Alice performs her measurement and before she sends the two classical bits to Bob.
There has been a real change in the physical properties of Bob’s system, as it acquires
one of four pure states. (Although note that at this stage the probability distributions
for measurements on Bob’s system will nonetheless not display any dependence on the
parameters characterizing |χ〉, in virtue of the no-signalling theorem. It is only once
the bits from Alice have arrived and Bob has performed the correct operation that
measurements on his system will display a dependence on the parameters α and β.)

5.2 No collapse and no extra values: Everett

It is possible to retain the idea that the wavefunction provides a complete description
of reality while rejecting the notion of collapse; this way lies the Everrett interpretation
(Everett (1957))24. The characteristic feature of the Everett interpretation is that the
dynamics is always unitary; and no extra values are added to the description provided
by the wavefunction in order to account for definite measurement outcomes. Instead,
measurements are simply unitary interactions which have been chosen so as to correlate
states of the system being measured to states of a measuring apparatus. Obtaining a
definite value on measurement is then understood as the measured system coming to have
a definite state (eigenstate of the measured observable) relative to the indicator states
of the measuring apparatus and ultimately, relative to an observer25. A treatment of
teleportation in the Everettian context was given by Vaidman (1994). Braunstein (1996)

23Of course, in a relativistic setting, rigid bodies would not be permissible, although they are in
non-relativistic quantum mechanics. This does not in any case affect the point of the analogy.

24It should be noted that there have been a number of different attempts to develop Everett’s original
ideas into a full-blown interpretation of quantum theory. The most satisfactory of these would appear
to be an approach on the lines of Saunders and Wallace (Saunders (1995, 1996b, 1998, 1996a); Wallace
(2002, 2003a)) which resolves the preferred basis problem and has made considerable progress on the
question of the meaning of probability in Everett (on this, see in particular Deutsch (1999); Wallace
(2003b)).

25This is the case for ideal first-kind (non-disturbing) measurements. The situation becomes more
complicated when we consider the more physically realistic case of measurements which are not of
the first kind; in some cases, for example, the object system may even be destroyed in the process of
measurement. What is important for a measurement to have taken place is that measuring apparatus
and object system were coupled together in such a way that if the object system had been in an
eigenstate of the observable being measured prior to measurement, then the subsequent state of the
measuring apparatus would allow us to infer what that eigenstate was. In this more general framework
the importance is not so much that the object system is left in a eigenstate of the observable relative
to the indicator state of the measuring apparatus, but that we have definite indicator states relative to
macroscopic observables.

14



provides a detailed discussion of the teleportation protocol within unitary quantum
mechanics without collapse.

With teleportation in an Everettian setting, and unlike teleportation under the or-
thodox account, it is clear that there will be no action-at-distance in virtue of collapse
when Alice performs her measurement, for the simple reason that there is no process
of collapse. Instead, the result of Alice’s measurement will be that Bob’s system comes
to have definite relative states related to the unknown state |χ〉, with respect to the
indicator states of the systems recording the outcome of Alice’s measurement (see Ap-
pendix B). (It is argued in Timpson and Brown (2002) and Timpson (2005) that this
does not amount to a new form of nonlocality.) Note, though, that at this stage of the
protocol, the reduced state of every system involved will now be maximally mixed26.
As Braunstein (1996) notes, this feature corresponds to the ‘disembodiment’ of the in-
formation characterizing the unknown state in the orthodox account of teleportation:
following Alice’s measurement, all the systems involved in the protocol will have become
fully entangled. Dependence on the parameters characterizing the unknown state will
only be observable with a suitable global measurement, not for any local measurements.
In particular, one can consider the correlations that now exist between the systems
recording the outcome of Alice’s measurement and Bob’s system. Certain of the joint
(and irreducible) properties of these spatially separated systems will depend on the
identity of the unknown state. In this sense, the information characterizing |χ〉 might
now be said to be ‘in the correlations’ between these systems. (This is the terminology
Braunstein adopts.)

Once Bob has been sent the systems recording the outcome of Alice’s measurement,
however, he is able to disentangle his system from the other systems involved in the
protocol. Its state will now factorise from the joint state of the other systems; and will
in fact be the pure state |χ〉. Dependence on the parameters α and β will finally be
observable for local measurements once more, but this time, only at Bob’s location.

In collapse versions of quantum mechanics, the nonlocal effect of collapse was the
main physical mechanism underlying teleportation. In the no-collapse Everettian set-
ting, the fundamental mechanism is provided by the fact that in the presence of en-
tanglement, local unitary operations—in this case, Alice’s measurement—can have a
non-trivial effect on the global state of the joint system.

So, has anything significant happened at Bob’s location before Alice sends him the
result of her measurement and he performs his conditional unitary operation? Well,
arguably not: nothing has happened other than all of the systems involved in the protocol
having become entangled, as a result of the various local unitary operations.

5.3 No collapse, but extra values: Bohm

The Bohm theory account provides us with an interesting intermediary view of tele-
portation, in which there is no collapse of the wavefunction, but nonlocality plays an
interesting rôle. We shall follow the analysis of Maroney and Hiley (1999).

The Bohm theory (Bohm (1952)) is a nonlocal, contextual, deterministic hidden vari-
able theory, in which the wavefunction Ψ(x1, x2 . . . xn, t) of an n-body system evolves
unitarily according to the Schrödinger dynamics, but is supplemented with definite val-
ues for the positions x1(t), x2(t) . . . xn(t) of the particles. Momenta are also defined
according to pi = ∇iS, where S is the phase of Ψ, hence a definite trajectory may be

26This would not in general be the case if the initial entangled state were not maximally entangled,
or if Alice’s measurement were not an ideal measurement; with these eventualities, the teleportation
would be imperfect (fidelity less than 1).

15



associated with a system, where this trajectory will depend on the many-body wave-
function (and thus, in general, on the positions and behaviour of all the other systems,
however far away). If the initial probability distribution for particle positions is assumed
to be given by |Ψ|2, then the same predictions for measurement outcomes will be made
as in ordinary quantum mechanics. For detailed presentations of the Bohm theory, see
Bohm and Hiley (1993) and Holland (1995).

The guiding effect of the wavefunction on the particle positions may also be under-
stood in terms of a new quantum potential that acts on particles in addition to the
familiar classical potentials. The quantum potential is given by

Q(x1, x2, . . . , xn) = −h̄2
n

∑

i=1

∇2i R
2miR

,

where R is the amplitude of Ψ and mi is the mass of the i-th particle. Among the ways
in which this quantity differs from a classical potential is that it will in general give rise
to a nonlocal dynamics (that is, in the presence of entanglement, the force on a given
system will depend on the instantaneous positions of the other particles, no matter how
far away); and it may be large even when the amplitude from which it is derived is small.
Bohm and Hiley (1993, §3.2) suggest that the quantum potential should be understood
as an ‘information potential’ rather than a mechanical potential, as a way of accounting
for its peculiar properties.

The determinate values for position in the Bohm theory are usually understood as
providing the definite outcomes of measurement27 that would appear to be lacking in a
no-collapse version of quantum mechanics, in the absence of an Everett-style relativiza-
tion. Following a measurement interaction, the wavefunction of the joint object-system
and apparatus will have separated out (in the ideal case) into a superposition of non-
overlapping wavepackets (on configuration space) corresponding to the different possible
outcomes of measurement. The determinate values for the positions of the object-system
and apparatus pointer variable will pick out a point in configuration space; and the out-
come that is observed, or is made definite, is the one corresponding to the wavepacket
whose support contains this point. The wavefunction for the total system remains as
a superposition of all of the non-overlapping waverpackets, however. Bohm and Hi-
ley (1993) introduce the notions of active, passive and inactive information to describe
this feature of the theory. If Ψ may be written as a superposition of non-overlapping
wavepackets, then they suggest that the definite configuration point of the total system
picks out one of these wavepackets (the one whose support contains the point) as ac-
tive. The evolution of the point is determined solely by the wavepacket containing it;
and in keeping with their conception of Q as an information potential, the information
associated with this wavepacket is said to be active. The information associated with
the other wavepackets is termed either ‘passive’, or ‘inactive’. ‘Passive’, if the different
wavepackets may in the future be made to overlap and interfere, ‘inactive’ if such inter-
ference would be a practical impossibility (as for example, if environmental decoherence
has occurred in a measurement-type situation — this corresponds to the case of ‘effective
collapse’ of the wavefunction).

In their discussion of the teleportation protocol, Maroney and Hiley adopt the ap-
proach in which a definite spin vector is also associated with each spin 1/2 particle,
in addition to its definite position. The idea is that with each system is associated an

27Note, though, that measurement may not usually be understood as revealing pre-existing values
in the Bohm theory. Interestingly, Brown and Wallace (2005) have recently argued that these definite
position values may not be so helpful in solving the measurement problem as is often supposed.

16



orthogonal set of axes (body axes) whose orientation is specified by a real three dimen-
sional spin vector, s, along with an angle of rotation about this vector; where these
quantities are determined by the wavefunction28.

The analysis of teleportation then proceeds much as in the Everett interpretation,
save that we may also consider the evolution of the determinate spin vectors associated
with the various systems. Initially, system 1 in the unknown state |χ〉 will have some
definite spin vector that depends on α and β, s(α,β), while it turns out that if Alice
and Bob share a singlet state, the spin vectors for their two systems will be zero (Bohm
and Hiley (1993, §10.6)). Now Alice performs her Bell-basis measurement. As in the
Everettian picture, the effect of measurement is to entangle the systems being measured
with systems recording the outcome of the measurement. But this is not the only
effect, in the Bohm theory. The total wavefunction is now a superposition of four terms
corresponding to the four possible outcomes of Alice’s measurement; and one of these
four terms will be picked out by the definite position value of the measuring apparatus
pointer variable. For each of these four terms taken individually, Bob’s system will be in
a definite state related to the state |χ〉, thus with each will be associated a definite spin
vector sj(α,β), j = 1, . . . , 4, pointing in some direction. When one of the four terms
is picked out as active, and the others rendered passive (or inactive), following Alice’s
measurement, the spin vector for Bob’s system will change instantaneously from zero to
one of the four sj (α,β) (Maroney and Hiley (1999)).

Thus in the Bohm theory, teleportation certainly involves nonlocality; and moreover,
something very interesting does happen as soon as Alice has made her measurement.
Bob’s system acquires a definite spin vector that depends on the parameters character-
izing the unknown state, as a result of a nonlocal quantum torque (Maroney and Hiley
(1999)). Furthermore, there is a one in four chance that this spin vector will be the
same as the original s(α,β); and all this while the total state of the system remains
uncollapsed, with all the particles entangled.

Finally, as we have seen before, once Alice sends Bob systems recording the outcome
of her measurement, he may perform the conditional unitary operation necessary to
disentangle his system from the others, and leave his system in the state |χ〉. The spin
vector of his system will now be s(α,β) with certainty.

5.3.1 A note on active information

The conclusion of Maroney and Hiley (1999) and Hiley (1999) is that according to the
Bohm theory, what is transferred from Alice’s region to Bob’s region in the teleportation
protocol is the active information that is contained in the quantum state of the initial
system. However questions may be raised about how apposite this description is.

Let us label the pointer degree of freedom of the measuring apparatus by x0. At
the beginning of the teleportation protocol, the state of system 1 factorises from the
entangled joint state of 2 and 3; and the state of the measuring apparatus will also
factorise. Accordingly, the quantum potential will be given by a sum of separate terms:

Q(x1, x2, x3,x0) = Q(x1,α,β) + Q(x2, x3) + Q(x0), (5)

where it has been noted that the first term, the one that will determine the motion of

28This is the approach to spin of Bohm et al. (1955). For a systematic presentation see Bohm and
Hiley (1993, §10.2-10.3) or Holland (1995, Chpt. 9). Other approaches to spin are possible, e.g., Bohm
and Hiley (1993, §10.4-10.5), Holland (1995, Chpt. 10), or the ‘minimalism’ of Bell (1966, 1981), in
which no spin values are added.

17



system 1, depends on the parameters characterizing the unknown state29.
Once Alice performs her Bell basis measurement, however, all the systems become

entangled; and the potential will be of the form:

Q(x1, x2, x3,x0) = Q(x1, x2,x0) + Q(x3,x0,α,β) (6)

The part of the quantum potential that will affect system 3 now depends on α and β.
Finally, at the end of the protocol, systems 1, 2 and the measuring apparatus are

left entangled; and system 3, in the pure state |χ〉3, factorises. The quantum potential
then takes the form:

Q(x1, x2, x3,x0) = Q(x1, x2,x0) + Q(x3,α,β) (7)

Maroney and Hiley say:

‘What we see clearly emerging here is that it is active information that has
been transferred from particle 1 to particle 3 and that this transfer has been
mediated by the nonlocal quantum potential.’(Maroney and Hiley (1999,
p.1413))

‘...it is the objective active information contained in the wavefunction that is
transferred from particle 1 to particle 3.’ (Maroney and Hiley (1999, p.1414))

Note that the part of the potential that is active on system 3 will already have
acquired a dependence on α and β before the end of the protocol; that is, as soon
as Alice has performed her measurement. So if active information depending on these
parameters is transferred at all, it will have been transferred before the end of the
protocol. However it is not until Alice has sent her message to Bob and he performs his
conditional operation that the term Q(x3,α,β) in eqn. (7) will take the same form as
the initial Q(x1,α,β).

The difficulties for the stated conclusion arise when we consider more closely what
is meant by ‘active information’. In Maroney and Hiley (1999); Hiley (1999), the con-
nection is made with a different sense of the word ‘information’ than the ones we have
considered so far. This is a sense that derives from the verb ‘inform’ under its branch
I and II senses (Oxford English Dictionary), viz. to give form to, or, to give formative
principle to (this latter, a Scholastic Latin offshoot).

Thus ‘information’ as it appears in ‘active information’ and company, means the
action of giving form to30. ‘The information of x’ (read: The in-formation of x) means
the action of giving form to x.

Now, while we may understand what is meant by Q being said to be an information
potential—it is a potential that gives form to something, presumably the possible tra-
jectories associated with particles (although note that the distinction with mechanical
potentials is now blurred, as these give form to the possible trajectories too)—and may
understand the term ‘active’ as picking out the part of the quantum potential that is
shaping the actual trajectory in configuration space of the total system, it does not
make sense to say that active information is transferred in teleportation. Because ‘in-
formation’ here refers to a particular action—the giving of a form to something—and

29The component of the force on the i-th system due to the quantum potential is given by miẍi =
−∇iQ (Holland (cf. 1995, §7.1.2)); therefore, only terms in the sum which depend on xi will contribute
to the motion of the i-th system.

30Cf. OED ‘information’, sense 7.

18



an action is not a thing that can be moved31. The same type of action may be taking
place at two different places, or at two different times, but an action may not be moved
from A to B.

Thus with ‘active information’ understood in the advertised way, all that can be said
is that an action of the same type is being performed (by the quantum potential) on
system 3 at the end of the teleportation protocol as was being performed on system 1
at the beginning, not that something has been transferred between the two. We may
not, then, understand ‘transfer’ literally. When all is said and done, it is perhaps clearer
simply to adopt the standard description and say that the quantum state of particle 1
has been ‘transferred’ in teleportation; that is (as a quantum state is a mathematical
object and therefore cannot literally be moved about either), that system 3 has been
made to acquire (is left in) the unknown state |χ〉.

To sum up: it perhaps looked as if the Bohmian notion of active information might
provide us with a sense of what is transported in teleportation if we insist that informa-
tion, ‘the information in the wavefunction’, is, in a literal sense, transported. But this
proves not to be the case.

5.4 Ensemble and statistical viewpoints

So far, in all the interpretations we have considered, the quantum state may describe
individual systems. Let us close this section by looking briefly at approaches in which
the state is taken only to describe ensembles of systems.

We may broadly distinguish two such approaches. The first I will term an ensemble
viewpoint. In this approach, the state is taken to represent a real physical property, but
only of an ensemble. Following a measurement, the ensemble must be left in a proper
mixture32, in order for there to be definite outcomes, i.e., the ensemble is left in an
appropriate mixture of sub-ensembles, each described by a pure state (eigenstate of the
measured observable). Thus there will be a real process of collapse, but only at the
level of the ensemble, not for individual systems (which are not being described by a
quantum state, if at all).

The second approach I call a statistical interpretation. (This is the interpretation
that would be adopted by instrumentalists, for example.) On this view, the quantum
formalism merely describes the probabilities for measurement outcomes for ensembles,
there is no description of individual systems and collapse does not correspond to any
real physical process.

On both these approaches, as the state is only associated with an ensemble, it is
not until an entire ensemble has been teleported to Bob (that is, Alice has run the
teleportation protocol on every member of an ensemble in the unknown state |χ〉) that
he acquires something in the state |χ〉. An ensemble or statistical viewpoint thus makes
a natural partner to conservative classical quantity surveying in teleportation.

Under the statistical interpretation, there is clearly no nonlocality involved in telepor-
tation, as there is no real process of collapse; and nothing of any interest has happened
before the required classical bits are sent to Bob. (The no-signalling theorem entails
that Alice’s measurement won’t affect the probability distributions for distant measure-
ments.) The end result of the completed teleportation process is that Bob’s ensemble
is ascribed the state |χ〉; where this merely means that the statistics one will expect for

31On some accounts, an action is the bringing about of some event or state of affairs by an agent
(Alvarez and Hyman (1998)); on others, an action is an event (Davidson (1980)). On no account is an
action something which can intelligibly be said to be moved about.

32See d’Espagnat (1976) for this terminology, also Timpson and Brown (2004).

19



measurements on Bob’s ensemble are now the same as those one would have expected
for measurements on the initial ensemble presented to Alice.

The ensemble viewpoint presents a rather different picture, as it does involve a
real process of collapse, even if only at the ensemble level. Let us suppose that Alice
has performed the Bell basis measurement on her ensembles, but has not yet sent the
ensemble of classical bits to Bob. The effect of this measurement will have been to leave
Bob’s ensemble in a proper mixture composed of sub-ensembles in the four possible
states a fixed rotation away from |χ〉. Thus there has been a nonlocal effect: that
of preparing what was an improper mixture into a particular proper mixture, whose
components depend on the parameters characterizing the unknown state. The use of
the flock of classical bits that Alice sends to Bob is to allow him to separate out the
ensemble he now has into four distinct sub-ensembles, on each of which he performs the
relevant unitary operation, ending up with all four being described by the state |χ〉.

6 Concluding remarks

The aim of this paper has been to show how substantial conceptual difficulties can arise
if one neglects the fact that ‘information’ is an abstract noun. This oversight seems to
lie at the root of much confusion over the process of teleportation; and this gives us very
good reason to pay attention to the logical status of the term. A few closing remarks
should be made.

Schematically, a central part of the argument has been of the following form:
Puzzles arise when we feel the need to tell a story about how something travels from

Alice to Bob in teleportation. In particular, it might be felt that this something needs to
travel in a spatio-temporally continuous fashion; and one might accordingly feel pushed
towards adopting something like the Jozsa/Penrose view.

But if ‘the information’ doesn’t pick out a particular, then there is no thing to take
a path, continuous or not, therefore the problem is not a genuine one, but an illusion.

We can imagine a number of objections. A very simple one might take the following
form: You have said that information is not a particular or thing, therefore it does not
make sense to inquire how it flows (but only inquire about the means by which it is
transmitted). But don’t we have a theory that quantifies information (viz. communi-
cation theory); and if we can say how much of something there is, isn’t that enough to
say that we have a thing, or a quantity that can be located?

This objection is dealt with quickly. Note that this form of argument will not work
in general—one can say how much a picture might be worth in pounds and pence, for
example, but this is not quantifying an amount of stuff, nor describing a quantity with a
location—and it does not work in this particular case either. The Shannon information
doesn’t quantify an amount of stuff that is present in a message, say, nor the amount
of a certain quantity that is present at some spatial location. The Shannon information
H(X) describes a specific property of a source (not a message), namely, the amount of
channel resources that would be required to transmit the messages the source produces.
This is evidently not to quantify an amount of stuff, nor to characterize a quantity that
has a spatial location. (The source certainly has a spatial location, but its information
does not.) Or consider the mutual information. Loosely speaking, this quantity tells
us about the amount we may be able to infer about some event or state of affairs from
the obtaining of another event or state of affairs. But how much we may infer is not a
quantity it makes sense to ascribe a spatial location to.

Another objection might be as follows: You have suggested that it is a mistake to

20



hypostatize information, to talk of it as a thing that moves about. How is this to be
reconciled with some of the ways we often talk about information in physics, especially
the example in relativity, where the most natural way of stating an important constraint
is to say that relativity rules out the propagation of information faster than the speed
of light?

The response is that one can admit this mode of talking without it entailing a
hypostatized conception of information. The constraint is that superluminal signalling
is ruled out on pain of temporal loop paradoxes (Rindler (e.g. 1991, §7.ix)). What this
means is that no physical process is permissible that would allow a signal to be sent
superluminally and thus allow information to be transmitted superluminally. What are
ruled out are certain types of physical processes, not, save as a metaphor, certain types
of motion of information33.

A final objection that might be raised to support the line of thought that inclines
one towards the Jozsa and Penrose conception of teleportation is just this: Well, don’t
we after all require that information be propagated in a spatio-temporally continuous
way? Even if this is not to be construed as a flow of stuff, or the passage of an entity?

The response illustrates part of the value of noting the features of the term ‘infor-
mation’ that have been emphasized here.

The genuine question we face is: what are the physical processes that may be used
to transmit information? Not the (obscure) question ‘How does information behave?’.
Once we see what the question is clearly, then the answer, surely, is to be given by our
best physical theory describing the protocol in question. To be sure, many of the most
familiar classical examples we are used-to use spatio-temporally continuous changes in
physical properties to transmit information (a prime example might be the use of radio
waves), but it is up to physical theory to tell us about the nature of the processes we
are using to transmit information in any given situation. And the examples we have
found in entanglement assisted communication seem precisely to be examples in which
global rather than local properties are being used to carry information; and there seems
not to be a useful sense in which information is being carried in a spatio-temporally
continuous way (although, see Timpson (2005) for further discussion of Deutsch and
Hayden’s opposing view).

It is not the nature of information that is at issue, but the nature of the physical
objects and the physical properties we may use to transmit information.

On a final note, the deflationary approach towards teleportation that I have advo-
cated should be compared with what may be termed the ‘nihilist’ approach of Duwell
(2003). While I am in broad sympathy with much of what Duwell has to say, we differ
on some important points. Duwell also advocates the view that quantum information is
not a substance, but reaches from this the strong conclusion that quantum information
does not exist. From the current point of view this conclusion is unwarranted. Certainly,
quantum information is not a substance or entity, but this does not mean that it doesn’t
exist, it is just a reflection of the fact that ‘information’ is an abstract noun. ‘Beauty’
for example, is an abstract noun, but no one would want to conclude that there is no
beauty in the world. Moreover, Duwell’s conclusion could only possibly be hyperbolical,
for if classical information can be said to exist, then so too can quantum information;
and contrapositively, if quantum information does not exist, then no more does clas-
sical information. The concept of classical information is given by Shannon’s noiseless

33The types of processes in question might not be identifiable without recourse to concepts of what
would count as successful transmission of information, but this does not mean that one has to conceive
of information as an entity or substance, just that one needs a concept of what it means to receive a
signal from which one can learn something.

21



coding theorem, the concept of quantum information, by the quantum noiseless coding
theorem. As we are by now vividly aware, these are not concepts of material quantities
or things. But rejecting the concept of quantum information would be akin to cutting
off one’s nose to spite one’s face; and is by no means necessary in order to get a proper
understanding of teleportation.

Teleportation is not rendered unproblematic by trying to do without the notion of
quantum information and facing the protocol equipped only with Shannon’s concept,
but simply by resisting the temptation to hypostatize an abstract noun; and, having
recognised the status of ‘information’ as an abstract noun, by realising that the only
genuine question one faces is the relatively straightforward one of describing the physical
processes by which information is transmitted.

Acknowledgements

Thanks are due to Jon Barrett, Harvey Brown and Peter Morgan for useful discussion,
to Jane Timpson for the Figure; and to John Christie and Joesph Melia for asking some
good questions.

22



A Elements of information theory

The Shannon information measure is defined as:

H(X) = −
n

∑

i=1

p(xi) log p(xi);

logarithms to base 2. Its primary role is in characterising the degree to which the output
of a source modelled as producing letters picked from a finite alphabet {x1, . . . ,xn} with
probabilities p(xi) may be compressed. Shannon (1948) showed in his noiseless coding
theorem that for very long messages of length N produced by such a source, it was
possible to encode them onto a string of NH(X) bits, thus effecting a compression from
2N log n (the number of possible messages of length N drawn from an alphabet of n
letters) to 2N H(X).

The value of H(X) depends on the probability distribution {p(xi)}, in fact:

0 ≤ H(X) ≤ log n,

where the minimum is achieved when all but one of the p(xi) are zero (the distribution
is maximally peaked); and the maximum when the distribution is flat, ∀i p(xi) = 1/n.
Thus, a source that always outputs the same letter has zero information (cf. footnote 14);
and one that produces each letter with equal probability has maximum information for
a given alphabet size.

H(X) may also be understood as a measure of uncertainty, that is, as a measure
of how concentrated the probability distribution {p(xi)} is (Uffink (1990)). But this
rôle is logically independent of its rôle in characterising information sources (ibid.), a
fact which is rarely recognised and has accordingly given rise to considerable confusion.
(Ibid. See Timpson (2003) for discussion.)

A communication channel (to be precise, a discrete memoryless channel) may be
characterised in terms of conditional probabilities p(yj|xi): given that input xi to the
channel is prepared, what is the probability of getting output yj ? If the channel is noise-
less, then the mapping from input to output states will be one-to-one; in the presence
of noise, a given input may give rise to a spread of outputs with various probabilities.
In his noisy coding theorem, Shannon (1948) proved the surprising result that even in
the presence of noise, it is possible to send messages over a channel with a probability
of error that tends to zero as the length of message, N, tends to infinity.

Given an input distribution p(xi) and the conditional probabilities p(yj|xi) charac-
terising the channel, define the mutual information H(X : Y ):

H(X : Y ) = H(X) − H(X|Y ),
where the ‘conditional entropy’ H(X|Y ) =

∑

j
p(yj )

(

−
∑

i
p(xi|yj) log p(xi|yj)

)

. It
follows from the noisy coding theorem that the mutual information governs the rate at
which it is possible to send information over a channel with input distribution p(xi): It
will be possible to send the output of any information source W with an information
H(W) = H(X : Y ) over the channel, with arbitrarily small error as N is increased. The
capacity of a channel is defined as the supremum over input distributions p(xi) of the
mutual information H(X : Y ).

A quantum source (Schumacher (1995)) produces systems in signal states ρxi with
probabilities p(xi). A density operator ρ =

∑

i
p(xi)ρxi may be associated with the

23



output of this source. Schumacher (1995) considered quantum sources with pure signal
states ρxi = |xi〉〈xi|. The von Neumann entropy S(ρ) is defined as:

S(ρ) = −Trρ log ρ.

The quantum noiseless coding theorem shows that for large N, the output of length N
of a source with signal states ρxi = |xi〉〈xi| may be encoded onto a quantum system of
2N S(ρ) dimensions, that is, using NS(ρ) qubits.

B Teleportation in the absence of collapse

The original Bennett et al. (1993) treatment of teleportation involved collapse following
Alice’s measurement in order to pick out, probabilistically, a definite state of Bob’s
system that depended in a fixed way on the identity of the unknown state. As we have
seen, however, it is quite possible to treat the teleportation protocol in a no-collapse
setting (Vaidman (1994); Braunstein (1996); Mermin (2001)). It might prove helpful to
see how this can work.

Recall the pertinent expression of the initial state in eqn. (4):

|χ〉1|ψ−〉23 =
1

2

(

|φ+〉12
(

−iσ3y|χ〉3
)

+ |φ−〉12
(

σ3x|χ〉3
)

+ |ψ+〉12
(

−σ3z|χ〉3
)

+ |ψ−〉12
(

−13|χ〉3
)

)

.

Alice is going to perform her Bell-basis measurement and she will need systems to record
the outcome of that measurement. As there are four possible outcomes, she may use two
qubits, call them c and d, with an indicator basis {|0〉, |1〉}. Her measurement interaction
UA may be chosen (in the ideal case) so that:

|φ+〉12|0〉c|0〉d 7→ |φ+〉12|0〉c|0〉d,
|φ−〉12|0〉c|0〉d 7→ |φ−〉12|0〉c|1〉d,
|ψ+〉12|0〉c|0〉d 7→ |ψ+〉12|1〉c|0〉d,
|ψ−〉12|0〉c|0〉d 7→ |ψ−〉12|1〉c|1〉d.

(8)

Before Alice’s measurement, then, the total state is |0〉c|0〉d|χ〉1|ψ−〉23. Note that
relative to the states of c and d, the state of system 1 at Alice’s locale is definite, but the
state of Bob’s system, 3, is not. Being maximally mixed (half of a maximally entangled
pair) it has no pure state of its own.

When Alice performs her measurement on 1 and 2, the mapping (8) will apply and
all the systems will now become entangled. The new total state is:

|Ψ〉tot =
1

2

(

|0〉c|0〉d|φ+〉12
(

−iσ3y|χ〉3
)

+ |0〉c|1〉d|φ−〉12
(

σ3x|χ〉3
)

+ |1〉c|0〉d|ψ+〉12
(

−σ3z |χ〉3
)

+ |1〉c|1〉d|ψ−〉12
(

−13|χ〉3
)

)

. (9)

24



We can see that system 1 is certainly no longer in the state |χ〉, it is in fact now
thoroughly entangled and has the reduced state 1/2 1. But also, crucially, relative to
definite measurement outcome states of systems c and d, Bob’s system now has a definite
state related to the initial |χ〉. As everything is entangled, though, the reduced state of
his system is still simply 1/2 1.

We need to disentangle system 3 from all the others; and preferably in such a way that
it is left in the unknown state |χ〉. To do this, Bob will need to perform a conditional
unitary operation, that is, an operation on his system that depends on the states of
the two qubits recording the outcome of Alice’s measurement. As he can only apply
operations locally (the standard assumption) systems c and d will need to be transported
to him before the operation may be performed.

Once c and d are with Bob, he may perform the following conditional unitary trans-
formation, UB:

UB = P
cd
|00〉 ⊗ (iσ

3
y) + P

cd
|01〉 ⊗ σ

3
x + P

cd
|10〉 ⊗ (−σ

3
z ) + P

cd
|11〉 ⊗ (−1

3),

where P cd
|00〉

is the projector onto |0〉c|0〉d, and so on. This will have the effect of disen-
tangling 3 and leaving it in the unknown state |χ〉, as desired (note that UB does not
depend on the identity of |χ〉):

UB|Ψ〉tot =
1

2

(

|φ+〉12|0〉c|0〉d + |φ−〉12|0〉c|1〉d + |ψ+〉12|1〉c|0〉d + |ψ−〉12|1〉c|1〉d
)

|χ〉3.
(10)

25



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