MOJ MOJES

Ecology & Environmental Sciences
Opinion
Volume 2 Issue 1 - 2017
The Kochen-Specker Theorem with Two Trials of Measurements
Nagata K1* and Nakamura T2
1Department of Physics, Korea Advanced Institute of Science and Technology, Korea
2Department of Information and Computer Science, Keio University, Japan
Received: February 02, 2017 | Published: February 13, 2017
*Corresponding author: Nagata K, Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea, Email:
Citation: Nagata K, Nakamura T (2017) The Kochen-Specker Theorem with Two Trials of Measurements. MOJ Eco Environ Sci 2(1): 00011. DOI: 10.15406/mojes.2017.02.00011

We review non-classicality of quantum datum. We consider whether we can assign the predetermined “hidden” result to numbers 1 and −1 as in results of measurements in a thought experiment. We assume the number of measurements is two. If we detect 1/ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa Gaai4laKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@39F1@ as 1 and detect | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGG8b qcfa4aaaGaaOqaaKqzGeGaey4KH8kakiaawQYiaaaa@3B90@ as −1, then we can derive the Kochen-Specker theorem. The same situation occurs when we use a new measurement theory that the results of measurements are either 1/ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa Gaai4laKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@39F1@ or − 1/ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa Gaai4laKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@39F1@  

PACS numbers: 03.65.Ud, 03.65.

Opinion

The quantum theory [1-5] is indeed successful physical theory. From the incompleteness argument of Einstein, Podolsky, and Rosen (EPR) [6], a hidden variable interpretation of the quantum theory has been as an attractive topic of research [2,3]. The no-hidden variables theorem of Kochen and Specker (KS theorem) [7] is very famous. In general, the quantum theory does not accept the KS type of hidden-variable theory. Greenberger, Horne, and Zeilinger discover [8,9] the so-called GHZ theorem for four-partite GHZ state. And, the KS theorem becomes very simple form (see also Refs. [10-14]). For the KS theorem, it is begun to research the validity of the KS theorem by using inequalities (see Refs. [15-18]). To find such inequalities to test the validity of the KS theorem is particularly useful for experimental investigation [19]. Many researches address non-classicality of observables. And non-classicality of quantum state itself is not investigated at all (however see [20]). Further, non-classicality of quantum datum is not investigated very well. Does finite-precision measurement nullify the Kochen-Specker theorem? Meyer discusses that finite precision measurement nullifies the Kochen-Specker theorem [21]. Cabello discusses that finite-precision measurement does not nullify the Kochen-Specker theorem [17]. We address the problem. Here we ask: Can we assign definite value into each quantum datum? We cannot assign definite value into each quantum datum. This gives the very simple reason why Kochen-Specker inequalities are violated in real experiments. Further, our discussion says that cannot assign definite value to each quantum datum even though the number of measurements is two. This gives the Kochen-Specker theorem in two trials of measurements. These argumentations would provide supporting evidence of the statement by Cabello.

In this paper, we review non-classicality of quantum datum. We consider whether we can assign the predetermined “hidden” result to numbers 1 and −1 as in results of measurements in a thought experiment. We assume the number of measurements is two. If we detect MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHrg sRaaa@3870@ as 1 and MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHtg YRaaa@3874@ detect as −1, then we can derive the Kochen-Specker theorem. The same situation occurs when we use a new measurement theory [22] that the results of measurements are either 1| 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIXaGaaiiFaKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@3B2B@ or 1| 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIXaGaaiiFaKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@3B2B@ . We consider a value V which is the sum of data in some experiments. The measured results of trials are either 1 or −1. We assume the number of −1 is equal to thenumber of 1. The number of trials is 2. Then we have

V=1+1=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaeyypa0JaeyOeI0IaaGymaiabgUcaRiaaigdacqGH9aqpcaaIWaaa aa@3D6B@                             (1)

First, we assign definite value into each experimental datum. In the case, we consider the Kochen-Specker realism. By using r 1 , r 2 , r 1' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaadbaGaaGymaaqabaqcLbsacaGGSaGaamOCaSWaaSbaaWqa aiaaikdaaeqaaKqzGeGaaiilaiaadkhajuaGdaWgaaqaaKqzadGaaG ymaiaacEcaaKqbagqaaaaa@41A0@  and r 2' , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGYb WcdaWgaaadbaGaaGOmaiaacEcaaeqaaKqzGeGaaiilaaaa@3A59@ we can define experimental data as follows r 1 =1, r 2 =1, r 1' = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaqcLbsacqGH9aqpcaaI XaGaaiilaiaadkhalmaaBaaajuaGbaqcLbmacaaIYaaajuaGbeaaju gibiabg2da9iabgkHiTiaaigdacaGGSaGaamOCaSWaaSbaaKqbagaa jugWaiaaigdacaGGNaaajuaGbeaajugibiabg2da9aaa@4C2B@ 1 and r 2' =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb WcdaWgaaadbaGaaGOmaiaacEcaaeqaaKqzGeGaeyypa0JaeyOeI0Ia aGymaaaa@3C58@

Let us write V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb aaaa@3760@  as follows

V=( l=1 2 r l ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaeyypa0tcfa4aaeWaaeaadaaeWbqaaiaadkhadaWgaaqaaKqzadGa amiBaaqcfayabaaabaqcLbmacaWGSbGaeyypa0JaaGymaaqcfayaaK qzadGaaGOmaaqcfaOaeyyeIuoaaiaawIcacaGLPaaaaaa@4752@ (2)

The possible values of the measured results rl are either 1 or −1. The same value is given by

V=( l=1 2 r l' ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaeyypa0tcfa4aaeWaaeaadaaeWbqaaiaadkhadaWgaaqaaKqzadGa amiBaiaacEcaaKqbagqaaaqaaKqzadGaamiBaiabg2da9iaaigdaaK qbagaajugWaiaaikdaaKqbakabggHiLdaacaGLOaGaayzkaaaaaa@47FD@ (3)

We change the label as ll' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGSb GaeyOKH4QaamiBaiaacEcaaaa@3AFF@ .The possible values of the measured results r l' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGSbGaai4jaaqabaaaaa@38B5@ are either 1 or −1. In the following, we evaluate a value V×V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaey41aqRaamOvaaaa@3A52@  and derive a necessary condition under an assumption that we assign definite value into each experimental datum.

We introduce an assumption that Sum rule and Product rule commute [23]. We have

V×V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaey41aqRaamOvaaaa@3A52@

=( l=1 2 r1 )×( l'=1 2 l' ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaqadaGcbaqcfa4aaabCaOqaaKqzadGaamOCaiaaigdaaSqa aKqzadGaamiBaiabg2da9iaaigdaaSqaaKqzadGaaGOmaaqcLbsacq GHris5aaGccaGLOaGaayzkaaqcLbsacqGHxdaTjuaGdaqadaGcbaqc fa4aaabCaOqaaKqzadGaamiBaiaacEcaaSqaaKqzadGaamiBaiaacE cacqGH9aqpcaaIXaaaleaajugWaiaaikdaaKqzGeGaeyyeIuoaaOGa ayjkaiaawMcaaaaa@573E@

= l=1 2 . l'=1 2 r l r l ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaaeWbGcbaqcLbsacaGGUaaaleaajugWaiaadYgacqGH9aqp caaIXaaaleaajugWaiaaikdaaKqzGeGaeyyeIuoajuaGdaaeWbGcba qcLbmacaWGYbWcdaWgaaadbaGaamiBaaqabaqcLbmacaWGYbWcdaWg aaadbaGaamiBaaqabaWcdaWgaaadbaGaai4jaaqabaaaleaajugWai aadYgacaGGNaGaeyypa0JaaGymaaWcbaqcLbmacaaIYaaajugibiab ggHiLdaaaa@5358@

l=1 2 . l'=1 2 | r l r l ' | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHKj YOjuaGdaaeWbGcbaqcLbsacaGGUaaaleaajugWaiaadYgacqGH9aqp caaIXaaaleaajugWaiaaikdaaKqzGeGaeyyeIuoajuaGdaaeWbGcba qcLbmacaGG8bGaamOCaSWaaSbaaWqaaiaadYgaaeqaaKqzadGaamOC aSWaaSbaaWqaaiaadYgaaeqaaSWaaSbaaWqaaiaacEcaaeqaaSGaai iFaaqaaKqzadGaamiBaiaacEcacqGH9aqpcaaIXaaaleaajugWaiaa ikdaaKqzGeGaeyyeIuoaaaa@5607@

= l=1 2 . l'=1 2 ( r l ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaaeWbGcbaqcLbsacaGGUaaaleaajugWaiaadYgacqGH9aqp caaIXaaaleaajugWaiaaikdaaKqzGeGaeyyeIuoajuaGdaaeWbqaam aabmaabaWaaSbaaWqaaKqzGeGaamOCaaadbeaalmaaBaaameaajugW aiaadYgaaWqabaaajuaGcaGLOaGaayzkaaaabaqcLbmacaWGSbGaai 4jaiabg2da9iaaigdaaKqbagaajugWaiaaikdaaKqzGeGaeyyeIuoa jugWaiaaikdaaaa@546A@

= 2((1)2 + (−1)2)
= 4. (4)

The inequality (4) can be saturated because the following case is possible

|{ l|rl=1 }||=||{ l' | rl' =1 }|| MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGG8b qcfa4aaiWaaeaajugibiaadYgacaGG8bGaamOCaKqzadGaamiBaKqz GeGaeyypa0JaaGymaaqcfaOaay5Eaiaaw2haaKqzGeGaaiiFaiaacY hacqGH9aqpcaGG8bGaaiiFaKqbaoaacmaabaqcLbsacaWGSbGaai4j aiaacYhajuaGdaWgaaqaaKqzGeGaamOCaKqzadGaamiBaiaacEcaaK qbagqaaKqzGeGaeyypa0JaaGymaaqcfaOaay5Eaiaaw2haaKqzGeGa aiiFaiaacYhaaaa@5931@

|{ l|rl=1 }||=||{ l' | rl' =1 }|| MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGG8b qcfa4aaiWaaeaajugibiaadYgacaGG8bGaamOCaKqzadGaamiBaKqz GeGaeyypa0JaeyOeI0IaaGymaaqcfaOaay5Eaiaaw2haaKqzGeGaai iFaiaacYhacqGH9aqpcaGG8bGaaiiFaKqbaoaacmaabaqcLbsacaWG SbGaai4jaiaacYhajuaGdaWgaaqaaKqzGeGaamOCaKqzadGaamiBai aacEcaaKqbagqaaKqzGeGaeyypa0JaeyOeI0IaaGymaaqcfaOaay5E aiaaw2haaKqzGeGaaiiFaiaacYhaaaa@5B0B@ (5)

Thus,

( V×V ) max =4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aajugibiaadAfacqGHxdaTcaWGwbaajuaGcaGLOaGaayzkaaWaaSba aeaaciGGTbGaaiyyaiaacIhaaeqaaKqzGeGaeyypa0JaaGinaaaa@423F@ (6)

Therefore we have the following assumption concerning the Kochen-Specker realism

( V×V ) max =4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aajugibiaadAfacqGHxdaTcaWGwbaajuaGcaGLOaGaayzkaaWaaSba aeaaciGGTbGaaiyyaiaacIhaaeqaaKqzGeGaeyypa0JaaGinaaaa@423F@ (7)

Next, we derive another possible value of the product V×V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaey41aqRaamOvaaaa@3A52@ of the value V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb aaaa@3760@  under an assumption that we do not assign definite value into each experimental datum. This is quantum mechanical case.

In this case, we have

V×V=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaey41aqRaamOvaiabg2da9iaaicdaaaa@3C12@ (8)

We have the following assumption concerning quantum mechanics

( V×V ) max =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamOvaiabgEna0kaadAfaaOGaayjkaiaawMcaaKqbaoaa BaaabaGaciyBaiaacggacaGG4baabeaajugibiabg2da9iaaicdaaa a@424F@ (9)

We cannot assign the truth value “1” for the two assumptions (7) and (9), simultaneously. We derive the KS paradox. Thus we cannot assign definite value into each experimental datum. The number of data is two. We can derive the similar KS paradox when we use a new measurement theory [22] that the results of measurements are either 1/ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa Gaai4laKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@39F1@ or 1/ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIXaGaai4laKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@3ADE@ . In conclusions, non-classicality of quantum datum has been investigated. We have considered whether we can assign the predetermined “hidden” result to natural number1 and −1 as in results of measurement in a thought experiment. The number of trials has been twice. If we detect | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaKqbao aaaiaabaGaeyyKH0kacaGLQmcaaaa@3A5A@ as 1 and detect | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGG8b qcfa4aaaGaaOqaaKqzGeGaey4KH8kakiaawQYiaaaa@3B90@  as −1, then we can have derived the Kochen-Speker theorem. The same situation has occurred when we use a new measurement theory [22] that the results of measurements are either 1| 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa GaaiiFaKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@3A3E@  or 1| 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIXaGaaiiFaKqbaoaakaaakeaajugibiaaikdaaSqabaaaaa@3B2B@ . Generally Multiplication is completed by Addition. Therefore, we think that Addition of the starting point may be superior to any other case.

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