Modal Interpretation of Repeated Measurement

Modal Interpretation of Repeated Measurement: A Rejoinder to Leeds and Healey Bas C. van Fraassen Philosophy of Science, Vol. 64, No. 4. (Dec., 1997), pp. 669-676. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28199712%2964%3A4%3C669%3AMIORMA%3E2.0.CO%3B2-R Philosophy of Science is currently published by The University of Chicago Press.

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Modal Interpretation

of Repeated Measurement:

A Rejoinder to Leeds and H e a l e y *



Bas C. van FraassentT Department of Philosophy, Princeton University

A recent article (Leeds and Healey 1996) argues that the modal interpretation (Copenhagen variant) of quantum mechanics does not do justice to ilnlnediately repeated nondisturbing measurements. This objection has been raised before, but the article presents it in a new, detailed, precise form. I show that the objection is mistaken.

A recent article (Leeds and Healey 1996) argues that my favorite modal interpretation of quantum mechanics does not do justice to immediately repeated non-disturbing measurements. This objection has been raised before, but never with such detailed, fair, and accurate appraisal. In this note I will show that the objection is mistaken. I do concede, wholeheartedly, that I failed to explain the matter well in my 1991. Their criticisms on that account are just. Specifically, I failed to make explicit all relevant implications of a condition on value states in many-body systems (see my 1991, Ch. 9, $8). I cannot very well fault my critics for missing a11 implication on which I reflected so poorly. But I believe I can now repair the matter for all of us. Leeds and Healey's article contains also an insightful critical discussion of empiricist approaches in general, which I will here leave aside. *Received September 1996. ?Send reprint requests to the author, Department of Philosophy, Princeton University,

1879 Hall, Princeton, NJ 08544-1006.

$The author wishes to thank Martin Jones, Tim Maudlin, Brad Monton, and Paul

Teller for helpful comments and correspondence. Philosophy of Science, 64 (December 1997) pp. 669-676. 0031-8248/97/6404-0008$2.00 Copyright 1997 by the Philosophy of Science Association. All rights reserved.

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1. The Interpretation's Minimalism. The Copenhagen Variant of the Modal Interpretation (henceforth CVMI) was explicitly designed to have as little packed into it as possible while still exemplifying certain salient elements of the traditional Copenhagen interpretation. What values observables have is logically independent of what the quantum mechanical states ("dynamic states") of the relevant systems are, although constrained by them in certain ways. (For example, if the state is eigenstate a,of observable A, then A has the corresponding (ith) eigenvalue-I shall follow Leeds and Healey's notation.) In addition, probabilities are assigned to values of observables which are outcomes of measurements (Born probabilities, the interpretation of Born's rule for calculating probabilities). The two most obvious consequences of the intended minimalism are these: probabilities are assigned to events only at the end stages of measurements, and there is no 'dynamics' for the values. The latter means that there are no additional conditions specifically for changes in value over time. In some models of the same situation, the value of an unmeasured observable may stay the same while in another it fluctuates madly. The hope was to convince myself, and all of us, that there is a way the world could be as quantum theory says it is, a way in accord with certain seminal ideas of the Copenhagen school. For this purpose, the more minimal the assumptions made, the better. Yet I realize that the resulting interpretation is for that very reason unsatisfying in certain respects. That is exactly because not everyone values those Copenhagen ideas as much, and because even those who do would like to add to them. Let us call this Question 1: on the CVMI, is it possible for the values of at least some observables to enjoy reasonable constancy and continuous evolution under suitable circumstances? There is relevant recent work which explores this sort of question for other varieties of modal interpretation. Bohm, Bub, Arntzenius, and lately especially Vink have shown us ways of adding such conditions in general. I would welcome, of course, all suggestions about possible strengthenings of my interpretative assumptions. It would be a matter of intellectual interest and possibly new insight to have on the books a well-defined set of variants of the Copenhagen variant, each with its own stronger constraints on evolution of values of observables. Question 1 does not concern measurement alone, but it can be strikingly illustrated by asking what happens to values of observables between measurements. At present, the CVMI does not rule out that the pointer on a measurement apparatus will change position between the end of a measurement and the moment when that pointer's position is

MODAL INTERPRETATION OF REPEATED MEASUREMENT

671

recorded (by a further measurement, e.g., a person who comes to look). If the set-up is well-constructed, the later moment may be taken to be the end of a measurement, of the same type, of the same observable, and the measurement outcome statistics will be the same. But there is no guarantee that the pointer observable's value was constant through that interval. That point could have been, all by itself, the basis for Leeds and Healey's complaint. But then they would not have gone beyond a discussion of Question 1 and its measurement illustrations. I am obliged to Leeds and Healey for presenting the problem of repeated measurement in a way that clearly raises it as an additional issue. Although they note the above point in passing (p. 94), Leeds and Healey ask a more searching question. Consider the results of two measurements of the same observable that have been initiated successively on the same system, and compare their outcomes; will those outcomes be the same? Let us call that Question 2. The disturbing conclusion they reach is that, based on CVMI, they need not be the same-in fact, need not be the same even if the record (outcome of a measurement to check for agreement) indicates that they are the same! This conclusion would indeed be disturbing. But I can show that it is not so. Whether or not an 'agreement' measurement is actually carried out, the two pointer observables will have the same value.

2. Repeated (von Neumann Type) Measurement. In order to address the precise situation discussed by Leeds and Healey, and to use the same notation and labels as they do, let me quote their description: Let the measured system I start off in a superposition Cciaiof eigenstates of an observable A corresponding to distinct eigenvalues, and let us measure A twice, using two measuring systems I1 and I11 (we will assume that I and I1 each evolve freely after their measurement interaction, that there is no interaction between I1 and 111, and that both A and the 'pointer-reading observable' B for I1 commute with the free Hamiltonians for I and I1 respectively). Then, under the familiar idealized assumptions of a von Neumann measurement, the combined system I + I1 will be in dynamic state Cc, (ai x pi) at the end of the first measurement; at the end of the second measurement the dynamic state of I I1 + I11 will be, ignoring phase factors, Cci (ai X Pi x Pi). At the end of the first measurement, the individual systems I and I1 are in dynamic states [Clci12P[ai]and Clci12P[Pi],respectively]. As for the individual value states, because I and I1 interact by a measurement interaction, I ends up in some ai, with I1 in the corresponding pi. At the conclusion of the first measurement, the pointer reading observable B on

+

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BAS C. VAN FKAASSEN

I1 thus has the value b,, we will say that its pointer reads b,. Likewise, at the end of the second measurement, I ends up in some aj, with I11 in the corresponding f3,; its pointer the reads b,. (Leeds and Healey 1996,93-94) (The phrase in square brackets corrects a typographical error; compare op. cit., 92.) The desirable conclusion, according to von Neumann, is that the two outcomes agree, i = j. Is there any way to verify this? The literature already contains much criticism of whether von Neumann was here pointing to a fact of experience, or even whether real sequential measurements are accurately modelled in this abstract representation. Let us, with Leeds and Healey, set at least the latter doubts aside for now. They proceed to formulate the problem for the CVMI--what I called Question 2 above-in the following precise terms: Let M be an observable for the combined system I1 + 111, which has eigenvalue 1 on the space spanned by all Pi x pi, and which has value 0 on all pi x pj, for j not equal to i. Then if I1 and I11 are in pure dynamic states Pi and P, respectively (always ignoring phase factors), the value of M will be 1 if and only if i = j. In the Copenhagen interpretation, this means that in the only case in which our pointer readings can have definite values, M will have the value 1 just in case these values agree. In the context of that interpretation, then, it is reasonable to speak of M as the observable which is, or registers, agreement between the two pointer readings. Even in the context of the modal interpretation, one can continue to speak of M as the 'agreement' observable. Here, however, the locution needs to be taken with a grain of salt-M may take up the value 1 even though the pointer readings do not agree. Our present example provides an illustration of this. The final dynamic state of I1 + I11 is Xlci12P[PiX Pi]. The value state of I1 + I11 is a pure state possible with respect to this mixture; since all summands of the mixture are eigenstates of M with eigenvalue 1, so is the value state. So M takes the value 1 on the system I1 + 111. There is, however, nothing in van Fraassen's formalism to guarantee that the pointer reading on I11 is equal to the pointer reading on 11. (op. cit., 94) Observable M can take different values at different times of course. That M takes value 1 on system I1 + III is correct for the final time, the end of the measurement by 111, which they go on to denote as "t,". They add, to make the last quoted sentence precise, that there is in the

CVMI no guarantee that at t,, the pointer reading on 11has any precise value at all, let alone a value equal to the pointer reading on 111: Rather at t, the dynamic states of I1 and I11 are each Zlci12P[Pi]; because 111 was used to measure I, its value state is some pi but because I1 did not participate in a measurement interaction, its value state can be some p,, j not equal to i, or even a superposition of various p,. (ibid.) This conclusion does follow from reasoning based solely on those parts of the CVMI which have been noted so far. But there is a further part of this interpretation which is being ignored here.

3. Relative Possibility in Many-Body Systems. The part of the CVMI noted so far (besides the status of the Born probabilities) is that the value state must be possible relative to the quantum mechanical state. If the latter is the mixture Zlci12P[Pi],then the value state is a pure state in the subspace spanned by the states {PI).But there is a further condition on value states for the various subsystems of a many-body system. It has no punch for 1-body and 2-body systems, but comes into its own thereafter. Switching for a moment to a more general notation, consider a composite systems X(l) + . . . + X(15), in quantum mechanical state W. There is a general recipe for deducing the quantum mechanical state of a given subsystem such as X(9) + X(13) or X(9) + X(11) + X(13)the so-called 'reduction of the density matrix'. (This was of course utilized in each of the passages I just quoted from Leeds and Healey.) Let us use the general notation #W(9;13) for the reduced state which is thus assigned to subsystem X(9) + X(13). In addition, let us use the notation h(9;13) for the value state of X(9) + X(13). These examples should suffice to explain the general notation conventions. Now the already noted condition on value states imply, for this example: value state 3L(9,13) is possible relative to #W(9; 13) value state h(9) is possible relative to #W(9) value state h(13) is possible relative to #W(13) As can be seen at once, such a reduced density matrix as #W(13) can be calculated in various ways (which are mutually consistent: #W(13) = #(#W(9;13))(13) for example). This helps to see what is meant by the further condition on value states of subsystems. In our example, it yields: h(9) is also possible relative to #1(9; 13)(9)

674

BAS

c. VAN

FRAASSEN

h(13) is also possible relative to #h(9; 13)(13) For the full exposition and the required consistency proof, to show that this condition can be satisfied by systematic assignments of value states, see my 1991, Ch. 9, 58 and 9.

4. Application to Repeated Measurement. With this further part of the CVMI in hand, we return to Leeds and Healey's discussion. Our first task will be to show that by the definition of von Neumann type measurements there are at least four such measurements in the situation described. Three of these end at the final time t,. Note well that in the CVMI the definition of each type of measurement is entirely in terms of the quantum mechanical states and evolution operators (Hamiltonians), with no reference to the value states (see my 1991, Ch. 7,94 and Ch. 8, 93). The four measurements are: a measurement of A by a measurement of A by a measurement of A by a measurement of A by

I1 ending at intermediate time 6 I11 ending at final time t, 11, also ending at t, I1 + 111, also ending at t,

These are all von Neumann type measurements (by definition; cf. van Fraassen 1991, 218). Let us look at the last three in detail. At the final time t,, the complete system I + I1 + I11 is in pure state Cci(a, x pi x P,). The quantum mechanical states of the subsystem are as follows:

I

+

I is in I1 and I11 are in I1 and I + I11 are in I1 + I11 is in

~Ici12P[~iI

Clci12P[Pi]

Clci12P[aix Pi]

zIciI2P[Pi X PI]

What about the value states? Since that final time is the end of the measurement by 111, we have I11 has value state p, (for a certain index m)

(1)

Because I1 has been left undisturbed by the evolution of the total system, the process involving I and I1 during the entire interval (ending at t,) is also a measurement of A by I1 on I. The process involving I and I1 which lasts the entire interval satisfies von Neumann's conditions on measurement: the evolution is such that necessarily, if 1's initial state is ai then the final state of system I + I1 + I11 is ai X pi x pi, and so the final state of I + I1 at t, is a, x pi. This measurement is not identical with the one that ended at the intermediate time, but is of exactly the same sort, of the same observable

MODAL INTERPRETATION OF REPEATED MEASUREMENT

675

with the same pointer observable, and is definitely a von Neumann type measurement. Therefore, by the same token we have I1 has value state p, (for a certain index n)

(2)

We do not know yet, of course, whether m = n. By the same reasoning as for 11, the 2-body system I1 + 111 is a measurement apparatus which measures A on I, during the total interval ending at t,. The evolution of the total process is such that, if I starts in ai, then I + (I1 + 111) must end in ai x (pi x Pi). The pointer observable is of course B x B. This pointer observable has as eigenstates all the members of the basis (p, x P,), with r and s not necessarily the same. Because the process is a measurement, we conclude that I1 + I11 has at the end a value state which is a pure eigenstate of B x B, hence: I1

+ I11 has value state

h

=

(p, x p,) (for certain indices r, s) (3)

But this value state must be possible relative to the quantum mechanical state of I1 + 111, namely Clci12P[PiX Pi]. The states possible relative to this mixture include superpositions of the pure states Pi x pi. But in this case these are excluded by (3). Therefore: r = s: I1

+ I11 has value state h = (p, x PJ

(4)

Now we begin to apply the further condition that relates subsystems. The value state P, of I11 must be possible relative to #h(III) = p,. Therefore m = r, that is, I11 has value state P,. The value state P, of I1 must be possible relative to #h(II) = p,. Therefore, n = r, that is, I1 also has value state P, .

(5) (6)

This ends our demonstration, as far as Leeds and Healey's objection goes. For contrary to their assertion, the outcomes of measurements of A by I1 and by 111, ending at t,, are the same. If an inspection (measurement of the described observable M) indicates that they are the same, then that result reveals the truth of the matter, on the CVMI. But the desired agreement in values (pointer readings of I1 and I11 at t,) obtains regardless of whether such a measurement is made. Let me add just two concluding comments, the first about the principles used in the argument and the second about its significance.First, the argument is a bit complex, so it may be helpful to note here the four principles which were being made to act conjointly. They are: (a) that value states are pure; (b) that the value state is possible relative to the dynamic state;

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(c) that the value state of a measurement apparatus at the end of a measurement is an eigenstate of the pointer observable; and fourth, (d) the constraint relating the value states of the various parts of a composite system. Finally, as to the significance of the argument: as Leeds and Healey note, there has been continuing disagreement about how von Neumann's idea of immediately repeated measurement can be realized, if at all. (My own feeling is that only Dicke's proposal-see my 1991, 257-258-has come near to being realistic.) But for the sort of set-up abstractly described by Leeds and Healey, which I take to be consistent with QM, the CVMI implies exactly the agreement (of actual outcomes at the end of the total process) which was deemed desirable. REFERENCES

Leeds, S. and R. Healey (1996), "A Note on van Fraassen's Modal Interpretation of Quantum Mechanics", Philosophy of Science 63: 91-104. van Fraassen, B.C. (1991), Quantum Mechanics: An Empiricist View. Oxford: Clarendon Press.