SJSU/TP-98-16 January 1998 8 9 9 1 n a J 7 1 v Yet another comment on 1 “Nonlocal character of quantum theory” 1 0 1 0 8 J. Finkelstein1 9 Department of Physics / h San Jos´e State University p - San Jos´e, CA 95192, U.S.A t n a Abstract u q There has been considerable discussion of the claim by Stapp [1] : that quantum theory is incompatible with locality. In this note I ana- v i lyze the meaning of some of the statements used in this discussion. X r a 1 Participating Guest, Lawrence Berkeley National Laboratory e-mail: [email protected] 1 Stapp has claimed to have proven that quantum theory is incompatible with relativistic causality [1]. This claim has been criticized by Unruh [2] and by Mermin [3, 4], and Stapp has replied to this criticism, in [5, 6, 7]. Stapp’s proof involves a statement (to be explained below) which I will denotebyS(L2). StappclaimstoshowthatS(L2)istrue,thatananalogous statementS(L1)isfalse,andthatthisdifferenceinthetruth-valuesofS(L2) and S(L1) constitutes a violation of a locality condition he calls LOC2. UnruhandMerminstatethattheyagreethatS(L2)istrue;thusthedispute betweenStappandhistwocriticswouldseemtoinvolvetherelationbetween S(L2)andS(L1),and,inparticular,themeaningandapplicabilityofLOC2. However, as I will point out in this note, the agreement on S(L2) is illusory; the version of S(L2) agreed to by Unruh and Mermin does not have the same meaning as does S(L2) as understood by Stapp. In this note I will discuss the meaning of S(L2). In ref. [1], Stapp considers two particles in the (entangled) Hardy state [8], on which measurements can be performed in two spacelike-separated regions called Right and Left. On the Right, measurement is made of either of two non-compatible properties called R1 and R2; similarly, on the Left either L1 or L2 is measured. The result of any given measurement is either + or −. Thestatement I call S(L2) is thestatement that, if L2is measured, a statement called S is true; in symbols S(L2) := [L2 ⇒ S] (1) (similarly, S(L1) := [L1 ⇒ S]), where S is defined by S := “If R2 is measured and yields the result +, then if R1 had been measured it would have yielded the result −”(2) Now, whatexactly doesS mean? Inthis(counterfactual) statement, one is describing an “actual” world, in which R2 is measured, and a “hypothet- ical” world in which, instead, R1 is measured; S then is the assertion that, in the hypothetical world, the result of R1 would necessarily be −. For this to make sense, itis necessary for thehypothetical world to bespecifiedmore fully. Roughly speaking, the idea is to specify that, except for the replace- ment of R2 by R1, the hypothetical world agrees closely with the actual world; S then is the assertion that, in every world that fits the specification, the result of R1 is −. And so to complete the definition of S, it is necessary to specify exactly in which ways the hypothetical world is required to agree with the actual world. 2 Here is one way to make the specification: to demand that the hypo- thetical world agree with the actual world on all events which are not in the invariant future of the measurement on the Right (that is, on all events either spacelike-separated from, or else on or within the backward lightcone from, that measurement). Let F denote this set of events (which are behind the Forward lightcone), and then define SF to be this version of S: SF := “If R2 is measured and yields the result +, then in every world which agrees with the actual world on F (in particular, which agrees with the actual Left result) and in which R1 rather than R2 is measured, the result of R1 is −.” (3) It is not important whether or not the definition (3) agrees with what most people would mean by the statement S; what is important is to have a clearly-defined definition for S. Of course, (3) might be motivated by phys- ical principles such as causality, and those principles might be relevant in proving that SF is true in some particular case; nevertheless, (3) is just a definition, and we are free to chose a different definition if we like. Here is an example of a different definition: we could specify that the hypothetical world must agree with the actual world on all events in the invariant past of the measurement on the Right. Let B denote that set of events (behind the Backward lightcone), and SB that version of S: SB := “If R2 is measured and yields the result +, then in every world which agrees with the actual world on B and in which R1 rather than R2 is measured, the result of R1 is −.” (4) SF and SB are not identical statements (truth of SB implies truth of SF, but not the other way around). No physical principle can tell us whether either (3) or (4) give the “correct” meaning for S, since definitions cannot be “correct”. I certainly do not mean to assert that SB captures the usual meaning of S, nor do I mean to suggest SB as a useful alternative to SF. I haveintroducedSB merelytoemphasizethat,tobeunambiguous,statement S must include a specification of the ways in which the hypothetical world is required to agree with the actual world, and to suggest that it is a good idea to spell out that specification as completely as possible. 3 So far, I have discussed the meaning of statement S; for that discussion, the quantum state of the two particles was completely irrelevant. To discuss the conditions under which S is true, it will be necessary to remember that the particles are in the Hardy state [8]. Let SF(L2) denote S(L2) in which S is understood as SF; that is, SF(L2) := [L2 ⇒ SF], (5) withanalogousdefinitionsforSB(L2),SF(L1),andSB(L1). Ithappensthat, for particles in the Hardy state, SF(L2) is true, and SB(L2), SF(L1), and SB(L1) are all false. • To see that SF(L2) is true, note that a quantum calculation2 shows, for the actual world of SF(L2) (in which L2 and R2 are measured and the result of R2 is +), that the result of L2 is necessarily +. Since SF constrains the hypothetical world to agree with the actual world on the Left, in the hypothetical world that result is also +. Then a quantum calculation for this hypothetical world (in which L2 and R1 are measured, and the result of L2 is +) requires the result of R1 to be −. Note that no locality assumption is needed here; the truth of SF(L2) follows simply from its definition and the quantum properties of the Hardy state. • To see that SB(L2) is false, note that, again, in the actual world the result of L2 is +. Now, however, we are free to consider a hypothetical world in which the result of L2 is −, and in such a world (L2 and R1 measured, result of L2 is −), a quantum calculation reveals a non-zero probability for the result of R1 to be +. Thus there is a hypothetical world consistent with the specification of SB(L2) in which the result of R1 is +; therefore SB(L2) is false. Now it may seem strange to allow a hypothetical world in which the result of L2 is −, while in the actual world that result is +; after all, how could the decision to measureR1rather than R2change theresulton theLeft? Itmay help to remember that a hypothetical world is, ipso facto, not the same as the actual world. If I choose to talk about a hypothetical world, then I get to choose what world to talk about; if I (perhaps foolishly) were to adopt SB as representing the meaning of S, then I could find an allowed hypothetical world in which the result of R1 is +. 2Thequantumpredictionsforthemeasurementswearediscussing, forparticlesinthe Hardy state, are presented in refs. [1, 2, 3]. 4 • For completeness, we can see that SF(L1) is false by noting that in theactual world (L1and R2measured,resultof R2is +), itis allowed that the result of L1 be −. Then in the hypothetical world (L1 and R1 measured, result of L1 is −), the result of R1 is allowed to be +. Finally, since SF(L1) is false, SB(L1) must be false also. Again, I am certainly not advocating adopting SB to represent the meaning of S. I am suggesting that, whatever we wish S to mean, it is useful to spell outthatmeaningexplicitly, byspecifyingthewaysinwhichthehypothetical world is required to agree with the actual world. Mermin, in ref. [3], denotes by (I) the statement here called S(L2). From his discussion of why he considers this statement to be true, it is clear that Mermin requires the hypothetical world to agree with the actual world on all events which, in some frame, occur earlier than the measurement on the Right. Since for every event in the set we have called F there is a frame in which that event does precede the measurement on the Right, Mermin’s understanding of S(L2) coincides with SF(L2). Although Unruh [2] does not give general criteria for the interpretation of counterfactuals, in his discussion of statement S(L2) (which appears in eqs. 12 and 13 of ref. [2]) he strongly emphasizes that this statement must be understood as requiring that the result of L2 be +, and that is the aspect of SF(L2) which is relevant for all the further discussion of locality. Stapp[1, 5, 6, 7], however, evidently interprets S(L2) differently. For him, the meaning of S (as opposed to the conditions required for its proof) involves only the Right region; he writes, for example [6] “This ‘meaning’ of statement S is strictly in terms of a relationship between the possibilities for the outcomes of alternative possible experiments both of which are confined to the region R(ight).” So statement S for Stapp is not the same as SF (and incidentally, notthesameasSB either); thusstatementS isusedtomeandifferentthings by Stapp and by his two critics. Of course, the issue is not which is the “correct” meaning of statement S—that is, after all, just a matter of definition.3 Theissue is whether quan- tum mechanics is incompatible with relativistic locality. But it is difficult to discuss that issue without unambiguous and agreed-upon definitions for the statements which are being discussed. 3Humpty Dumpty has remarked [9] “When I use a word it means just what I choose it to mean—neithermore norless.” 5 Acknowledgement: I have benefited from conversations with (but not necessarily agreement from) Henry Stapp. I would also like to acknowledge the hospitality of the Lawrence Berkeley National Laboratory. References [1] H. P. Stapp, “Nonlocal character of quantum theory,” Am. J. Phys. 65, 300 (1997). [2] W. Unruh, “Is Quantum Mechanics Non-Local?,” quant-ph/9710032 (1997). [3] N. D. Mermin, “Nonlocal character of quantum theory?,” quant- ph/9711052 (1997). [4] N. D. Mermin, “Nonlocality and Bohr’s reply to EPR,” quant- ph/9712003 (1997). [5] H. P. Stapp, “Mermin’s suggestion and the Nature of Bohr’s Action-at- a-Distance Influence,” quant-ph/9711060 (1997). [6] H. P. Stapp, quant-ph/9712036 (1997). [7] H. P. Stapp, “Reply to Unruh,” quant-ph/9712043 (1997). [8] L. Hardy, “Quantum mechanics, local realistic theories, and Lorentz invariant realistic theories,” Phys. Rev. Letters 68, 2981 (1992). [9] H. Dumpty, quoted in L. Carroll, Through the Looking Glass, and What Alice Found There, Macmillan and Co., London (1872). 6