Locality and measurements within the SR model for an objective 4 0 interpretation of quantum mechanics 0 2 n a Claudio Garola and Jaros law Pykacz J ∗ † 0 3 2 v One of the authors has recently propounded an SR (semantic 5 2 realism) model which shows, circumventing known no-go theo- 0 rems, that an objective (noncontextual, hence local) interpreta- 4 tion of quantum mechanics (QM) is possible. We consider here 0 3 compound physical systems and show why the proofs of nonlo- 0 cality of QM do not hold within the SR model, which is slightly / h simplified in this paper. We also discuss quantum measurement p - theory within this model, note that the objectification problem t n disappears since the measurement of any property simply reveals a its unknown value, and show that the projection postulate canbe u q considered as an approximate law, valid FAPP (for all practical : v purposes). Finally, we provide an intuitive picture that justifies i X someunusualfeaturesoftheSRmodelandproves itsconsistency. r a KEY WORDS: quantum mechanics, objectivity, realism, local- ity, quantum measurement, semantic realism. ∗Dipartimento di Fisica dell’Universit`a and Sezione INFN, 73100 Lecce, Italy; e-mail: [email protected] †Instytut Matematyki, Uniwersytet Gdan´ski, 80-952, Gdan´sk, Poland; e-mail: [email protected] 1 1. INTRODUCTION One of us has recently proposed an SR model which provides an interpreta- tion of quantum mechanics (QM) that is objective.(1) Intuitively, objectivity means here that any measurement of a physical property of an individual sample of a given physical system reveals a preexisting value of the measured property, that does not depend on the measurements that are carried out on the sample.1 The SR model is inspired by a series of more general papers aiming to supply anSR interpretation of QMthat is realistic ina semantic sense, inthe framework of an epistemological position called Semantic Realism (briefly, SR; see, e.g., Refs. 2-5): indeed, it shows how an SR interpretation can be consistently constructed. However, the SR model is presented in Ref. 1 by using only the standard language of QM, in order to make it understandable even to physicists that are not interested in the conceptual subtelties of the general theory. But the treatment in Ref. 1 does not deal explicitly with the special case of compound physical systems, hence neither the measurement problemnorthelocality/nonlocalityproblemareconsidered, even thoughthe locality of the interpretation of QM provided by the SR model is anticipated. Therefore, we intend to discuss briefly these topics in the present paper. Our analysis begins with some preliminaries. We discuss in Sec. 2 the concept of physical property from a logical viewpoint, stress that properties 1More rigorously, objectivity can be intended as a purely semantic notion, as follows. Any physical theory is stated by means of a generallanguage which contains a theoretical language LT andanobservativelanguage LO. Theformerconstitutestheformalapparatus of the theory and contains terms denoting theoretical entities (as probability amplitudes, electromagnetic fields, etc.). The latter is linked to LT by means of correspondence rules that provide an indirect and partial interpretation of LT on LO. Furthermore, LO is interpreted by means of assignment rules which make some symbols of LO correspond to macroscopic entities (as preparing and measuring devices, outcomes, etc.), so that the elementarysentences ofLO are verifiable, or testable, since they state verifiable properties of individual objects of the kind considered by the theory (note that this does not imply that also the molecular sentences of LO are testable). On the basis of these assigments, a truththeoryis(oftenimplicitly)adoptedthatdefinestruthvaluesforsomeorallsentences of LO. Then, we say that physical properties are objective in the given theory if the truth values of all elementary sentences of LO are defined independently of the actual determinationofthem thatmaybe done byanobserver(for instance,the correspondence theory of truth reachesthis goalby means of a set-theoreticalmodel; by the way,we note that this truth theory entails only a form of observative, or macroscopic, realism, even if it is compatible with more demanding forms of realism). 2 having different logical orders correspond to different kinds of experimental procedures, and note that the properties represented by projection operators in standard QM or in the SR model are first order properties only; we also point out that, even if every state S of the physical system can be associated with a first order property (the support F of S), recognizing an unknown S state requires experimental procedures corresponding to higher order prop- erties, which is relevant to the treatment of the measurement problem, as we show in Sec. 6. Furthermore, we briefly analyze in Sec. 3 some typical proofs of nonlocality of standard QM and individuate in them a common general scheme, notwithstanding their differences. Bearing in mind the above preliminaries, we deal with the locality prob- lem from the viewpoint of the SR model in Sec. 4. We provide firstly a slightly simplified version of the model, and then note that the objective in- terpretation of QM provided by it supplies an intuitive local picture of the physical world and avoids a number of paradoxes, since objectivity implies locality. But this entails that the arguments examined in Sec. 3 must fail to hold, otherwise one would get a contradiction. Thus, we dedicate the rest of Sec. 4 to show that the proofs of nonlocality in Sec. 3 are actually invalid within the SR model, so that no inconsistency occurs. As a byprod- uct of our analysis, we get that Bell’s inequalities do not provide a test for distinguishing local realistic theories from QM. We then come to quantum measurements and observe in Sec. 5 that the SR model avoids the main problem of standard quantum measurement the- ory, i.e., the objectification problem; we also note that measurements still play a nonclassical role according to the SR model, since choosing a specific measurement establishes which properties can be known and which remain unknown, but point out some relevant differences between this perspective and the standard QM viewpoint. Moreover, we show in Sec. 6 that the fur- ther problem of double (unitary/stochastic) evolution of quantum measure- ment theory disappears within the SR model, since stochastic evolution can be considered as an approximate law that is valid for all practical purposes; we also discuss some consequences of the projection postulate that illustrate further the differences existing between the interpretation of the measuring process according to the SR model and the standard interpretation. Finally, we provide in Sec. 7 an intuitive picture that justifies some relevant features of the SR model and proves its consistency by modifying the extended SR model(1) in which microscopic properties are introduced as theoretical entities. 3 2. PHYSICAL PROPERTIES, STATES AND SUPPORTS Consider the following sets of statements in thestandard languageof physics. (i) “The energy of the system falls in the interval [a,b]”. “The system has energy and momentum ~p at time t”. E (ii) “The energy of the system falls in the interval [a,b] with frequency f whenever the system is in the state S”. “If the system has energy , then its momentum is p~ with frequency f”. E (iii) “The energy of the system falls in the interval [a,b] with a frequency that is maximal in the state S”. “If the system has energy , then its momentum is p~ with a frequency E that is maximal whenever the system is in the state S”. All these statements express, in some sense, “physical properties” of a physical system. But these properties have not the same logical status, cor- respond to conceptually different experimental apparatuses, and a careful analysis of their differences is useful if one wants to discuss the objective interpretation of QM provided by the SR model in the case of compound physical systems. Therefore, let us preliminarily observe that the word sys- tem in the above statements actually means individual sample of a given physical system, or physical object according to the terminology introduced in the SR model (indeed the term physical system is commonly used in the standard language of physics for denoting both classes of physical objects and individual samples, leaving to the context the charge of making clear the specific meaning that is adopted). Then, let us note that the first statement in (i) assigns the property F=having energy that falls in the interval [a,b] to a physical object, while the second statement assigns the properties E=having energy at time t, E P=having momentum ~p at time t. The properties F, E, P are first order properties from a logical viewpoint, since they apply to individual samples, and each of them can be tested (in a given laboratory) by means of a single measurement performed by a suitable ideal dichotomic registering device having outcomes 0 and 1 (of course, E and P can be tested conjointly only if they are commeasurable). Let us come to the statements in (ii). These assign second order proper- ties to ensembles of physical objects. To be precise, in the first statement one considers the ensemble of objects that possess the property F and the ensemble of objects that are in the state S, and the second order property 4 regards the number of objects in their intersection, which must be such that its ratio with the number of the objects in S is f. Analogously, in the second statement one considers the ensemble of objects that possess the property E andtheensemble ofobjects thatpossess theproperty P,andthesecondorder property regards the number of objects in their intersection, which must by such that its ratio with the number of objects that possess the property E is f. The first of these properties can be tested by producing a given number of physical objects in the state S, performing measurements of the first order property F on its elements, counting the objects that have the property F, and then calculating a relative frequency. The second property can be tested by means of analogous procedures (which require measurements of first order properties onanumber ofobjects) ifE andParecommeasurable, while there is no procedure testing it in standard QM if E and P are not commeasurable. Finally, the statements in (iii) assign third order properties to sets of ensembles. The property in the first statement can be tested (in a given laboratory)by producing sets of ensembles, performing measurements of first orderpropertiesonallelementsofeachensemble, calculatingfrequencies, and finally comparing theobtained results. The property in thesecond statement requires analogous procedures, which may exist or not, depending on the commeasurability of E and P. It is now apparent that one could take into account further statements containing properties of still higher order. Our discussion however is suf- ficient to prove the main point here: properties of different logical orders appear in the common language of physics, and properties that are different when looked at from this logical viewpoint are also different from a phys- ical viewpoint. Of course, nothing prohibits that a first order property F be attributed to some or all elements of an ensemble of physical objects: but first order properties must be distinguished from higher order proper- ties, and, in particular, from correlation properties, which usually are second order properties that establish relations among first order properties (the ex- ample above shows that the measurement of a property of this kind requires the comparison of sets of results obtained by measuring first order proper- ties). We shall see that this distinction is relevant when dealing with the measurement problem in Sec. 6. From a mathematical viewpoint, only first order properties are repre- sented directly within standard QM. To be precise, let ( ( ), ) be the L H ≤ lattice of all orthogonal projection operators on the Hilbert space of a H physical system, and let be the set of all first order properties of the L 5 system. According to standard QM, every element of ( ( ), ) represents L H ≤ bijectively (in absence of superselection rules) an element of . For the sake L of brevity, we call any element of physical property, or simply property, in L the following, omitting the reference to the logical order. The set can be endowed with the partial order induced on it by the L mathematical order defined on ( ) (that we still denote by ), and the ≤ L H ≤ lattice( , )isusuallycalledthe lattice of properties ofthesystem. Itfollows L ≤ that every pure state S can be associated with a minimal property F S ∈ L that is often called the support of S in the literature (see, e.g., Ref. 6). To be precise, if S is represented in by the vector ϕ , F is the property S H | i represented by the one-dimensional projection operator P = ϕ ϕ , which ϕ | ih | obviously is such that P P for every P ( ) such that P ϕ 2= ϕ ≤ ∈ L H || | i || 1. It is then apparent that F can be characterized as the property that is S possessed by a physical object x with certainty (i.e., with probability 1) iff x is in the state S. Indeed, for every vector ϕ′ representing a pure state S′, | i one gets P ϕ′ 2= 1 iff ϕ′ = eiθ ϕ , hence iff S′ = S. ϕ || | i || | i | i The existence of a support for every pure state of a physical system is linked with the problem of distinguishing different pure states, or pure states from mixtures, in standard QM. Indeed, there is no way in this theory for recognizing experimentally the state S of a single physical object x whenever this state is not known (for the sake of brevity, we assume here that S is a pure state): even if one measures on x an observable that has S as an A eigenstate corresponding to a nondegenerate eigenvalue a, and gets just a (equivalently, if one tests the support F of S and gets that F is possessed S S by x), one cannot assert that the state of x was S before the measurement, since there are many states that could yield outcome a and yet are different from S (for instance, all pure states that are represented by vectors that are not orthogonalto the vector representing S). But if one accepts the definition of states as equivalence classes of preparing devices propounded by Ludwig (and incorporated within the SR model(1)) one can know whether a given preparing device π prepares physical objects in the state S (briefly, one can recognize S) by measuring mean values of suitable observables, which is ob- viously equivalent to testing second order properties. The simplest way of doing that is testing F on a huge ensemble of objects prepared by π by S means of an ideal dichotomic device r: indeed, one can reasonably assume that π belongs to the state S whenever r yields outcome 1 on all samples, that is, whenever F is possessed by every physical object x prepared by π S or, equivalently, the mean value of F is 1. In particular, if S is an entangled S 6 state of a compound physical system made up by two subsystems, this pro- cedure allows one to distinguish S from a mixture M corresponding to S via S biorthogonal decomposition (see, e.g., Ref. 7). Also this remark is relevant to the quantum theory of measurement (see Sec. 6). 3. NONLOCALITY WITHIN STANDARD QM The issue of nonlocality of QM was started by a famous paper by Einstein, Podolski and Rosen (EPR),(8) which however had different goals: indeed, it aimed to show that some reasonable assumptions, among which locality, imply that standard QM is not complete (in a very specific sense introduced bytheauthors),henceitcannotbeconsideredasafinaltheoryofmicroworld. Later on, the thought experiment proposed by EPR, regarding two physical systems that have interacted in the past, was reformulated by Bohm(9) and a number of further thought experiments inspired by it were suggested and used in order to point out the conflict between standard QM and locality. Hence, one briefly says that standard QM is a nonlocal theory. As anticipated in Sec. 1, we want to schematize some typical proofs of nonlocality in this section, in order to prepare the ground to our criticism in Sec. 4. For the sake of clearness, we proceed by steps. (1)The existing proofsofnonlocalityofQMcanbegroupedintwo classes (see, e.g., Ref 10). (i)The proofs showing that deterministic local theories are inconsistent with QM. (ii)The more general proofs showing that stochastic local theories (which include deterministic local theories) are inconsistent with QM. For the sake of brevity, we will only consider the proofs in (i). It is indeed rather easy to extend our analysis and criticism to the proofs in (ii). (2) We denote by QPL in the following a set of empirical quantum laws, which may be void (intuitively, a physical law is empirical if it can be directly checked, at least in principle, by means of suitable experiments, such as, for instance, the relations among compatible observables mentioned in the KS condition;(1) amoreprecisedistinctionbetween empiricalandtheoretical laws will be introduced in Sec. 4, (2)). We denote by LOC the assumption that QM is a local theory (in the standard EPR sense,2 that can be rephrased by 2“Since at the time of measurementthe two systems no longerinteract,no realchange can take place in the second system in consequence of anything that can be done to the first system”.(8) 7 saying that a measurement on one of many spatially separated subsystems of a compound physical system does not affect the properties of the other subsystems). Finally, we denote by R the following assumption. R. The values of all physical properties of any physical object are prede- termined for any measurement context. (3) Bearing in mind the definitions in (2), the general scheme of a typical proof of nonlocality is the following. Firstly, one proves that QPL and LOC and R (not QM), ⇒ or, equivalently, QM (not QPL) or (not LOC) or (not R). ⇒ Secondly, since QM QPL, one gets ⇒ QM (not LOC) or (not R). ⇒ Finally, one proves that QM and (not R) (not LOC), ⇒ so that one concludes QM (not LOC). ⇒ (4)Let usconsider someproofsofnonlocalityandshow thatthey actually follow the scheme in (3). Bell’s original proof.(11) Here, the Bohm variant of the EPR thought ex- periment is considered (which refers to a compound system made up by a pair of spin-1/2 particles formed somehow in the singlet spin state and mov- ing freely in opposite directions). Then, a Bell’s inequality concerning some expectation values (hence a physical law linking second order properties, see Sec. 2) is deduced by using assumptions LOC and R together with a perfect correlation law (PC: if the measurement on one of the particles gives the re- sult spin up along the u direction, then a measurement on the other particle gives the result spin down along the same direction), which is an empirical law linking first order properties and following from the general theoretical laws of QM. The deduction is based onthe fact that assumption R allows one to introduce hidden variables specifying the state of a physical system in a more complete way with respect to the quantum mechanical state. Then, the expectation values predicted by QM are substituted in the Bell’s inequality and found to violate it. The above procedure can besummarized by the implication PC and LOC and R (not QM), which matches the first step in the general scheme, with ⇒ PC representing QPL in this particular case. One thus obtains QM (not ⇒ 8 LOC) or (not R) and concludes that QM contradicts local realism. The last step in the scheme was not done explicitly by Bell and can be carried out by adopting, for instance, the proof that PC and LOC R (hence PC and ⇒ (not R) (not LOC)) propounded by Redhead.(12) ⇒ Weaddthatthesame paradigm, withPC asa specialcase ofQPL,occurs in different proofs, as Wigner’s(13) and Sakurai’s.(14) In these proofs, however, an inequality is deduced (still briefly called Bell’s inequality) that concerns probabilities rather than expectation values. Clauser et al.’s proof.(15) This proof introduces a generalized Bell’s in- equality, sometimes called BCHSH’s inequality, that concerns expectation values (hence it expresses a physical law linking second order properties, as Bell’s inequality). This inequality is compared with the predictions of QM, finding contradictions. BCHSH’s inequality is deduced by using LOC and R only, so that one proves that LOC and R (not QM), hence QPL is void in ⇒ this case. The rest of the proof can be carried out as in the general scheme. Greenberger et al.’s proof.(16) Here no inequality is introduced. A system of four correlated spin-1/2 particles is considered, and the authors use di- rectly a perfect correlation law PC (that generalizes the PC law mentioned 1 above), R and LOC3 in order to obtain a contradiction with another perfect correlation law PC , hence with QM. Thus, the authors prove that PC and 2 1 LOC and R (not QM), which matches the first step in the general scheme, ⇒ with PC representing QPL in this case. Again, the rest of the proof can be 1 carried out as in the general scheme. Mermin’s proof.(17) Also this proof does not introduce inequalities. The author takes into account a system of three different spin-1/2 particles, as- sumes a quantum physical law linking first order properties (the product of four suitably chosen dichotomic nonlocal observables is equal to -1) together with LOC and (implicitly) R, and shows that this law cannot be fulfilled to- gether with other similar laws following from QM. Thus, also Mermin proves an implication of the form QPL and LOC and R (not QM), from which ⇒ the argument against LOC can be carried out as in the general scheme. (5) The analysis in (4) shows that the scheme in (3) provides the general structure of the existing proofs of nonlocality. In this scheme, assumptions R and LOC play a crucial role. Let us therefore close our discussion by 3To be precise, the authors introduce, besides LOC, realism and completeness in the EPR sense. These assumptions are however equivalent, as far as the proof is concerned, to assumption R. 9 comparing R and LOC with the assumption of objectivity (briefly, O), which plays instead a crucial role in the proofs of contextuality of standard QM.(5) Let us notefirstly that assumption R expresses a minimalformof realism. This realism can be meant in a purely semantic sense, as objectivity (see Sec. 1), hence it is compatible with various forms of ontological realism (as the assumption about the existence of elements of reality in the EPR argument4) but does not imply them.(4) Yet, R is weaker than O. Indeed, O entailsthatthevaluesofphysicalpropertiesareindependent ofthemeasuring apparatuses (noncontextuality), while R may hold also in a contextual theory (as Bohm’s), since it requires only that the values of physical properties are not brought into being by the very act of measuring them.5 Let us note then that O also implies LOC, since it entails in particular that the properties of the subsystems of a compound physical system exist independently of any measurement. By putting this implication together with the implication O R, one gets O LOC and R. However, the ⇒ ⇒ converse implication does not hold, since R and LOC are compatible with the existence of measurements that do not influence each other at a distance but influence locally the values of the properties that are measured. Thus, we conclude that R and LOC are globally weaker than O. 4. RECOVERING LOCALITY WITHIN THE SR MODEL It iswell known that nonlocality ofstandardQMraises a number ofproblems and paradoxes. However, it has been proven in several papers (see, e.g., Refs. 4 and 18-20) that the general SR interpretation of QM invalidates some typical proofs of nonlocality. Basing on our analysis in Sec. 3, we want to attain in this section a similar result within the framework of the SR model, which has the substantial advantage of avoiding a number of logical and epistemological notions, making things clear within the standard language of QM. To this end, we use throughout in the following the definitions and concepts introduced in Ref. 1. 4“If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to 1) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity”.(8) 5In order to avoid misunderstandings, we note explicitly that assumption R coincides with assumption R in Ref. 4 and not with assumption R in Ref. 5, which is instead the assumption of objectivity. 10