logic and probability in quantum mechanics || the einstein-podolsky-rosen paradox

The Einstein-Podolsky-Rosen ParadoxAuthor(s): Bas C. Van FraassenSource: Synthese, Vol. 29, No. 1/4, Logic and Probability in Quantum Mechanics (Dec., 1974),pp. 291-309Published by: SpringerStable URL: http://www.jstor.org/stable/20115001 .

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BAS C. VAN FRAASSEN

THE EINSTEIN-PODOLSKY-ROSEN PARADOX*

The first part of this article analyzes the 'paradoxical' implications of

elementary quantum theory described by Einstein, Podolsky, and Rosen

(1935; henceforth, EPR). At the end of the analysis we are left with a

dilemma for the interpretation of quantum mechanics.

In the second part, I embrace one horn of the dilemma and try to show

that this can be done with consistency and adequacy. The interpretation I offer was introduced in a previous paper on quantum measurement,

and generalized in a subsequent paper on quantum logic (van Fraassen,

1972; 1973). I call it a modal interpretation; it is emphatically not of the

variety of 'quantum-logical' interpretations espoused by Putnam and,

lately, by Bub. On the contrary, what I try to develop is the orthodox

statistical interpretation, cleansed from inconsistencies and dubious in

terpretative principles. The EPR paradox is a critical touchstone for any such attempt.

I. ANALYSIS OF THE PARADOX

I begin by outlining the main arguments in the EPR debate, and attempt to isolate the main postulates used.1 My history of the debate is deliber

ately biased, in order to present my own perspective.

1. Einstein, Podolsky, and Rosen

Information about a physical system is normally summed up by at

tributing a state to that system. If system X is in pure state 4>, and ob

servable A is such that A(j> =

a<?>, then we say that X is in an eigenstate of A corresponding to eigenvalue a and write <?>

= \a}. In that case we

can predict with certainty that if A is measured (on X) the value found

will (would) be a.

The paper by EPR demonstrates that sometimes the value found in

a measurement of A can be predicted with certainty while X is not in

an eigenstate of A.

Synthese 29 (1974) 291-309. All Rights Reserved

Copyright ? 1974 by D. Reidel Publishing Company, Dordrecht-Holland



292 BAS G VAN FRAASSEN

The argument is as follows. Let X and Y be two systems (mutually and severally) isolated at t and at T, but interacting during the interval

(t, T). Suppose that at t, X and Y were in pure states (j) and \j/, respec

tively. Then we can calculate the states of the complex system X+ Y by means of two principles.

Composition. If X and Y are (mutually and severally) isolated, then

X+ Y is in state 9?t] if and only if X is in state 6 and Y in state rj.

This provides the state of X + Y at t as (f)?\?/ - the tensor product of the

two vectors, which belongs to the relevant product space. From this

initial state, the final state at T can be calculated via Schr?dinger's equa

tion, which I here state in abstract form.

Evolution. The state #, at time t, of an isolated system, evolves in ac

cordance with the equation <Pt+m=Um<Pt, where {Um} is a

certain one-parameter group of linear operators.

Hence if m = T-1, the state of X + Y at T is Um(^>?^) = 0.

Now let A be an observable pertaining to Y, with spectrum {a?} and

complete set {|ar>} of unit eigenvectors.2 Then there is for every vector

0 in the product space a unique set of vectors {</>,} such that 4> has an

'orthogonal expansion'

(1) * = Icl^?|fll>

for certain coefficients ct. (The set {<?>J may not be orthogonal.) Suppose now that we make an A measurement on Y; what happens? Here EPR

accept the following postulate:

Projection. If and O measurement is made on some system in state

0=YJdi\oiy, then the system undergoes a transition to a

state \oky with probability dl.

To apply this postulate to the complex system X + Y in state <P, it is

further assumed that measuring A on Y is exactly the same as measuring I?A on X+Y, where / is the identity operator. The observable I?A has eigenvectors 0?? 1^), hence the transition will now be to some state

<?>k?\ak>> with probability cl.



THE EINSTEIN-PODOLSKY-ROSEN PARADOX 293

What do we know then, if we have performed an ^-measurement on

Y at time T, and have registered value ak? Well, X and Y are mutually and severally isolated again; X+Y is in state <^?|ak>; hence by the

composition principle, Y must be in state \ak} and X in state </>k. The crucial point is of course that the measurement on Y gave us

information about X even though we had no physical access to X. To

highlight the strangeness of this, EPR ask us to consider incompatible observables A and B, and incompatible observables C and D, with eigen values ah bh ch dh respectively, in the circumstances that both the fol

lowing hold:

(2) $ = YJei\ciy?\aiy

(3) * = Iff,|?i>?|fc|>.

This is quite possible, for example if all the coefficients et are equal. But

then the above reasoning shows that after an ^-measurement on Y we

can predict with certainty the outcome of a C-measurement on X, and

alternatively, after a B-measurement on Y we could predict with cer

tainty the outcome of a D-measurement on X. And this cannot be because

X is in an eigenstate of both C and D, for these are incompatible observ

ables with no eigenstates in common.

Subsequent writers pointed out that the certain predictions in question were only conditional', for A and B cannot both be measured at T -

they are incompatible too. Anticipating this, EPR made some derogatory remarks about what may depend on the experimenter's whim. But the

conclusion is startling in any case: the conditional certainties in question constitute information that cannot be conveyed or summed up by the

attribution of states to X and Y separately. No single attribution of states

to X and Y at T can convey the information given by the attribution of

0 to (X+Y).

2. Schr?dinger

Schr?dinger (1935) began by drawing the moral with which I ended the

preceding section:

When two systems, of which we know the states by their respective representatives, enter

into temporary physical interaction due to known forces between them, and when after

a time of mutual influence the systems separate again, then they can no longer be described




in the same way as before, viz. by endowing each of them with a representative of its own.

I would not call that one but rather the characteristic trait of quantum mechanics

[p. 555].

But then he notes the 'sinister' corollary that by physical action on system

Y, the experimenter can steer system X into one type of state or other.

For consider the above Equations (2) and (3): by making an A-measure

ment, on Y, system X is steered into an eigenstate of C; if instead a B

measurement is made on Y, X is steered into an eigenstate of D.

Schr?dinger also demonstrated two further facts in this first paper. The first is that there will in the EPR case be an infinite set of pairs related as A to B or C to D. The second is that no essential changes occur if we consider measurement on Y at T and on X at a later time

T*, for then A will be similarly paired with a new calculable observable

B*.

In his second paper, Schr?dinger asks: how much control does the

experimenter on Y have over the state of XI And the answer is astonish

ing: by suitable choice of measurement on Y, he can in general steer X

into any state he likes, with nonvanishing probability. Schr?dinger shows this via a theorem on mixed states and a further postulate, the

reduction postulate.3 Since these are important, I shall outline them

here.

Mixed states were first introduced to represent ignorance. Suppose

{fa} is a set of states, and we know of a system only that it is in one of

these, and is in state fa with probability vvf. In that case we say that the

system has mixed state p = ? wtP \_fa~], where P[</>?] is the projection

along fa. The operator p is called a statistical operator. If the above set

{fa} happens to be a set of mutually orthogonal states, then they are

eigenvectors of p -

pfa =

wtfa - and the subspace S [p] spanned by the

proper eigenstates fa (i.e., such that w^O) is the image space of p. Once familiar, statistical operators may be used to represent any state;

the states which may equally be represented by a vector (?> or projection P [</>] are called pure. Because the first use of statistical operators was

to represent ignorance, to attribute p = Y? w?P[0?] is to say that the

system is in one of those pure states, but we do not know which. Let us

say that any pure state fa which appears in such an equation for p, with

w?#0, is possible relative to p. (Note that in this definition, I do not

require the states fa to be mutually orthogonal.) The theorem about




mixtures proved by Schr?dinger can then be stated in part as

(4) l/, =

S[p]

where Up is the set of pure states possible relative to p, and S [p] the

image space of p. The reduction postulate constitutes a second and prima facie indepen

dent introduction of mixtures. When X + Y is in state 4>, can we find a

state p for Y such that measurements of A on Y receive the same pre dictions whether we treat them as A-measurements on a system in state

p or as /(^-measurements on a system in state $? The answer is yes, there is a unique such state p, but it is in general not a pure state but a

mixture. The reduction postulate says that this p is the state of Y. I give it here in the special but useful form

Special Reduction. If X+ Y is in state 4> = ? et \c?y?\a?) then X is in

state Pi =? efP[}Ci}~\ and Y in state

/>2 = Ie?P[h>].

This does not get around the EPR paradox; it is still the case that the

attribution of # to X+Y yields conditional certainties which are not

conveyed by the attribution of px to X and p2 to Y. Hence anyone who

accepts the reduction postulate must say that the states of the parts do

not determine the state of the whole.

It has further been maintained that even if the mixed state is attributed

to a system by virtue of the reduction postulate, this attribution simply

represents ignorance of the pure state. I will call this the ignorance inter

pretation of mixtures. It is now usually stated in the following form:4

Ignorance. If a system is in mixed state p, then it is really in one of the

proper eigenstates of p.

It may be noted that this accords a privileged status to the orthogonal

decompositions p = Xw?/>[0i]- Nancy Cartwright has arguments to

suggest that this privileged status has no justification in physics. Since

it is a hotly debated topic (see van Fraassen, 1972, Sec. 12; Hooker, 1972,

pp. 102-106; Cartwright, 1972; Grossman, forthcoming), I here add only that the quote with which I opened this section suggests that Schr?dinger did not accept the ignorance interpretation (reading 'representative' as

'state' and noting his reliance on the reduction postulate).




3. Margenau

Independently of Schr?dinger, Margenau (1936) also noted the corollary about actio in distans to the EPR paradox. Apparently experiments on

Y after separation from X can determine, in the sense of causally influ

ence, the state of system X. But instead of taking this as an anomaly inherent in quantum mechanics, Margenau saw the EPR paradox, and

especially this corollary, as a powerful argument for rejecting the projec tion postulate.

Margenau has also given independent arguments against that postu late. The first argument is that projection is blatantly inconsistent with

the law expressed in Schr?dinger's equation and here stated abstractly as the evolution postulate. The 'consistency proofs' offered by von Neu

mann and Groenewold do not show there is no such inconsistency. In

fact they may be read either as showing a way to restore consistency by

restricting the scope of applicability of these postulates ('measurement interactions are sui generis') or as showing that predictive calculations

made using the projection postulate could have been made without it

(van Fraassen, 1972, pp. 333-335). The second argument Margenau gives is this: if projection were true, then a single measurement would divulge the state of a system; but the state is a compendium of much statistical

information. As a third argument we may list the above corollary of

miraculous actio in distans.

When the projection postulate is removed, we need another postulate to connect attributions of state with measurement results. But that can

be the familiar Born rules, which I state here in special and general form:

Born (a) If X is in state ]T ct |a?>, the probability that an ^-measure

ment on X will (would) yield value ak equals cl (if A non

degenerate; in general, equals ? {cj:aj =

ak}).

(b) If X is in state p, the expectation value for an A-measure

ment on X is Trace (pA).

There are at this point two problems confronting Margenau. The first is the original reason given by von Neumann for adopting

the projection postulate: an immediate repetition of a measurement will

yield the same value the second time. If however, we simply apply the

Born rule twice, whether assuming that the state of the measured system




changes or remains the same, we do not get that conclusion unless there

is a change in accordance with the projection postulate. Margenau argues that such repetition is usually not possible; but von Neumann (1955, pp.

212-214) explicitly discussed an experiment to illustrate the case.5 In an

experiment by Compton and Simons, light is scattered by electrons and

the scattered light and scattered electrons are intercepted and have their

energy and momentum measured. It was concluded from this experiment that the mechanical laws of collision hold. But von Neumann reformu

lates the conclusion as follows: if we assume that the laws of collision

are valid, the position and central line of the collision may be calculated

from the measurement of the path of either the light quantum or the

electron after the collision. It is an empirical fact that the two calcula

tions always agree. But the two measurements do not occur simulta

neously; the measurement apparatus may be arranged so that either

process may be observed first. So we have two measurements, M1 and M2, the second after the first; beforehand, their outcome is only statistically

determined, but after Ml5 the outcome of M2 may be inferred. From this,

plus the fact that M1 and M2 are in effect (i.e., via calculation) measure

ments of the same observable (say, a coordinate of the place of collision

or of the direction of the central line), von Neumann infers that, if an

observable is measured twice in succession, the second measurement 'is

constrained to give a result which agrees with that of the first.' And since

the outcome of the second measurement can be predicted with certainty, von Neumann infers that after the first measurement, the measured

system must be in an eigenstate of that observable.

Upon what slender support dogma may be founded! In the experi ment described, measurements are made directly on two objects (an electron and a photon) which have interacted and then separated again. The observables directly measured are ones which have become corre

lated by the interaction (as in the EPR thought experiment). And on the

basis of this, an inference is made about what would happen if a single measurement could be immediately repeated upon the same object !

The second problem confronting Margenau is this: what does it mean

to say that a measurement apparatus shows value akl The projection

postulate is naturally taken as supplying the answer: during the mea

surement, the apparatus also transits to an eigenstate of a corresponding observable (the 'indicator observable'). If we reject the projection postu




late, the most we can get is this: let X be the system on which we measure

observable B, and Y the apparatus with indicator observable D. At the

final time T, the complex system X+ Y is in correlated state <P = Ytgi \bt}

?\dty. Using either th,e projection postulate or the reduction postulate

plus ignorance interpretation we get: F transits to some pure state \dk}. If we took this course we could say: that is what it means for apparatus Y to show value dk. But Margenau rejects this. So what does it mean to

say that apparatus Y shows value dkl As I noted parenthetically, reduction plus ignorance generally gives

the same result as projection. Accordingly, Margenau rejected the ig norance interpretation of mixtures in later publications. To sum up,

Margenau removes what he calls 'the real difficulty inherent in Einstein

Podolsky-Rosen's conclusion,' but he leaves a serious open question in

the theory of measurement. If a measurement outcome is not to be de

scribed as a transit to a new pure state, how shall it be described?

4. Reisler

Following Schr?dinger, I have distinguished two features in the EPR

thought experiment, which we may call conditional certainty and physical

influence at a distance. It seems to me that there can be no doubt that

the second feature hinges on the acceptance of the projection postulate. Once the latter is seen as controversial, because of Margenau's argu ments, there is no compelling reason to believe that some actio in distans occurs.

In his dissertation written under Margenau, Reisler (1967, especially pp. 30-31) pointed out, in effect, that the conditional certainties do not

disappear when the projection postulate is discarded. Take the case of the biorthogonal expansion (Equation (2)), so that # =

?iei |?i>?|tf;> =

Y,ij efiij |fr/>?la?>- If we measure A on system Y, we might equivalently say that we have measured I? A on X+ Y. If we measure A on Y and

B on X, we have measured B?A on X-\-Y, yielding a value pair (bm; an). But the experimenter looking at the gauges need look no further when he has noted value an. For the probability that value bm^bn is zero: by the Born rule, the probability of finding pair (bm; an) is - barring degen eracy

- (el?mn), which equals el when m = n and equals zero otherwise.6

So even when the projection postulate is removed, conditional cer

tainties remain. Hence so does the fact that information about the com




posite system goes beyond any attribution of states to its components.

5. The Dilemma

In the course of this discussion I have noted a number of principles :

composition, evolution, reduction, Born, projection, and ignorance. The

last two are essentially interpretative postulates: they tell us how 'mea

surement outcome' in the Born rule should be understood, namely as a

transition to a new pure state. (As I pointed out above, the transition

required by the projection postulate will often be implied by reduction

plus ignorance.) Now we have a dilemma.

If we accept the projection postulate, the Born rule probabilities can

be understood as state-transition probabilities, and measurement reports as attributions of states. Hence the statement 'Observable A has value a'

is unambiguous; it always means that the system is in eigenstate of A

corresponding to value a.7 But the projection postulate also implies the

causal anomalies explained by Schr?dinger. If we deny the projection postulate (and also the ignorance interpre

tation, so as not to get the same result by another route), then the causal

anomalies disappear. But then we can no longer interpret measurement

reports as attributions of states. Indeed, we can then only say that, at

the end of a measurement, apparatus and system alike are in mixed

states; the statement 'the apparatus shows value a' has at this point no

interpretation. (It has been argued that we could keep ignorance if we

reject reduction. But then the same horn impales us, for then we cannot

attribute any state at all to the apparatus.) A rejection of the projection and ignorance postulates must therefore

be followed by another interpretation. And any such interpretation must

accommodate the mysterious conditional certainties exhibited by EPR.

II. THE MODAL INTERPRETATION

I now accept the postulates called composition, evolution, reduction, and Born, with no restrictions on their scope; reject projection and ig

norance; and develop an interpretation according to which the phe nomena are as if projection and ignorance were true. Because of space

limitations, I must refer to my earlier articles for reasons to adopt this

interpretation.




1. Models of Physical Situations

To recapitulate without argument: in the modal interpretation we dis

tinguish two kinds of statements - state attributions and value attribu

tions. The former are the kind dealt with in quantum logic, and have the

form

(m, E) is true (about system X) exactly if X is in a state p such

that (by the Born rule) the probability equals 1 that a mea

surement of observable m on X would (will) yield a value in

Borel set E - in symbols, P (E)= 1.

The second kind are not dealt with in quantum logic; they can be used

to symbolize observation reports, and have the form

(m, E} is true (about system X) exactly if observable m actu

ally has a value in Borel set E.

If X is in a state p such that Ppn(E) =

1, then both (m, E) and <m, ?> are

true, and I say that p makes these statements true in this case. But in

principle <m, ?> may be true even though it is not made true by the state

of the system. The difference between Copenhagen (or a Copenhagen variant of the

modal interpretation) and a more classically minded interpretation is

this: on the Copenhagen position, (m, E} is hardly ever true unless

(m, E) is true. The exception comes with mixed states, for if the measure

ment apparatus is, at the end of the measurement, in mixture p =

Z c?^[l^i)] where the |ft?> are 'indicator states', eigenstates of observ

able B, then it may actually be true that the pointer indicates value bk -

although this cannot be deduced from state p. In that case (B, {bk}} is true, although (B, {bk}) is not true. The exact principle8 is this:

Copenhagen. In each physical situation in which the system is in state

p there is a pure state fa in Up such that:

(m, E) is true exactly if p makes it true

<m, E} is true exactly if (?> makes it true.

To read this, note the definition of 'makes true' in the preceding para

graph, and of 'Up in Part I, Section 2. This Copenhagen principle spells out the famed 'transition from the possible to the actual'. The state p of

the system describes what is possibly the case about values of observ




ables; what is actual is only possible relative to the state and not deduc

ible from it. Second, note that many observables will not have 'sharp' values in a given situation, in that the least Borel set E such that P (E)

= 1

is often not a unit set. For example, we can deduce here that if <position,

{q}} is true, then {momentum, {/?}> cannot be true for any single value/?. The interpretation of the Born rules is now as follows. Suppose that

an ^-measurement is performed on X in initial state fa =

Yjci\ai)i by

apparatus Y. The end-state of X+ Y is ? ct ?a^?^} where the states

{\bi)} are the 'indicator states' of apparatus Y. By the reduction postulate, X is in mixed state px =? cfP[|af>] and Y in mixed state ? cf P[|ftf>]. All this we arrive at before considering the Born rule (I shall make this

precise in the next section). Note that when cl ^ 1, the statements (A, {ak}) and (B, {bk}) are not true about X and Y, respectively, at this end stage. I now interpret the Born rule as saying that nevertheless, at this final

time, given that this is the end stage of an ^-measurement, the prob

ability that {A, {ak}} is true equals cl, and likewise for <?, {bk}} about

Y. So one of the statements 'the apparatus shows value ak - which is

here represented by \B, \bk}y - is true, though none of the statements

'the apparatus must (with probability 1) be showing value ak is true. I

must emphasize that I am not interpreting the Born probabilities as

state-transition probabilities (projection postulate) nor as ignorance-of real-state probabilities (ignorance interpretation) but as ignorance-of actual-value probabilities (conditional on the appropriate measurement

setup). And I consider such ignorance-of-actual-value probability state

ments as testable in terms of relative frequencies of actual values ob

served.

2. What Is a Measurement?

It was called to my attention by Dr. Jon Dorling that the EPR paradox

may plague the above interpretation of the Born rule, for what if we say: in the EPR experiment, if Equations (2) and (3) (in Pt. I, Sec. 1) are both

true, then Y acts as a measuring instrument for X with respect to both

observables C and D. Hence we are apparently attributing probability

el to <C, cky and probability gk to <D, dk} ; but C and D are incompatible, so they cannot have sharp simultaneous values; hence all these prob abilities add up to more than 1.

The way out of this problem is to specify very precisely what a mea




s?rement is (which I failed to do in my earlier articles, in the manner re

quired). Consider an interaction between X and Y, between times t and

T, while these systems are (mutually and severally) isolated at the initial

and final time. Under what conditions is this interaction an ^-measure

ment performed on X by (or with) apparatus 7?

My answer is that Y must be an A-measuring apparatus, which is in

its groundstate at time t. This requires that I further define the italicized

terms. Well, Y is an A-measuring apparatus with a set of indicator states

{fa} and a groundstate \j/Q if the evolution operator UA, which governs the interaction of Y with any system X during any interval A

= (T?t),

satisfies the equation

(5) Mfli>?<M=k>?ifc

and in addition, the states fa are mutually orthogonal. From Equation (5) we can deduce that

(6) li<t> = YJci\aiy then

^(</>?^o)=Zc;k*i>?^

which therefore allows us to deduce the final state of X + Y at T, if we

are given the initial states <j> and \?/Q of X and Y at t.

Equations (5) and (6) will be familiar from any discussion of measure

ment. What is different is that I will not call all interactions that satisfy

(6) measurement processes, but insist that Y must be a measurement

apparatus. This means no more and no less than that (6) must hold

regardless of what the initial state fa o? X is, and not just in the instance

under inspection. Now I can establish the consistency of the interpreta tion of the Born rules. For suppose that Y is at the same time an A-mea

suring device with indicator states {t/^} and also a ^-measuring device

with indicator states {0f}, in each case with groundstate i//0. Then I shall

establish that A and B cannot be incompatible. For the supposition entails that

(7) if 0=2]c( !?,> = ? ?W then

UA(<t>?il/0)=Y,ci\ai>?[l'i =

Iidi\bi>?ei

By the reduction principle, when X + Y is at time T in state <P = UA (<?> ? ij/0)




then

(8) X is in state p = ?c?P[|a,>]=?d?P[!&,>].

Now let the vector fa be chosen such that all the values cf are positive and distinct. (That there is such a vector (?> is clear.) In that case, as von

Neumann (1955, p. 329) already pointed out, the orthogonal decomposi tion of p is unique. But this orthogonal decomposition is at the same

time Y, c?P[|a?>] and ? ^?^[l^?)]- Therefore the two sets of eigenvectors {\aty} and {!&*>} are the same. That does not mean that observables A

and B are the same, but it does mean that they are functionally related :

they are calculable as functions of the same maximal observable. And

hence compatible. I consider this the precise formulation of the adage that incompatible

observables (in the mathematical sense) cannot be simultaneously mea

sured. It also establishes that my interpretation of the Born rule does not

translate the (logically consistent) conditional probabilities in an EPR

situation into (logically inconsistent) absolute probabilities.

3. Representation of Two-Body Systems

This section is slightly more technical, to explain the details of the rep resentation. Consider first a single system X in state p. To say it is in

this state is to describe the situation only partly; it tells us which of the

statements (m, E) is true. To know in addition which statements <m, ?> are true, we want a second factor, X\ this will be a pure state in Up. So

the set of models of possible situations is

M(p)={<p,A>:Ae[/p}.

If- and only if- this situation is the end of a measurement, then the Born

rule specifies a probability measure on M(p). Suppose Y was an ^-mea

surement apparatus, and the measurement interaction was that between

X and Y in interval (t, T) with (j)?\?/0 as initial state and ? c? \ais)?\?/i as final state. Then at T, Y is in state p =

? cfP[}j/i'] and l/p =

S[p] is the

subspace spanned by the vectors i/^ such that cf #0. Then the Born rule

specifies probabilities which may be formalized as a measure P on M(?) such that

P{(p,?}:? is a multiple of i/^} = cf.




Since the cf sum to 1,

P{(p, >l> :A is not a multiple of \?ft for any i} = 0.

(Mutatis mutandis for the system X at the final stage of this measure

ment.) That there is a probability measure P with these properties is

easily seen. I regard the Born rule as totally conditional - it does not

apply except in the context of measurement - and as the only fulcrum

for the interpretation of probabilities in quantum mechanics.

Turning now to many-body systems, the symbolism needs to be ex

tended. In each statement we must indicate the system to which it refers, and so the notation '(m, E)' must be replaced by '(m, X, ?)'; similarly for

'<m, E}\ A model of a physical situation must specify a state and a

second factor for each system in that situation. And because of the non

classical EPR correlations, X + Y must be considered a third system distinct from X and Y. So we get a model that looks like this:

w = (p, *p, p*;k, kl9 k2}

where * p and p * are the states the reduction postulate assigns to X and to Y when X + Y is in state p.

In model vv, (m, X + Y, E) is true if p makes it so, and (m, X, E) is true

if * p makes it so. (Note that such a statement has meaning only if m is an observable defined on the appropriate Hilbert space.) A statement

<rn, X, E} is true exactly if kt makes it so - equivalently, if kt makes

(m, E) true. We must clearly insist that <*p, Ax> is a model of a one-body situation, with *p the state of Xand kx the second factor needed to specify actual values. So kt must be possible relative to *p; similarly, k must

be possible relative to p and k2 possible relative to p *. Should any further

requirement be imposed on models of two-body situations?

Yes, there is a further requirement: a value of A?I in X+ Y must be a value of A in X; similarly for I?B and Y. In my terminology:

If (A?I, X+Y,E) is true, so is {A, X, E}

If (I?B, X+Y,E} is true, so is (B, Y, ?>.

This is taken care of by noting that the reduction postulate allows us to

calculate (mixed) states *k and k* directly from k, and insisting that kx and k2 be possible relative to *k and A*, respectively. (All this is auto




matic if p itself is pure.) Then we reason as follows: if {A?I, X-\-Y,E} is true, then k makes (A?I, E) true; but then *k makes (A, E) true -

by the calculation used in the reduction postulate. Now this is the case only if every pure state which is possible relative to * k also makes (A, E) true

(for the proof J refer to van Fraassen, 1973, p. 98, Equation (6)). Hence

kt makes (A, E) true; and so {A, X, E} is true here. Mutatis mutandis

for k2. The simplest example is this: if k = \ak)?\bm) then kx and k2 ave

(scalar multiples of) \ak} and |bm>, respectively. So the models of the physically possible situations for state p of X+ Y

are

(9) M(p) =

{(p, *p,p*;k, ku A2>:AeS[p] and

k^S^X] and /l2eS[,l*]}.

Let us now look at the special case in which p happens to be pure state

$ = Yj c? la?>?|fr?>

- such as in the final state of the EPR experiment or

of a measurement interaction. The models in M(<P) are meant to rep resent the bare logical possibilities for the physical situation when 3> is

the state of X+Y. In that case k = & also; and one possible situation

would be

w=(&,*&, <*>*, <*>, \aky, \bkyy

when cl /0. The family M(<P) also contains models in which the factors

kx and k2 are not so neatly correlated. But we do not have to make a

special effort to rule them out, as I now show.

There are here two cases : first, <P is the end-state of X + Y in an A

measurement by Y on X; second, as in the EPR case, <P is being con

sidered itself as possible subject for an A?B measurement by appa ratus Z.

In the first case, the Born rules give us a probability assignment to the

statements {A, X, {ak}} and (B, X, {bk}}, which may be formalized in

the complex situation as follows :

P{<4>, *#, <P*; 0, k^g^^y-.g scalar and kteS[*<P]} =

cjt

P{(<P, *$,$*; 0, e\ak}, k2}:escalar and k2eS[$*~]} =

cl.

This does not imply that if B has value bk in Y, then A must have value

ak in X. So have we lost the conditional certainties of EPR? The answer

is no; the bare logical possibility of such a lack of correlation does not




mean that there is a nonzero probability of ever finding it.9 For every instance of the first case is also an instance of the second case; so read on.

In the second case, if an A?B measurement is performed then $ be

comes transformed (via the interaction and the reduction postulate) into

p'=Yj cfP ?\ai}? l^?)]- The Born rule now gives a probability assignment to the statements (A?B, X+ Y, akbk} of ? {cfn'.ambm

= akbk}

- which is

just cl if the eigenvalues at and bt were chosen nicely. Since the values

cf sum to 1, this leaves only zero probability for all other cases (i.e., out

comes that can only be written as akbm with fe/m). This assignment we

can formalize as

P{<p\ *p\ P'*; d(\aky?\bk}), kx, k2}:da scalar,

i1mSl*d{\a??\bk>)lX2in

S[d(\ak>?\bk})*]}=c2k.

But now, notice that *d(\ak}?\bky) = \ak} and d(\ak}?\bk))*

= \bk}. Hence this fixes kx and k2 exactly, namely as scalar multiples of \ak} and \bk},

respectively. Since, as I said, all remaining cases get probability zero, we

have here the conditional certainty that if <?, Y, {bk}} is true, then so is

{A, x, K}>.

4. Conclusion

Unlike most participants in the debate which EPR initiated, I have not

treated their thought experiment as a paradox to be solved or dissolved.

I have followed Margenau in holding that the conditional certainties ex

hibited by EPR are objective features of the physical situation, but that

the putative consequence of action at a distance must be taken as a major

argument against the projection postulate. That argument applies mutatis

mutandis to the combination of the reduction postulate and the ignorance

interpretation; and again I have followed Margenau by rejecting the latter

in favor of the former.

Instead of debating EPR's conclusion that quantum mechanics is

somehow incomplete, I have taken the conditional certainties they ex

hibit as a crucial touchstone for the interpretation of quantum mechanics.

Once the projection postulate and the ignorance interpretation are re

jected, the whole first formulation of the orthodox statistical interpreta tion is gone. I do not believe that it can be restored simply by due at




tention to frequencies in statistical ensembles.10 But something very much like that works, in my opinion, and that is the modal interpreta

tion.

According to the modal interpretation, there are two kinds of state

ments about physical systems. The first kind is about their state; these

are essentially statistical, and have predictive value (via the theory). The

second kind is about what is actually the case in the physical situation,

and they comprise especially reports of measurement outcomes. The

Born rules are interpreted as linking the two: attributions of state imply, via the Born rules, probabilities attaching to measurement outcomes

when measurements are actually performed. This differs from the original view only in that a measurement outcome report is not read as implying a state for the system, and hence not as having direct predictive value.

But the models of physical situations which the modal interpretation

provides, have actuality, possibility, and probability so interlaced that

the phenomena are 'as if the projection postulate and ignorance interpre tation were true'. The 'reduction of the wave packet' is interpreted as a

transition from the possible to the actual: possible values of observables

to actual values of observables, but not an 'acausal' transition to a new

state. It is for this reason that I take the modal interpretation to be a

new formulation of the orthodox statistical interpretation. And this new

formulation accommodates such phenomena as the nonclassical EPR

correlations in a consistent and nonanomalous fashion.* *

University of Toronto

NOTES

* The research for this article was supported by Canada Council Grants S72-0810 and

S73-0849.1 wish to acknowledge gratefully my debt to Professor H. Margenau, Yale Uni

versity; and also to Professor N. Cartwright, Stanford University, Dr. J. Dorling, Chelsea

College of Science and Technology, and Professor C. A. Hooker, University of Western

Ontario. 1

The mathematical background for the discussion can be found in van Fraassen (1972,

Appendix on tensor products). 2 I am using subscripts so as not to rule out degeneracy: a{

= a? does not imply

\a\} =

\ap. The set {\a?)} is an orthonormal base. 3

The reduction postulate is used and motivated by Schr?dinger (1936, p. 450), and then

stated explicitly in the summary on page 452. For the complete reduction postulate, and

the calculations of the states, see van Fraassen (1972, Appendix).



308 BAS C. VAN FRAASSEN

4 Stated in this way, the ignorance interpretation ignores the 'equal weights' problem, at the

cost of not assigning probabilities (see van Fraassen, 1972, Sec. 12). This means that

projection is no longer strictly deducible from reduction plus ignorance. 5

See also the careful analysis of the argument by Sneed (1966). 6 Not barring degeneracy, the probability is ? {ei?ij:bj

= bm and ?e?=?e?} in which the

conditional certainty will limit bn to some Borel set. 7

Broadly construed, this means Tr(pP)=l, where p is the state, possibly mixed, and

P the projection on the subspace spanned by the eigenvectors {\a?):ai =

a}. 8 This is what I called the 'Copenhagen variant' of the modal interpretation in

van Fraassen (1973). A more extreme 'Copenhagen position' is possible, using super

valuations; see van Fraassen (1972, Sec. 18). The basic idea of the modal interpretation I

seem to recognize in many writings; cf. Jauch (1968, p. 173) and Post (1971, pp. 279-280). 9 It would be quite easy to rule out the bare possibility : impose on the models the restric

tion that the so-called 'transition probability' from X to k1?k2 t>e nonzero (i.e., insist that

AA ?A2 not be orthogonal to k). This would yield conditional certainties about actual values

when not outcomes of measurements. Hence it would go beyond what the Copenhagen school would accept or assert. I have preferred to keep the interpretative principles

minimal. 10

See van Fraassen (1973, Appendix 'Ensemble Models of Mixtures'). This does not im

ply that I regard probability statements as anything but (theoretical) statements about

frequencies. 11A final note to explain the relations between my three essays on the modal interpretation. In van Fraassen (1971) I gave a preferential status to orthogonal decompositions of mix

tures. The arguments of Nancy Delaney Cartwright convinced me that this was a mis

take. Accordingly, I generalized my treatment in van Fraassen (1973); but there I did not

discuss probability assignments to measurement outcomes. I had meanwhile been con

vinced by Jon Dorling that if we give equal status as 'measurement interactions' to all

interactions whose evolution operator is of the requisite form, then the EPR paradox spells serious difficulty for the assignment of probabilities to measurement outcomes. The present

essay reintroduces probabilities, for the 'Copenhagen variant' of the modal interpretation which was developed in van Fraassen (1973), and in a way that avoids the difficulty pointed out by Dorling.

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logic and probability in quantum mechanics || the einstein-podolsky-rosen paradox

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