Download Four strategies for dealing with the counting anomaly

Four strategies for dealing with the counting anomaly in spontaneous collapse
theories of quantum mechanics
PETER J. LEWIS
Department of Philosophy, University of Miami, Coral Gables, Florida, USA
Abstract A few years ago, I argued that according to spontaneous collapse theories of quantum
mechanics, arithmetic applies to macroscopic objects only as an approximation. Several authors
have written articles defending spontaneous collapse theories against this charge, including
Ghirardi and Bassi, Clifton and Monton, and now Frigg. The arguments of these authors are all
different and all ingenious, but in the end I think that none of them succeed, for reasons I
elaborate here. I suggest a fourth line of response, based on an analogy with epistemic
paradoxes, which I think is the best way to defend spontaneous collapse theories, and which
leaves my main thesis intact.
1. The counting anomaly
Discussion of the counting anomaly in quantum mechanics centres around the following
example. Suppose we have n marbles in a box. According to the standard interpretation rule for
quantum mechanical states (the eigenstate-eigenvalue link), a marble counts as being in the box
if and only if its state is an eigenstate for being in the box. But the dynamics of spontaneous
collapse theories entails that the states of marbles are generally not such eigenstates; rather, they
are states very close to the eigenstates. Such a state can be written
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a|in + b|out
(1)
where |in is the eigenstate for being in the box, |out is the eigenstate for being outside the box,
|a|2 + |b|2 = 1 and |a|2 >> |b|2. In order to interpret such a state as one in which the marble is in the
box, it is necessary to relax the eigenstate-eigenvalue link somewhat. The most straightforward
way to do this is to stipulate that a marble counts as being in the box if and only if the squared
inner product of its state and the eigenstate for being in the box is sufficiently close to 1 (Lewis,
1997, p. 316). Clifton and Monton have dubbed this relaxed rule the fuzzy link (1999, p. 699).
Now consider the state of all n marbles. If each of them is in a state like (1), the state of
the whole n-marble system can be written
(a|in1 + b|out1)  (a|in2 + b|out2)    (a|inn + b|outn)
(2)
Since the eigenstate of all n marbles being in the box is |in1  |in2    |inn, the squared
inner product of state (2) with the eigenstate of all n marbles being in the box is |a|2n. By making
n large, we can make this quantity as small as we like, and in particular, we can make it small
enough that state (2) does not count as a state in which all n marbles are in the box according to
the fuzzy link. Hence state (2) violates standard arithmetic; it is true of each marble that it is in
the box, but it is not the case that there are n marbles in the box. This is what Clifton and Monton
(1999, p. 700) call the counting anomaly.
Of course, the fuzzy link is only one possibility for interpreting quantum states in the
context of a spontaneous collapse theory. But in the original paper I argued that the choice of an
interpretation rule presents the defender of spontaneous collapse theories with a dilemma (Lewis,
1997, p. 318). Any such interpretation rule must either say one of two things: Either it says that
state (2) is not a state in which all n marbles are in the box (either because it says that some
number of marbles other than n is in the box, or because it says that there is no fact about the
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number of marbles in the box); or it says that state (2) is a state in which all n marbles are in the
box. If you choose the first option, then you are faced with the counting anomaly, as described
above. On the other hand, if you choose the second option, then you have to be willing to count
state (2) as a state in which all n marbles are in the box even though it will almost certainly not
behave as if all n marbles are in the box; in particular, there is only a very small chance that you
will find all n marbles in the box if you look. The would-be defender of spontaneous collapse
theories, it seems, has to find a way to deal with one horn or other of this dilemma.
2. Previous strategies
Let me briefly describe the two previously published strategies for dealing with the problem just
described. Clifton and Monton (1999) take on the first horn of the dilemma. They endorse the
fuzzy link, and hence admit that the counting anomaly affects states like (2). However, they
argue that this anomaly can never be made manifest; whenever you observe it, it goes away. In
order to observe the number of marbles in the box, you must correlate the positions of the
marbles with something—a pointer reading or a brain state. According to the dynamics of
spontaneous collapse theories, such an interaction will cause state (2) to evolve rapidly into a
state in which it is true of m of the n marbles considered individually that they are in the box, and
in which it is also true of the entire n-marble system that m marbles count as being in the box.
Hence the post-measurement state no longer manifests the counting anomaly, and an
unobservable anomaly, they argue, should not worry us very much.
I respond in detail to this strategy elsewhere (Lewis, 2003). Put briefly, I argue that the
measurements Clifton and Monton describe are precisely the means by which the counting
anomaly is made manifest. When one measures the number of marbles in the box for state (2),
one will almost certainly get a result m that differs from n, and if one repeats the experiment one
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will almost certainly not get the same result twice. This strikes me as prima facie evidence that
the pre-measurement state (2) is not a state in which all n marbles are in the box, and hence
prima facie evidence that (2) suffers from the counting anomaly.
Bassi and Ghirardi (1999b) take on the second horn of the dilemma. They appeal to a
fairly elaborate interpretation scheme for quantum states (Ghirardi, Grassi and Benatti, 1995),
according to which (2) is a state in which all n marbles are in the box, and hence for which the
counting anomaly does not arise. In fact, they show that according to their interpretation scheme,
no quantum state that lasts for more than a tiny fraction of a second can suffer from the counting
anomaly.
This proposal, however, fails to deal with the problem of the behaviour of systems in
states like (2); again, I respond in detail elsewhere (Lewis, 2003). In brief, I argue that since the
job of an interpretation rule is to map the language of quantum mechanics onto everyday
language, an interpretation rule fails if it assigns a description to a system that is not endorsed by
everyday language practices. The interpretation rule proposed by Ghirardi and Bassi stipulates
that state (2) is a state in which all n marbles are in the box, despite the fact that you will almost
certainly not find all the marbles in the box if you look. This, it seems to me, is a violation of
everyday language practices, and hence their proposed interpretation rule is untenable.
3. Frigg’s strategy
Like Ghirardi and Bassi, Frigg (2003) contends that (2) is a state in which all n marbles are in the
box, and hence that there is no counting anomaly. However, Frigg’s argument strategy is very
different from Ghirardi and Bassi’s; rather than replacing the fuzzy link with a new interpretation
rule for quantum states, Frigg relies on a more general argument concerning the property
structure of spontaneous collapse theories. Furthermore, this strategy allows the author to argue
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that spontaneous collapse theories in fact escape my dilemma, since the property structure entails
that not only is state (2) a state in which all n marbles are in the box, it is also a state which
behaves as if all n marbles are in the box.
Frigg begins by showing that the composition principle fails for fuzzy quantum
mechanics—spontaneous collapse theories interpreted according to the fuzzy link. The
composition principle is the principle that if every object in an ensemble has property P, then the
ensemble itself has property P. This is a nice result, but by itself it might appear to support rather
than refute the existence of the counting anomaly. After all, it looks like the counting anomaly is
just a special case of the failure of the composition principle; in state (2), each marble in the
ensemble of n marbles has the property of being in the box, but it is not the case that the
ensemble itself has the property of being in the box. If the counting anomaly is entailed by the
failure of the composition principle, then how can the failure of the composition principle be
advanced as a solution to the counting anomaly?
But there is a second important element to Frigg’s argument. Given the failure of the
composition principle, the state of affairs in which each of the n marbles has the property of
being in the box is different from the state of affairs in which the entire n-marble ensemble has
the property of being in the box. A question then arises as to which state of affairs is the one
corresponding to our everyday-language claim that all the marbles are in the box. Frigg insists
that it is the former, since for non-interacting marbles, all it takes for the everyday-language
claim that all the marbles are in the box to be true is that marble 1 is in the box, marble 2 is in the
box, and so on through marble n. Since state (2) satisfies this requirement by hypothesis, there is
no counting anomaly. Furthermore, state (2) behaves as if all the marbles are in the box when we
count them: “Since counting is a process that is concerned with individual objects rather than
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with ensembles as a whole, the thing we need in order to count is that the conjunction of all [the
individual marble properties] is true” (Frigg, 2003, pp. 16–17). The question of whether the nmarble ensemble as a whole has the property of being in the box is not relevant, he claims, either
to the truth of the everyday-language claim that all the marbles are in the box, or to whether the
system behaves as if all the marbles are in the box.
4. Assessment of Frigg’s strategy
Frigg’s argument is ingenious, and improves on previous strategies for dealing with the counting
anomaly in a couple of ways. First, it improves on the strategy of Ghirardi and Bassi in that it
does not rely on the introduction of new and elaborate interpretive machinery to bridge the gap
between quantum states and everyday language. Frigg retains the fuzzy link as the interpretation
rule for states, which, as I have argued elsewhere (Lewis, 2003), provides a simple and natural
link between states and language. Second, Frigg suggests a way of escaping my dilemma for
interpretations of states like (2). I maintained that any interpretation of state (2) must either fall
prey to the counting anomaly, or completely divorce the truth of claims about the state from its
behaviour. Frigg’s interpretation, if successful, entails that the claim “All the marbles are in the
box” is true of state (2), and also that state (2) behaves as if all the marbles are in the box,
thereby avoiding the counting anomaly without separating truth from behaviour.
My worry about this strategy is the same as my worry about Ghirardi and Bassi’s
strategy, namely that it violates everyday language practices. Consider what the everydaylanguage claim “All the marbles are in the box” entails about the behaviour of the system. Is it,
as Frigg maintains, simply a claim about each individual marble, namely that if you look for it,
you will almost certainly find it inside the box rather than outside the box? Or is it a claim about
the ensemble of marbles, namely that the whole ensemble will almost certainly be found inside
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the box and none of it outside? It seems to me that it is both, since in everyday language we do
not distinguish between the two.
Now this is an empirical claim about everyday language, and requires justification. But I
think that the very fact that the counting anomaly strikes us as paradoxical provides strong
evidence for this claim. Where our language practices admit violations of the composition
principle, there is no paradox; it does not strike us as self-contradictory to say that each
individual marble is small but the n-marble ensemble is not small. However, it does strike us as
self-contradictory to say that each individual marble is in the box but the n-marble ensemble is
not in the box, and the most natural explanation seems to be that we presuppose the composition
principle in our talk about the locations of ordinary macroscopic objects. I think that Frigg in
effect concedes this point when he notes that the failure of the composition principle is a
violation of common sense (Frigg, 2003, p. 10). But if these observations about everyday
language are correct, then contra Frigg, state (2) cannot be interpreted using the phrase “All n
marbles are in the box”, and neither does state (2) behave as this everyday language locution
leads us to expect.
Frigg tries to muster support for his position from the epistemic paradoxes. For example,
suppose I adopt a subjective probability account of belief, according to which I believe p if and
only if my subjective probability for p is greater than 0.95. If I have 100 propositional beliefs
each with a subjective probability of 0.98, then each proposition taken individually counts as a
belief, but the conjunction of all 100 propositions does not. Although this example provides a
familiar case in which the composition principle fails, I think it ultimately counts against the
point Frigg is trying to make rather than in favor of it. Consider the everyday language claim “I
believe q”, where q is a conjunction of propositions. Does this imply that each conjunct has a
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high subjective probability, or that the conjunction as a whole does? Again, I think it implies
both, since everyday language about beliefs presupposes the composition principle, and this is
precisely what makes the epistemic paradoxes paradoxical. The paradox cannot be solved simply
by stipulating that in everyday language the locution “I believe q” entails nothing about q as a
whole, since if that were the case, there would be no paradox to address in the first place. Yet
this is precisely Frigg’s strategy with regard to the counting anomaly in fuzzy quantum
mechanics.
5. Living with the counting anomaly
Despite my critical comments above concerning Frigg’s appeal to epistemic paradoxes, I do
think that the analogy between fuzzy quantum mechanics and epistemic paradoxes can provide
some insight into how to deal with the counting anomaly. Suppose, for whatever reason, one
becomes seriously committed to a subjective probability account of belief. What attitude should
one take about the resulting epistemic paradox? What should one say about a conjunction q of
100 individual propositions, each with a subjective probability of 0.98?
As I argued in the previous section, one cannot simply deny that there is such a paradox
on the grounds that the everyday claim “I believe q” carries no implications about my attitude
towards q as a whole; this strategy clearly misrepresents the everyday use of the term “believe”.
For similar reasons, one cannot interpret the paradox away by adopting a suitable rule for
translating between subjective probability language and everyday belief language. If the rule
interprets my state as one in which I believe q, then it violates the everyday use of “believe”, and
cannot be counted as a translation into everyday language. On the other hand, if the rule
interprets my state as one in which I don’t believe q, then the paradox remains. But if everyday
belief talk entails paradox, and this paradox cannot be interpreted away, perhaps the best
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response is to reject everyday belief talk in favor of the more tractable language of subjective
probabilities. If one is seriously committed to a subjective probability account of belief, that is,
one should regard this account not as a theoretical underpinning of our everyday belief talk, but
as a theoretical replacement for it.
This suggests a parallel line of reasoning for the counting anomaly. I have argued, contra
Frigg, that there really is a counting anomaly; the anomaly cannot be dissolved by claiming that
the everyday claim “All n marbles are in the box” refers only to the states of the individual
marbles and not to the ensemble, because the everyday claim carries implications about the
ensemble. Similarly, and contra Ghirardi and Bassi, one cannot interpret the paradox away by
adopting a suitable rule for translating between quantum mechanical language and everyday
location language. If the rule says that state (2) is a state in which all n marbles are in the box,
then it violates the ordinary use of location terms, and hence cannot count as a translation into
everyday language. But if the rule says that state (2) is not a state in which all n marbles are in
the box, then the counting anomaly remains.
But if any attempt to translate the language of quantum mechanics into everyday location
language either fails to respect everyday language or results in anomaly, perhaps it is everyday
language that is to blame. In other words, if one is seriously committed to a spontaneous collapse
theory of quantum mechanics, one should regard this theory not as the underpinning of our
everyday belief talk, but as a theoretical replacement for it. Everyday location language, while
adequate for ordinary uses, is seriously flawed as a descriptive apparatus, even for macroscopic
objects. When systems get complicated, as in state (2), one is compelled to retreat to the
language of quantum mechanics.
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In my original article, I tried to portray the counting anomaly as a serious problem for
spontaneous collapse theories of quantum mechanics (Lewis, 1997, pp. 320–24). But perhaps the
rhetoric of this section was overblown; the fact that a quantum mechanical description of a
macroscopic system is not translatable into everyday language is not a knockdown objection to
the quantum mechanical theory. However, I stand by the main thesis of the article, namely that
the counting anomaly is a feature of spontaneous collapse theories that cannot be interpreted
away.
References
BASSI, A. & GHIRARDI, G. C. (1999b) More about Dynamical Reduction and the Enumeration
Principle, British Journal for the Philosophy of Science, 50, pp. 719–34.
BASSI, A. & GHIRARDI, G. C. (2001) Counting Marbles: Reply to Clifton and Monton, British
Journal for the Philosophy of Science, 52, pp. 125–30.
CLIFTON, R. & MONTON, B. (1999) Losing Your Marbles in Wavefunction Collapse Theories,
British Journal for the Philosophy of Science, 50, pp. 697–717.
CLIFTON, R. & MONTON, B. (2000) Counting Marbles with ‘Accessible’ Mass Density: A Reply
to Bassi and Ghirardi, British Journal for the Philosophy of Science, 51, pp. 155–64.
FRIGG, R. (2003) On the Property Structure of Realist Collapse Interpretations of Quantum
Mechanics and the So-Called ‘Counting Anomaly’, International Studies in the
Philosophy of Science, forthcoming.
GHIRARDI, G. C. & BASSI, A. (1999a) Do Dynamical Reduction Models Imply that Arithmetic
Does Not Apply to Ordinary Macroscopic Objects?, British Journal for the Philosophy of
Science, 50, pp. 105–20.
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GHIRARDI, G. C., GRASSI, R. & BENATTI, F. (1995) Describing the Macroscopic World: Closing
the Circle within the Dynamical Reduction Program, Foundations of Physics, 25, pp. 5–
38.
LEWIS, P. J. (1997) Quantum Mechanics, Orthogonality and Counting, British Journal for the
Philosophy of Science, 48, pp. 313–28.
LEWIS, P. J. (2003) Counting Marbles: A Reply to Critics, British Journal for the Philosophy of
Science, 54, pp. 165–170.
Note on contributor
Peter J. Lewis is Assistant Professor of Philosophy at the University of Miami. He has published
articles on the foundations of quantum mechanics, inluding spontaneous collapse theories and no
collapse theories, as well as on scientific realism. Corresdpondence: Department of Philosophy,
University of Miami, P. O. Box 248054, Coral Gables, FL 33124-4670, USA. Email:
[email protected].
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