On the Bell- Kochen -Specker paradox

Transcription

1 On the Bell- Kochen -Specker paradox Koji Nagata and Tadao Nakaura Departent of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea E-ail: Departent of Inforation and Coputer Science, Keio University, Hiyoshi, Kohoku-ku, Yokohaa, Japan E-ail: Keywords: Quantu nonlocality, Algebra, Set theory, Foralis Abstract We use the validity of Addition and Multiplication for a hidden variables theory. First, we provide an exaple that the two operations Addition and Multiplication do not coute with each other as revealed by the analyses that are perfored in a finite set of nubers. Our discussion leads to an initial conclusion that Su rule and Product rule do not coute with each other in a hidden variables theory. If we accept this conclusion, we do not get the Bell- Kochen -Specker paradox. In ore detail, quantu echanics ay accept the hidden variables theory. Next, we discuss the validity of operators under an assuption that Su rule and Product rule coute with each other. In this case, we indeed get the Bell- Kochen -Specker paradox. We got the non-classicality of acroscopic experiental data observed in the Stern-Gerlach experient and the double-slit experient. If we detect > and then we detect >, the experients cannot accept the hidden variables theory. We considered whether we can assign the predeterined hidden result to nubers and - as in results of easureents with the nuber of easureents finite (e.g., twice) in the experients. It turned out that we cannot assign the predeterined hidden result to such results of easureents. The next conclusion indicates interestingly that the Stern-Gerlach experient cannot accept classical echanics. The double-slit experient had led to the sae situation, and they were indeed quantu echanical phenoena.. Introduction The quantu theory (cf. [-5]) gives accurate and at ties rearkably accurate

2 nuerical predictions. Much experiental data has explained well the quantu predictions for long tie. On the other hand, the incoplete arguent of Einstein, Podolsky, and Rosen (EPR) [6] for a hidden variable interpretation of the quantu theory has been an attractive topic of research [,3]. There are two ain approaches to study the hidden-variable interpretation of the quantu theory. One is the Bell-EPR theore [7]. This theore says that the quantu predictions violate the inequality following the EPR-locality condition. The EPR-locality condition tells that a result of easureent pertaining to one syste is independent of any easureent perfored siultaneously at a distance on another syste. The dependency has been proved by the entangleent. However, the locality is set by in priori. The other is the no-hidden-variables theore of Kochen and Specker (KS theore) [8]. The original KS theore says the non-existence of a real-valued function which is ultiplicative and linear on couting operators. In general, the quantu theory does not accept the KS type of hidden-variable theory. The proof of the original KS theore relies on intricate geoetric arguent. And, the KS theore becoes very siple for (see also Refs. [9-3]). Merin considered the Bell-EPR theore in a ultipartite state. He derived the ultipartite Bell inequality [4]. Greenberger, Horne, and Zeilinger discovered [5,6] the so-called GHZ theore for four-partite GHZ state. The quantu predictions by n-partite GHZ state violate the Bell-Merin inequality by an aount that grows exponentially with n. After all, several ultipartite Bell inequalities were reported [7-5]. They also revealed that the quantu predictions violate the local hidden-variable theories by an aount that grows exponentially with n. The KS theore was begun with research for the validity of itself by using the inequalities (see Refs. [6-9]). To find such inequalities to test the validity of the KS theore is particularly useful for experiental investigation [30]. The KS theore is related to the algebraic structure of a set of quantu operators. The KS theore is independent of a quantu state under study. One of the authors derives an inequality [9] as tests for the validity of the KS theore. The quantu predictions violate the inequality when the syste is in an uncorrelated state. An uncorrelated state is defined in Ref. [3]. The quantu predictions by n-partite uncorrelated state violate the inequality by an aount that grows exponentially with n. We cannot assign definite value into each quantu datu. This gives the very siple reason why Bell-Kochen-Specker inequalities are violated in real experients.

3 Leggett-type nonlocal hidden-variable theory [3] is experientally investigated [33-35]. The experients report that the quantu theory does not accept Leggett-type nonlocal hidden-variable theory. These experients are done in four-diensional space (two parties) in order to study nonlocality of hidden-variable theories. Many researches address non-classicality of observables. And non-classicality of quantu state itself is not investigated at all (however see [36,37]). Non-classicality of acroscopic quantu datu is not investigated very well. In this paper, we use the validity of Addition and Multiplication for a hidden variables theory. First, we provide an exaple that the two operations Addition and Multiplication do not coute with each other as revealed by the analyses that are perfored in a finite set of nubers. Our discussion leads to an initial conclusion that Su rule and Product rule do not coute with each other in a hidden variables theory. In this case, we do not get the Bell- Kochen -Specker paradox. Next, we discuss the validity of operators under an assuption that Su rule and Product rule coute with each other. In this case, we get the Bell- Kochen -Specker paradox. We got the non-classicality of acroscopic experiental data observed in the Stern-Gerlach experient and the double-slit experient. If we detect > and then we detect >, the experients cannot accept the hidden variables theory. We considered whether we can assign the predeterined hidden result to nubers and - as in results of easureents with the nuber of easureents finite (e.g., twice) in the experients. It turned out that we cannot assign the predeterined hidden result to such results of easureents. The next conclusion indicates interestingly that the Stern-Gerlach experient cannot accept classical echanics. The double-slit experient had led to the sae situation, and they were indeed quantu echanical phenoena.. The two operations Addition and Multiplication do not coute with each other We consider a value V which is the su of the results of trials. Result of trials is or -. We assue the nuber of - is equal to the nuber of. The nuber of trials is. We have V - 0. (.7) We derive the possible value of the product V 0. (.8) V We assign the truth value for the following proposition V 0. (.9) We have V ) 0. (.0) ( ax V V of the value V. It is

4 The value (V0) which is the su of the results of the trials is given by V r l l. (.) We assue that the possible value of the actually easured results r l is or -. We have V +. (.) The sae value is given by V r l '. (.3) We only change the notation as l. The possible value of the actually easured results r is or -. We have { l l N r } { N } and r' (.4) { l l N r } { N } r'. (.5) Here N {,,..., }. By using these facts we derive a necessary condition for the value given in (.). We derive the possible value of the product V of the value V given in (). We have the following under the assuption that the two operations Addition and Multiplication coute with each other. V ( l l l l l l r ) ( l r r,(.7) ( r r ),(.8) l ' ( r ) (( ) l ' 4 l l ' l l r ),(.6),(.9) + (),(.). ),(.0)

5 The step (.6) to (.7) is OK. The step (.7) to (.8) is valid under the assuption that the two operations Addition and Multiplication coute with each other. We insert parentheses. The step (.8) to (.9) is true since we have only changed the notation as l. The above inequality (.9) is saturated since { l l N r } { N r } and ' (.) { l l N r } { N r } '. (.3) We derive a proposition concerning the value given in (.) under the assuption that the possible value of the actually easured results is or -, that is derive the following proposition ( V ) ax 4. (.4) V 4. We We do not assign the truth value for the two propositions (.0) and (.4) siultaneously. We are in a contradiction. Thus we have to give up the assuption that the two operations Addition and Multiplication coute with each other. Our discussion leads to an initial conclusion that Su rule and Product rule do not coute with each other in a hidden variables theory. If we accept this conclusion, we do not get the Bell- Kochen -Specker paradox. In ore detail, quantu echanics ay accept the hidden variables theory. In what follows, we discuss the validity of operators under an assuption that Su rule and Product rule coute with each other. In this case, we indeed get the Bell- Kochen -Specker paradox and quantu echanics does not accept the hidden variables theory, in general. 3. Easy exaple that we cannot assign definite value into each experiental datu We consider a ean value V to an expected value which is the su of data in soe experients. The actually easured results of trials are or -. We assue the nuber of - is equal to the nuber of. The nuber of trials is. Then we have V-+0. (3.) First, we assign definite value into each experiental datu. In this case, we consider the Bell-Kochen-Specker realis. By using,, and, we can define experiental data as follows:,, and. Let us write V as follows. (3.)

6 The possible value of the actually easured results is or -. The sae value is given by. (3.3) We change the notation as. The possible value of the actually easured results is or -. In the following, we evaluate the axiu value of the product and derive a necessary condition under an assuption that we assign definite value into each experiental datu. We have V V ) ( ) ( ( ) (( ) +() ) () 4 (3.4) The above inequality (3.4) is saturated since { l N r } { N r } l (3.5) and ' { l N r } { N r } l. (3.6) ' Hear N {,}. We derive an assuption concerning the value given in (3.) under an assuption that the possible value of the actually easured results is or -, that is ( ) 4. We derive the following assuption concerning the Bell-Kochen-Specker realis ( ) 4. (3.7) Next, we derive the other axiu value of the product of the value V under an assuption that we do not assign definite value into each experiental datu. This is quantu echanical case. Fro V0, we have ( )!" 0. (3.8) Thus, ( )!" 0. (3.9) We have the following assuption concerning quantu echanics ( )!" 0. (3.0) We cannot assign the truth value ``'' for the two assuptions (3.7) and (3.0). We are in the BKS contradiction. Thus we cannot assign definite value into each experiental

7 datu even though the nuber of data is two. 4. More general exaple that we cannot assign definite value into each experiental datu We consider a ean value V to an expected value which is the su of data in soe experients. The actually easured results of trials are or -. We assue the nuber of - is equal to the nuber of. The nuber of trials is.,, We have V-0. (3.) First, we assign definite value into each experiental datu. Let us write the value (V0) as follows. (3.) We assue that the possible value of the actually easured results is or -. The sae value is given by. (3.3) We change the notation as. The possible value of the actually easured results is or -. By using these facts we derive a necessary condition for the value given in (3.). We derive the axiu value of the product V V of the value V given in (3.). We have the following under an assuption that we assign definite value into each experiental datu: V V ) ( ) ( ( ) (( ) +() ) () 4%. (3.4) The above inequality (3.4) is saturated since { l l N r } { N r } and ' (3.5) { l l N r } { N r } '. (3.6) Here N {,,..., }.We derive an assuption concerning the value given in (3.) under an assuption that the possible value of the actually easured results is or -, that is ( ) 4%. We derive the axiu value concerning the

8 Bell-Kochen-Specker realis ( ) 4%. (3.7) Next, we derive the other axiu value of the product of the value under an assuption that we do not assign definite value into each experiental datu. This is quantu echanical case. Fro V0, we have ( )!" 0. (3.8) Thus, ( )!" 0. (3.9) We have the following assuption concerning quantu echanics ( )!" 0. (3.0) We cannot assign the truth value for the two assuptions (3.7) and (3.0). We are in the contradiction. Thus we cannot assign definite value into each experiental datu. We say the following stateent: We cannot assign definite value into each experiental datu when the nuber of easureent is even and finite. Otherwise we are in the BKS contradiction. We are not in the BKS contradiction if we cannot assign definite value into each experiental datu when the nuber of easureent is even and finite. 5. Conclusions In conclusions, we have used the validity of Addition and Multiplication for a hidden variables theory. First, we have provided an exaple that the two operations Addition and Multiplication do not coute with each other as revealed by the analyses that are perfored in a finite set of nubers. Our discussion has led to an initial conclusion that Su rule and Product rule do not coute with each other in a hidden variables theory. We do not have got the Bell- Kochen -Specker paradox if we accept the first conclusion. In ore detail, quantu echanics ay have accepted the hidden variables theory. It has been not easy to contest the Boh s hidden variable theory which is still in a great debate process. We have tried to validate our point on the hidden variable theory using the operators: Additive and Multiplicative. It has been hoped that this article induces further debates on our approach and siplistic deterinis. Next, we have discussed the validity of operators under an assuption that Su rule and Product rule coute with each other. In this case, we indeed get the Bell- Kochen

9 -Specker paradox. We have got the non-classicality of acroscopic experiental data observed in the Stern-Gerlach experient and the double-slit experient. If we detect > and then we detect >, the experients cannot have accepted the hidden variables theory. We have considered whether we can assign the predeterined hidden result to nubers and - as in results of easureents with the nuber of easureents finite (e.g., twice) in the experients. It has turned out that we cannot assign the predeterined hidden result to such results of easureents. The next conclusion indicates interestingly that the Stern-Gerlach experient cannot accept classical echanics. The double-slit experient had led to the sae situation, and they were indeed quantu echanical phenoena. Acknowledgeent We thank the reviewer of the paper. References [] J. J. Sakurai, Modern Quantu Mechanics (Addison-Wesley Publishing Copany, 995), Revised ed. [] A. Peres, Quantu Theory: Concepts and Methods (Kluwer Acadeic, Dordrecht, The Netherlands, 993). [3] M. Redhead, Incopleteness, Nonlocality, and Realis (Clarendon Press, Oxford, 989), nd ed. [4] J. von Neuann, Matheatical Foundations of Quantu Mechanics (Princeton University Press, Princeton, New Jersey, 955). [5] M. A. Nielsen and I. L. Chuang, Quantu Coputation and Quantu Inforation (Cabridge University Press, 000). [6] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (935). [7] J. S. Bell, Physics, 95 (964). [8] S. Kochen and E. P. Specker, J. Math. Mech. 7, 59 (967). [9] C. Pagonis, M. L. G. Redhead, and R. K. Clifton, Phys. Lett. A 55, 44 (99). [0] N. D. Merin, Phys. Today 43(6), 9 (990). [] N. D. Merin, A. J. Phys. 58, 73 (990). [] A. Peres, Phys. Lett. A 5, 07 (990). [3] N. D. Merin, Phys. Rev. Lett. 65, 3373 (990). [4] N. D. Merin, Phys. Rev. Lett. 65, 838 (990). [5] D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell s Theore, Quantu

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