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Analysis of non locality proofs in QuantumMechanicsTo cite this article Giuseppe Nisticograve 2012 J Phys Conf Ser 343 012088

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Analysis of non locality proofs in Quantum

Mechanics

Giuseppe NisticoDipartimento di Matematica Universita della Calabria - ItalyandINFN ndash gruppo collegato di Cosenza Italy

E-mail gnisticounicalit

Abstract Two kinds of non-locality theorems in Quantum Mechanics are taken into accountthe theorems based on the criterion of reality and the quite different theorem proposed byStapp In the present work the analyses of the theorem due to Greenberger Horne Shimonyand Zeilinger based on the criterion of reality and of Stapprsquos argument are shown The resultsof these analyses show that the alleged violations of locality cannot be considered definitive

1 IntroductionThe task accomplished by Quantum Mechanics as an empirical theory is to establishwhich relationships occur in Nature between physical events ndash including the occurrences ofmeasurementsrsquo outcomes ndash if the physical system is assigned a given state vector | ψ〉 Allexperimental observations so far performed have confirmed the quantum theoretical predictionswhich per se entail no violation of the locality principle we can express as follows

(L) Principle of Locality Let R1 and R2 be two space-time regions which are separated space-like The reality in R2 is unaffected by operations performed in R1

Conflicts between Quantum Mechanics and locality arise only if further conditions which donot belong to the genuine set of quantum postulates are required to hold

In the ldquoclassicalrdquo non-locality theorems [1]-[3][5]-[7] these further conditions bring back tothe criterion of reality introduced by Einstein Podolsky and Rosen (EPR) [4]

(R) Criterion of Reality If without in any way disturbing a system we can predict withcertainty the value of a physical quantity then there exists an element of physical realitycorresponding to this physical quantity

EPR argued that under certain circumstances more non-commuting observables must havesimultaneous physical reality as a consequence of (L) and (R) without being all measuredwhile Quantum Mechanics is unable to describe such a reality This lack prompted to seek fora theory more complete than Standard Quantum Mechanics able to ascribe reality to theseunmeasured observables But Bell [1] first and other authors later [2][3] proved that such alocal realistic theory cannot exist More precisely they found that contradictions arise just inthe attempt of assigning values to non measured observables in agreement with (L) (R) andwith Quantum Mechanics Since the empirical validity of Quantum Mechanics cannot be denied

7th International Conference on Quantum Theory and Symmetries (QTS7) IOP PublishingJournal of Physics Conference Series 343 (2012) 012088 doi1010881742-65963431012088

Published under licence by IOP Publishing Ltd 1

these contradictions imply a violation of the condition (L)and(R) ie locality joined the criterionof reality

A different approach leading to the need of faster than light transfer of information wasfollowed by Stapp He pointed out that the alleged demonstrations of Bell and followers abovecited suffer a serious shortcoming they ldquorest explicitly or implicitly on the local-hidden-variableassumption that the values of the pertinent observables exist whether they are measured or notThat assumption conflicts with the orthodox quantum philosophyrdquo [8]

Then he developed [9] a non-locality proof which requires neither hidden variable hypothesisnor criteria of reality According to this proof locality is not consistent with the predictions ofQuantum Mechanics about Hardyrsquos physical setting [3] if the following further assumptions areadded to the standard quantum postulates

- one assumption asserts that once a measurement outcome has actually occurred no actionin a space-like separated future region can change its value

- the other assumption establishes that given a concrete specimen of the physical system thechoice of what observable to measure among the possible alternatives is free

The strategy pursued by this different non-locality theorem is to prove that the validity of aspecific statement (SR) having the status of a physical law within Stapprsquos approach whichconcerns with the outcomes of measurements confined in a space-time region Rβ depends uponwhat it is freely chosen to do in a space-time region Rα separated space-like from Rβ

In the present work we show that the violation of locality which results from the two kindsof theorems considered here is not a definitive conclusion

In the case of the theorems based on the criterion of reality [1]-[3] a recent analysis [10]highlighted that if the criterion of reality is interpreted according to its strict meaning thentheir proofs fail Instead their proofs are valid if a wide interpretation of the criterion of realityis assumed to hold Therefore these non-locality theorems can be interpreted as argumentsagainst the wide interpretation and supporting the strict one rather than as localityrsquos violations

As stated in [10] the methods therein used for the analysis of the lsquoclassicalrsquo non-localitytheorems which restores locality to Quantum Mechanics become ineffective with respect toStapprsquos theorem because of the profound difference between the proof strategies In this articlewe develop a methodology allowing for an analysis of Stapprsquos proof on a logical ground Theresults of our analysis show that a logical pitfall affects the proof thus the conclusion thatlocality is violated is not even reached by Stapprsquos argument

Section 2 of the present article introduces the basic theoretical concepts which enter thetheorems at issue

In section 3 we analyze the impact of the two different interpretations of the criterion ofreality (R) on the proof of a non-locality theorem based on such a criterion namely the theoremof Greenberger Horne Shimony and Zeilinger (GHSZ) [2] In so doing first the differentconsequences of the strict and of the wide interpretation are identified in subsection 31 Insubsection 32 it is shown that if the strict interpretation is assumed then the proof of GHSZis not successful

In section 4 the logical structure of Stapprsquos non-locality argument is described in detail Insection 5 we endow Stapprsquos new assumptions (FC) and (NBITI) with formal content Thisis necessary in order that a logico-mathematical analysis of the proofs of Property 1 and ofProperty 2 essential for the validity of the theorem can be performed The analysis of theproof of Property 1 shown in subsection 51 proves that it is correct But since the proof ofProperty 2 turns out of be not valid as we show in subsection 52 we conclude that accordingto the present analysis the non-locality theorem at issue is not valid


2

The conclusive section 6 is devoted to relate the present work to other disproofs present inthe literature

2 Basic FormalismGiven a quantum state vector |ψ〉 of the Hilbert space H which describes the physical systemlet S(|ψ〉) be a support of ψ ie a concrete set of specimens of the physical systems whosequantum state is represented by |ψ〉 Let A be any two-value observable ie an observablehaving only two possible values denoted by minus1 and +1 and hence represented by a self-adjointoperator A with purely discrete spectrum σ(A) = minus1+1 Fixed any support S(|ψ〉) everytwo-value observable A identifies the following subsets S(|ψ〉)- the set A of the specimens in S(|ψ〉) which actually undergo a measurement of A- the set A+ of the specimens of A for which the outcome +1 of A has been obtained- the set Aminus of the specimens of A for which the outcome of A is minus1

On the basis of the meaning of these concepts we can assume that the following statements hold(see [10] p1268)

(2i) If A is a two-value observable then for all |ψ〉 a support S(|ψ〉) exists such that A 6= empty(2ii) A+ capAminus = empty and A+ cupAminus = A(2iii) If 〈ψ|Aψ〉 6= minus1 then S(|ψ〉) exists such that A+ 6= empty and

if 〈ψ|Aψ〉 6= +1 then S(|ψ〉) exists such that Aminus 6= emptyAccording to standard Quantum Theory two observables A and B can be measured together ifand only if [A B] = 0 therefore also the following statements hold for every pair of two-valueobservables A B

(2iv) [A B] 6= 0 implies A capB = empty for all S(|ψ〉)(2v) [A B] = 0 implies forall|ψ〉 existS(|ψ〉) such that A capB 6= emptyGiven a pair AB of two-value observables such that [A B] = 0 we say that the correlationA rarr B holds in the quantum state |ψ〉 if whenever both A and B are actually measured ie ifx isin A capB then x isin A+ implies x isin B+ so we have the following definition

(3i) A rarr B if [A B] = 0 and x isin A+ implies x isin B+ whenever x isin A capB

This correlation admits the following characterization [11]

(3ii) A rarr B iff1 + A

21 + B

2ψ =

1 + A

2ψ

Two observables A and B are separated written A B if their respective measurements requireoperations confined in space-like separated regions Rα and Rβ

3 ldquoClassicalrdquo non locality theoremsIn this section we show how the non-locality theorems based on the criterion of reality fail ifthe criterion of reality (R) is interpreted according to its strict meaning In so doing we limitourselves to GHSZ theorem because the ldquodisproofsrdquo [10] for the other theorems [1][3] exploitthe same ideas and methods

In subsection 31 we deduce the implications which follow from the criterion of realityinterpreted according to its strict meaning Moreover it is shown that stronger implicationslike those required by the theorems of Bell and followers can be deduced if a wide interpretationof the criterion is adopted

In subsection 32 we show how the proof of GHSZ non-locality theorem cannot be successfullycarried out if the strict interpretation of (R) instead of the wider one is assumed to hold


3

31 Strict and Wide interpretation of EPRrsquos criterionWe explain the two different interpretations of the criterion of reality by looking at the physicalsituation considered by EPR in [4] where they consider a system made up of two separatedand non interacting sub-systems I and II One of two non commuting observables A and B canbe measured on system I with non-degenerate eigenvalues an bn and respective eigenvectorsψn ϕn Similarly sub-system II possesses two non commuting observables P and Q with non-degenerate eigenvalues pk qk and respective eigenvectors uk vk The quantum state of the entiresystem I+II satisfies Ψ =

sumn ψnotimes un =

sumk ϕk otimes vk so that according to Quantum Mechanics

the following perfect correlations occur if we actually measure A (resp B) on I obtaining theoutcome an (resp bn) then the outcome of an actual measurement of P (resp Q) on II is pn

(resp qn) ldquoThus by measuring either A or B we are in a position to predict with certaintyand without in any way disturbing the second system either the value of the quantity P []or the value of the quantity Q []rdquo Now since A and B are non commuting they cannot bemeasured together therefore the strict application of the criterion (R) leads to the followinginterpretation

Strict Interpretation Reality can be ascribed either to P or to Q according to which observableeither A or B is actually measured and whose outcome would allow for the prediction

Instead EPRrsquos attitude was different ldquoOn the other hand since at the time of measurementthe two systems no longer interact [] we arrived at the conclusion that two physical quantities[P and Q] with non-commuting operators have simultaneous realityrdquo This means that in orderto attain the simultaneous reality of P and Q EPR interpreted the criterion of reality as follows

Wide Interpretation For ascribing reality to P (or Q) it is sufficient the ldquopossibilityrdquo ofperforming the measurement of A (or B) whose outcome would allow for the prediction withcertainty of the outcome of a measurement of P (or Q)

In order to express the two different interpretations within the theoretical apparatus theformalism should be able to describe the reality besides of the results of actually performedmeasurements also of the ldquoelements of realityrdquo stemming from (R) hence given |ψ〉 and fixedany support S(|ψ〉) we introduce the set A of the specimens in S(|ψ〉) which objectively possessa value of the observable A without being measured by A+ (resp Aminus) we denote the set ofspecimens of A which possess the objective value +1 (resp minus1) of A hence we can assumethat A+ cap Aminus = empty and A+ cup Aminus = A hold Then we define A = A cup A A+ = A+ cup A+Aminus = Aminus cup Aminus Of course the ldquosizerdquo of A depends in general on which interpretation of(R) the strict or the wide one is adopted Once defined the mappings a A rarr 1minus1 anda A rarr 1minus1 by

a(x) =

1 if x isin A+

minus1 if x isin Aminus and a(x) =

1 if x isin A+

minus1 if x isin Aminus

the correlation A rarr B can be equivalently expressed in terms of the mapping a

A rarr B if (a(x) + 1)(b(x)minus 1) = 0 for all x isin A capB

Now we can infer the implications of the strict interpretation of (R) we express as formalstatements

Let us suppose that A B holds and that A is measured on x isin A obtaining a(x) = 1ie x isin A+ If the correlation A rarr B also holds then the prediction of the outcome 1 can beconsidered valid for a measurement of B on the same specimen Now by (L) the act of actuallyperforming the measurement of A does not affect the reality in Rβ hence the criterion (R) couldbe applied to conclude that x isin B and b(x) = 1

if A B and A rarr B then x isin A+ rArr x isin B+


4

It is evident that this implication simply follows from the strict interpretation of the criterion(R) it can be more formally stated as follows

(sR) if A B and A rarr B we can predict with certainty the value of an eventual measurementof B and ascribe reality to it once a measurement of A with concrete outcome a(x) = 1 isperformed If x isin A+ no prediction about B is allowed by (R) and (L)

Hence according to (sR) A B and A rarr B imply A+ sube B+ sube B and the correlation(a(x) = 1) rArr (b(x) = 1) besides holding for all x isin A cap B also holds for all x isin A+Analagously if an actual measurement of B yields the outcome minus1 ie if x isin Bminus then thestrict interpretation of (R) leads us to infer that x isin A and a(x) = minus1 Therefore it follows thatBminus sube Aminus sube A and that the correlation (a(x) = 1) rArr (b(x) = 1) also holds for every x isin BminusHence the correlation extends to A+ cupBminus Thus from (R) (L) and Quantum Mechanics weinfer the following statement

(4i) Extension of quantum correlations Let A and B be space-like separated 2-valueobservables If A rarr B then

(a(x) + 1)(b(x)minus 1) = 0 forallx isin (A+ cupBminus) cup (A capB)

The quantum correlation A harr B ie A rarr B and B rarr A in the state ψ means that thecorrelation (a(x) = 1) hArr (b(x) = 1) holds for all x isin AcapB for all S(|ψ〉) In this case from (4i)we can deduce that (a(x) = 1) hArr (b(x) = 1) holds for all x isin (A+cupBminus)cup(B+cupAminus)cup(AcapB) =A cup B for all S(|ψ〉) Hence the strict interpretation of (R) also entails the followingimplications

A B A harr B imply A cupB sube A cap B ie a(x) = b(x) forallx isin A cupB forallS(|ψ〉) (4ii)

The wide interpretation of criterion (R) allows for larger extensions Indeed it leads us toinfer the following wider extensions of quantum correlations

If A B and A rarr B then A+ sube B+ and Bminus sube Aminus forallS(|ψ〉) (5i)

If A B and A harr B then A+ = B+ Bminus = Aminus and A = S(|ψ〉) forallS(|ψ〉) (5ii)

32 GHSZ theorem does not work with the strict interpretationIn this subsection we show how strong statements (5) implied by the wide interpretation play adecisive role in the non-locality theorem of GHSZ But we show also that if we assume the strictinterpretation so that only the weaker statements (4) can be considered valid then GHSZ prooffails

GHSZ theorem makes use of seven two-value observables of a particular quantum systemdivided into four classes

ωA = Aα Aβ ωB = B ωC = Cα Cβ ωD = Dα Dβ

These observables have been singled out by GHSZ in such a way that

(6i) two observables in two different classes commute and are separated from each other

(6ii) [Aα Aβ] 6= 0 [Cα Cβ] 6= 0 [Dα Dβ] 6= 0

In general provided that [A B] = 0 by A middot B we denote the observable represented by theoperator AB according to Quantum Theory the product of the simultaneous outcomes of A


5

and B is the outcome of A middotB The state vectors |ψ〉 is chosen so that the following correlationsbetween actually measured outcomes hold according to Quantum Mechanics

i) aα(x)b(x) = minuscα(x)dα(x) forallx isin (Aα capB) cap (Cα capDα) equiv Xii) aβ(y)b(y) = minuscβ(y)dα(y) forally isin (Aβ capB) cap (Cβ capDα) equiv Yiii) aβ(z)b(z) = minuscα(z)dβ(z) forallz isin (Aβ capB) cap (Cα capDβ) equiv Ziv) aα(t)b(t) = cβ(t)dβ(t) forallt isin (Aα capB) cap (Cβ capDβ) equiv T

(7)

Equations (7i) (7ii) (7iii) (7iv) express the perfect quantum correlations Aα middotB harr minusCα middotDαAβ middotB harr minusCβ middotDα Aβ middotB harr minusCα middotDβ Aα middotB harr Cβ middotDβ respectively

According to the wide interpretation (5ii) holds and therefore correlations (7) can beextended to the following correlations between objective values

i) aα(x)b(x) = minuscα(x)dα(x)ii) aβ(x)b(x) = minuscβ(x)dα(x)iii) aβ(x)b(x) = minuscα(x)dβ(x)iv) aα(x)b(x) = cβ(x)dβ(x)

forallx isin S(|ψ〉) (8)

The contradiction proved by GHSZ lies just in (8) Indeed given any x isin S(|ψ〉) 6= empty from(8i) and (8iv) we get

cα(x)dα(x) = minuscβ(x)dβ(x) (9)

From (8ii) and (8iii) the equality cα(x)dβ(x) = cβ(x)dα(x) follows which is equivalent to

cα(x)dα(x) = cβ(x)dβ(x) (10)

which contradicts (9)

Now we prove that this GHSZ proof of inconsistency does not work if we replace theimplications (5) by the weaker (4) allowed by the strict interpretation The extension ofcorrelations (7) implied by (4ii) is the following

i) aα(x)b(x) = minuscα(x)dα(x) forallx isin (Aα capB) cup (Cα capDα) equiv X

ii) aβ(y)b(y) = minuscβ(y)dα(y) forally isin (Aβ capB) cup (Cβ capDα) equiv Y

iii) aβ(z)b(z) = minuscα(z)dβ(z) forallz isin (Aβ capB) cup (Cα capDβ) equiv Z

iv) aα(t)b(t) = cβ(t)dβ(t) forallt isin (Aα capB) cup (Cβ capDβ) equiv T

(11)

In order that the GHSZ argument ndash which leads to the contradiction from (8) to (10) through(9)ndash can be successfully repeated starting from (11) the first step requires that (11i) and (11iv)should hold for the same specimen x0 therefore the condition XcapT 6= empty should hold the secondstep requires that also (11ii) and (11iii) should hold for such a specimen x0 Thus the condition

X cap Y cap Z cap T 6= empty (12)

should be satisfied Now from (6ii) and (2i) we derive

empty = (Aα capB) cap (Aβ capB) = (Cα capDα) cap (Cβ capDα) =

= (Cα capDα) cap (Cα capDβ) = (Cα capDα) cap (Cβ capDβ) = (13)

= (Cβ capDα) cap (Cα capDβ) = (Cβ capDα) cap (Cβ capDβ) =

= (Cα capDβ) cap (Cβ capDβ)

By making use of (11) and (13) we deduce X cap Y cap Z cap T = empty which refutes condition (12)necessary to prove the inconsistency Thus GHSZ proof fail if the strict interpretation replacesthe wide one


6

4 A different non locality theoremIn this section we formulate in detail the argument proposed by Stapp to show that Quantummechanics violates locality without making use of hidden variable hypotheses or criteria of reality

Let us first establish the three hypotheses of Stapprsquos theorem

(FC) Free Choices ldquoThis premise asserts that the choice made in each region as to whichexperiment will be performed in that region can be treated as a localized free variablerdquo[9]

(NBITI) No backward in time influence ldquoThis premise asserts that experimental outcomes thathave already occurred in an earlier region [] can be considered fixed and settled independentlyof which experiment will be chosen and performed later in a region spacelike separated from thefirstrdquo[9]

The third premise of Stapprsquos theorem affirms the existence as established by Hardy[3] offour two-value observables A(1) A(2) B(1) B(2) and of a particular state vector |ψ〉 for a certainphysical system which satisfy the following conditions

(hi) A(1) A(2) are confined in a region Rα separated space-like from the region Rβ wherein theobservables B(1) and B(2) are confined with Rα lying in time earlier than Rβ Hence inparticular A(j) B(k) j k isin 1 2

(hii) [A(1) A(2)] 6= 0 [B(1) B(2)] 6= 0 minus1 6= 〈ψ|A(j)ψ〉 6= +1 minus1 6= 〈ψ|B(j)ψ〉 6= +1

(hiii) [A(j) B(k)] = 0 j k isin 1 2 and in the state vector |ψ〉 the following chain of correlationsholds

a) A(1) rarr B(1) b) B(1) rarr A(2) c) A(2) rarr B(2)

(hiv) S(|ψ〉) and x0 isin S(|ψ〉) exist such that x0 isin A(1)+ capB(2)

minus

In fact this last condition is implied from the following non-equality satisfied by Hardyrsquos settinglang

ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)

Since the lhs is nothing else but the quantum probability that a simultaneous measurement ofA(1) and B(2) yields respective outcomes +1 and minus1 the non-equality states that the correlationA(1) rarr B(2) does not hold Therefore by (3i) it implies (hiv)

The logical mechanism of the non-locality proof at issue is based on the following pivotalstatement

(SR) ldquoIf [B(1))] is performed and gives outcome [+1] then if instead [B(2)] had been performedthe outcome would have been [+1]rdquo[9]

By leaving out for the time being the question of its validity we have to recognize followingStapp that (SR) has the status of a physical law about outcomes of measurements completelyperformable within region Rβ Then Stapp introduces the following statements

Property 1 If a measurement of A(2) is performed in region Rα then (SR) is validIn formula

x isin A(2) rArr (SR) holds for this x

Property 2 If a measurement of A(1) is performed in region Rα then (SR) is not validIn formula



7

If both these properties actually followed from the premises (FC) (NBITI) (hi-iv) then thevalidity of statement (SR) would depend on what is decided to do in region Rα separatedspace-like from Rβ hence a violation of the following locality principle would happen

ldquoThe free choice made in one region as to which measurement will be performed there haswithin the theory no influence in a second region that is spacelike separated from the firstrdquo [9]

In fact Stapp gives his own proofs [9] that both property 1 and property 2 do hold Thus weshould conclude that the above locality principle is violated if the three premises hold

5 Logical analysisIn this section we shall examine from a mere logical point of view the proofs of property 1 andproperty 2 as drawn by Stapp Let us begin by considering property 1

Property 1 x isin A(2) implies (SR) holds for this x

Stapprsquos Proof ldquoThe concept of lsquoinsteadrsquo [in (SR)] is given a unambiguous meaning by thecombination of the premises of lsquofreersquo choice and lsquono backward in time influencersquo the choicebetween [B(2)] and [B(1)] is to be treated within the theory as a free variable and switchingbetween [B(2)] and [B(1)] is required to leave any outcome in the earlier region [Rα] undisturbedBut the statements [(hiiia) and (hiiib)] can be joined in tandem to give the result (SR)rdquo [9]

We see that the steps of this proof are carried out by appealing to their intuitiveness ratherthan by means of the usual logico-mathematical methods so that in this form the proof unfits foran analysis on a logical ground In particular the possibility of such an analysis would requirethat the ldquounambiguous meaning of the concept of lsquoinsteadrsquo rdquo be endowed with a mathematicalcounterpart within the theoretical apparatus in order to make explicit its role and formallyverifiable the proof

We provide such a mathematical counterpart by means of a precise implication which can beinferred from the premises (FC) and (NBITI) for two separated observables A and B respectivelyconfined in space-like separated regions Rα and Rβ with Rα lying in time earlier than Rβ suchthat the empirical implication A rarr B holds in the state |ψ〉

Given any concrete specimen x isin S(|ψ〉) the validity of condition (FC) makes sensible thequestion

ldquowhat would be the outcome of a measurement of Brdquo

also in the case that B is not measured on that particular specimen x independently of whichif any observable is measured in region Rα This meaningfulness forces the introduction of twofurther extensions IB+ and IBminus in S(|ψ〉) of any two-value observable B confined in Rβ

The extension IB+ (resp IBminus) is defined to be the set of the specimens x isin S(|ψ〉) such thatif B had been measured even instead of an actually measured observable C in Rβ thenoutcome +1 (resp minus1) would have occurred

In general a prediction of which specimens belong to IB+ or to IBminus is not possible but thecoherence of the new concepts requires that the following statement hold

(15i) IB+ cap IBminus = empty

(15ii) x isin Bminus rArr x isin IB+ and x isin B+ rArr x isin IBminus

Now we make use of (NBITI) by taking into account that the correlation A rarr B holds If A isactually measured on x isin S(|ψ〉) and the outcome +1 is obtained ie if x isin A+ such a valuedoes not depend because of (NBITI) on the choice of what is decided to measure in Rβ Since


8

A rarr B we have to conclude that if B were measured on that specimen x then the outcome +1would be obtained Thus we have inferred the following implication from the premises (FC)and (NBITI)

(15iii) If A B and A rarr B then x isin A+ rArr x isin IB+

The new theoretical concepts just introduced make possible to re-formulate the crucial statement(SR) of Stapprsquos argument in the following very simple form

(SR) x isin B(1)+ implies x isin IB(2)

+

51 Property 1Now we can analyze the proof of Property 1 by expanding it in the following sequence ofstatements

(E1) Let us suppose that the antecedent of Property 1 holds

x isin A(2) (16i)

(E2) Let us suppose that the antecendent of (SR) holds too

x isin B(1)+ (16ii)

(E3) Hence (16i) and (16ii) implyx isin B(1) capA(2) (16iii)

(E4) Then (hiii) (16ii) and (16iii) imply

x isin A(2)+ (16iv)

(E5) (hiiic) (16iv) and (15iii) implyx isin IB(2)

+

In order that this re-worded proof be correct it is sufficient to prove that specimens satisfying(16i) and (16ii) actually exist since the steps from (E3) to (E5) are correctly demonstratedNow by (hiiib) (3ii) and (hii) we have 1+B(1)

21+A(2)

2 ψ = 1+B(1)

2 ψ 6= 0 Therefore 〈ψ |1+B(1)

21+A(2)

2 ψ〉 6= 0 But this last is just the probability that a simultaneous measurement ofB(1) and A(2) yields respective outcomes +1 and +1 being it non vanishing we have to concludethat a specimen x satisfyng (16i) and (16ii) actually exists

Thus our analysis does agree with Stapprsquos conclusion that (SR) holds if A(2) is measured inRα

52 Property 2Now we submit the proof of property 2 to our analysis

Property 2 x isin A(1) does not imply (SR) holds for this x

Hence this time Stapprsquos scope is to show that

x0 isin A(1) exists such that the antecedent of (SR) is true but the consequent is falseie that

existx0 isin A(1) x0 isin B(1)+ but x0 isin IB(2)

+ (17)


9

Stapprsquos Proof ldquoQuantum theory predicts that if [A(1)] is performed then outcome [+1] appearsabout half the time Thus if [A(1)] is chosen then there are cases where [x isin A(1)

+ ] is true Butin a case where [x isin A(1)

+ ] is true the prediction [(hiiia)] asserts that the premise of (SR)is true But statement [(hiv)] in conjunction with our two premises that give meaning tolsquoinsteadrsquo implies that the conclusion of (SR) is not true if [B(2)] is performed instead of [B(1)]the outcome is not necessarily [+1] as it was in case [A(2)] So there are cases where [A(1)] istrue but (SR) is falserdquo [9]

Conclusion (17) is attained by Stapp through the following sequence of statements we translatefrom his proof

(S1) A support S(|ψ〉 exists such that A(1)+ 6= empty

(S2) x isin A(1)+ rArr x isin B(1)

+

(S3) The antecedent of (SR) holds forallx isin A(1)+

(S4) existx0 isin A(1)+ such that x0 isin B(2)

minus

(S5) x0 isin IB(2)+

Let us now check the validity of each step

Statement (S1) holds by (2iii) and (hii)

Statement (S3) is implied from (S1) and (S2)

Statement (S4) holds because of (hiv)

Statement (S5) holds because of (S4) and (15ii)

We see that all steps (S1) (S3) (S4) (S5) hold true according to a logical analysis

What about step (S2) Statement (S2) is nothing else but the translation into our languageof the phrase ldquoBut in a case where [x isin A(1)

+ ] is true the prediction [(hiiia)] asserts that thepremise of (SR) is truerdquo stated by Stapp in his proof Hence according to Stapprsquos proof (S2)holds because of (hiiia) A(1) rarr B(1) But the implication

x isin A(1)+ rArr x isin B(1)

+

follows from A(1) rarr B(1) if isin A(1)capB(1) holds too because of (3i) However this last conditioncannot hold for the specimen x0 considered in (S4) because it has been characterized by thetwo conditions x0 isin A(1)

+ and x0 isin B(2)minus But if x0 isin B(2)

minus holds then x0 isin B(2) obviously holdstoo so that the premise of (SR) x0 isin B(1)

+ cannot hold because B(1) and B(2) do not commutewith each other and therefore B(1) capB(2) = empty by (hii) and (2iv)

6 Conclusive remarksIn this work we have analyzed two kinds of theorems proposed in the literature for proving thatthe principle of locality is not consistent with Quantum Mechanics Since Quantum Mechanicsper se ie without adding further assumptions to the genuine quantum postulates does notconflict with locality every non-locality theorem can reach the aimed inconsistency only byintroducing some other conditions besides the standard ones

In the first kind of non-locality theorems like the theorem of Bell [1] these further conditionscan be identified with the criterion of reality established by EPR in their famous 1935 paper


10

[4] Now in [10] it has been put forward that the interpretation of this criterion is notunique As shown in section 3 the interpretation of EPR goes beyond the strict meaningof the criterion The non-locality theorems assuming the criterion of reality are successful if thiswide interpretation is adopted But we show in section 32 that if the criterion is interpretedaccording to its strict meaning then the non-locality proof of GHSZ [2] becomes unable to reachthe inconsistency Similar disproofs for the other non-locality theorems based on the criterionof reality can be found in [10]

The argument proposed by Stapp aims to prove inconsistency between Quantum Mechanicsand locality by avoiding the use of criteria of reality or hidden variable hypotheses because theyentail contrary to quantum philosophy the assignment of pre-existing values to observableswhich are not measured In the present work we have analyzed the final version of Stapprsquosproof published in [9] the author recognizes as the more effective In fact such a final form isthe result of a number of works started in 1975 [12] submitted to various improvements overthe years These works received severe criticisms [13]-[16] all answered by Stapp [17]-[19]

However the debate has not reached a definitive conclusion because the criticisms enter thecounterfactual character of the concept of ldquoinsteadrdquo used in Stapprsquos argument and their aim isto check the validity of the proof within counterfactuals theory ie modal logic [20] On theother hand in his replies Stapp maintains that his proof contrary to the earliest versions doesnot make use of modal logic

The analysis presented in the present works does not make use of counterfactuals theoryIndeed our disproof proceeds

bull first by translating the consequences of Stapprsquos further assumptions (FC) and (NBITI) intothe formal statements (15i)-(15iii) within a suitable theoretical apparatus able to describeStapprsquos approach No counterfactual concepts such as ldquopossible worldsrdquo or ldquonearness ofpossible worldsrdquo are involved in such a translation

bull then the proofs of property 1 and property 2 as drawn by Stapp are analyzed from anordinary (not modal) logico-mathematical point of view

Since the proof of property 2 at the end of the analysis turns out to be not valid we have toconclude that Stapprsquos argument fails within our theoretical apparatus

Thus Stapprsquos refusals of previous criticisms do not apply to the disproof presented in thepresent work

References[1] Bell J S 1964 Physics 1 165[2] Greenberger D M Horne M A Shimony A and Zeilinger A 1990 AmJPhys 58 1131[3] Hardy L 1993 PhysRevLett 71 1665[4] Einstein A Podolsky B and Rosen N 1935 Phys Rev 47 777[5] Clauser J F Horne M A Shimony A and Holt R A 1969 PhysRev Lett 23 880[6] Mermin N D 1993 RevModPhys 65 803[7] Mermin N D 1995 PhysRevLett 74 831[8] Stapp H P 2006 FoundPhys 36 73[9] Stapp H P 2004 AmJPhys 72 30[10] Nistico G and Sestito A 2011 FoundPhys 41 1263[11] Nistico G 1995 FoudPhys 25 1757[12] Stapp H P 1975 Nuovo Cimento 29 270[13] Shimony A 2006 FoundPhys 36 61[14] Clifton R K and Dickson M 1994 PhysRev A 49 4251[15] Shimony A and Stein H 2001 AmJPhys 69 848[16] Mermin N D 1998 AmJPhys 66 920[17] Stapp H P 1994 PhysRev A 49 4257[18] Stapp H P 1998 AmJPhys 66 924


11

[19] Stapp H P 2006 FoundPhys 36 73[20] Lewis D 1973 Counterfactuals (Cambridge MA Harvard University Press)


12

Analysis of non locality proofs in Quantum

Mechanics

Giuseppe NisticoDipartimento di Matematica Universita della Calabria - ItalyandINFN ndash gruppo collegato di Cosenza Italy

E-mail gnisticounicalit

Abstract Two kinds of non-locality theorems in Quantum Mechanics are taken into accountthe theorems based on the criterion of reality and the quite different theorem proposed byStapp In the present work the analyses of the theorem due to Greenberger Horne Shimonyand Zeilinger based on the criterion of reality and of Stapprsquos argument are shown The resultsof these analyses show that the alleged violations of locality cannot be considered definitive

1 IntroductionThe task accomplished by Quantum Mechanics as an empirical theory is to establishwhich relationships occur in Nature between physical events ndash including the occurrences ofmeasurementsrsquo outcomes ndash if the physical system is assigned a given state vector | ψ〉 Allexperimental observations so far performed have confirmed the quantum theoretical predictionswhich per se entail no violation of the locality principle we can express as follows

(L) Principle of Locality Let R1 and R2 be two space-time regions which are separated space-like The reality in R2 is unaffected by operations performed in R1

Conflicts between Quantum Mechanics and locality arise only if further conditions which donot belong to the genuine set of quantum postulates are required to hold

In the ldquoclassicalrdquo non-locality theorems [1]-[3][5]-[7] these further conditions bring back tothe criterion of reality introduced by Einstein Podolsky and Rosen (EPR) [4]

(R) Criterion of Reality If without in any way disturbing a system we can predict withcertainty the value of a physical quantity then there exists an element of physical realitycorresponding to this physical quantity

EPR argued that under certain circumstances more non-commuting observables must havesimultaneous physical reality as a consequence of (L) and (R) without being all measuredwhile Quantum Mechanics is unable to describe such a reality This lack prompted to seek fora theory more complete than Standard Quantum Mechanics able to ascribe reality to theseunmeasured observables But Bell [1] first and other authors later [2][3] proved that such alocal realistic theory cannot exist More precisely they found that contradictions arise just inthe attempt of assigning values to non measured observables in agreement with (L) (R) andwith Quantum Mechanics Since the empirical validity of Quantum Mechanics cannot be denied


Published under licence by IOP Publishing Ltd 1














2










21 + B

2ψ =

1 + A

2ψ






3


sumn ψnotimes un =








a(x) =

1 if x isin A+


1 if x isin A+








4
















(6ii) [Aα Aβ] 6= 0 [Cα Cβ] 6= 0 [Dα Dβ] 6= 0



5



(7)















(11)










6











minus


ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)










7


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12














2










21 + B

2ψ =

1 + A

2ψ






3


sumn ψnotimes un =








a(x) =

1 if x isin A+


1 if x isin A+








4
















(6ii) [Aα Aβ] 6= 0 [Cα Cβ] 6= 0 [Dα Dβ] 6= 0



5



(7)















(11)










6











minus


ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)










7


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12










21 + B

2ψ =

1 + A

2ψ






3


sumn ψnotimes un =








a(x) =

1 if x isin A+


1 if x isin A+








4
















(6ii) [Aα Aβ] 6= 0 [Cα Cβ] 6= 0 [Dα Dβ] 6= 0



5



(7)















(11)










6











minus


ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)










7


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12


sumn ψnotimes un =








a(x) =

1 if x isin A+


1 if x isin A+








4
















(6ii) [Aα Aβ] 6= 0 [Cα Cβ] 6= 0 [Dα Dβ] 6= 0



5



(7)















(11)










6











minus


ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)










7


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12
















(6ii) [Aα Aβ] 6= 0 [Cα Cβ] 6= 0 [Dα Dβ] 6= 0



5



(7)















(11)










6











minus


ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)










7


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12



(7)















(11)










6











minus


ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)










7


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12











minus


ψ | 1 + A(1)

21minus B(2)

2ψ

rang6= 0 (14)










7


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12


















8





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12





+



x isin A(2) (16i)


x isin B(1)+ (16ii)



x isin A(2)+ (16iv)


+


21+A(2)

2 ψ = 1+B(1)


21+A(2)








+ (17)


9







+



minus

(S5) x0 isin IB(2)+










+








10











11



12







+



minus

(S5) x0 isin IB(2)+










+








10











11



12











11



12



12

analysis of non locality proofs in quantum mechanics - iopscience

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