Bell s inequality is violated in classical systems as well as quantum systems

Transcription

1 Bell s inequality is violated in classical systems as well as quantum systems Shiro ISHIKAWA Department of Mathematics, Faculty of Science and Technology, Keio University, , Hiyoshi, Kouhoku-ku Yokohama, Japan ishikawa@mathkeioacjp Abstract Bell s inequality is usually considered to belong to mathematics and not quantum mechanics We think that this makes it difficult to understand Bell s theory Thus in this paper, contrary to Bell s spirit which inherits Einstein s spirit), we try to discuss Bell s inequality in the framework of quantum theory with the linguistic Copenhagen interpretation And we clarify that whether or not Bell s inequality holds does not depend on whether classical systems or quantum systems, but depend on whether a kind of simultaneous measurements exist or not And further we assert that our argument based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell s philosophical argument based on Einstein s spirit) Key phrases: Bohr-Einstein debates, Bell s inequality, Combined observable, Linguistic Copenhagen interpretation, Quantum Language 1 Review: Quantum language = Measurement theory =MT) ) 11 Introduction Following refs [6, 7, 8, 9, 10, 11], we shall review quantum language ie, quantum theory with the linguistic Copenhagen interpretation, or measurement theory ), which has the following form: Quantum language = measurement theory) = Measurement Axiom 1) + Causality Axiom 2) + Linguistic Copenhagen ) interpretation how to use Axioms 1 and 2) 1) We think that the location of quantum language in the history of world-description is as follows the realistic world view monism, realism) Parmenides Socrates 0 :Greek philosophy Plato Aristotle Schola- 1 sticism monism) ewton realism) 2 dualism) Descartes Locke, 6 Kant idealism) relativity theory 3 quantum mechanics 4 linguistic view) linguistic philosophy statistics system theory language A 1 ) language A 2 ) 7 8 language 9 A 3 ) 5 10 unsolved) theory of everything quantum phys) =MT) quantum language language) the linguistic world view dualism, idealism ) Figure 1: The history of the world-description 1

2 And in Figure 1, we think that the following four are equivalent cf ref [10]): A 0 ) to propose quantum language cf 10 in Figure 1) A 1 ) to clarify the Copenhagen interpretation of quantum mechanics cf 7 in Figure 1) A 2 ) to clarify the final goal of the dualistic idealism cf 8 in Figure 1, see ref [11]) A 3 ) to reconstruct statistics in the dualistic idealism cf 9 in Figure 1) In Bohr-Einstein debates refs [2, 5]), Einstein s standing-point is on the side of the realistic view in Figure 1 On the other hand, we think that Bohr s standing point is on the side of the linguistic view in Figure 1 though Bohr might believe that the Copenhagen interpretation proposed by his school) belongs to physics) In this paper, contrary to Bell s spirit which inherits Einstein s spirit), we try to discuss Bell s inequality refs [1, 13]) in quantum language ie, quantum theory with the linguistic Copenhagen interpretation) And we clarify that whether or not Bell s inequality holds does not depend on whether classical systems or quantum systems in Section 3), but depend on whether a kind of simultaneous measurements exist or not in Section 2) And further we assert that our argument based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell s philosophical argument based on Einstein s spirit) 12 Quantum language: Mathematical preparations ow we briefly introduce quantum language as follows Consider an operator algebra BH) ie, an operator algebra composed of all bounded linear operators on a Hilbert space H with the norm F BH) = sup u H =1 F u H ), and consider the pair [A, ] BH), called a basic structure Here, A BH)) is a C - algebra, and A BH)) is a particular C -algebra called a W -algebra) such that is the weak closure of A in BH) The measurement theory =quantum language= the linguistic interpretation) is classified as follows B 1 ): quantum system theory when A = CH)) B) measurement theory = B 2 ): classical system theory when A = C 0 Ω)) That is, when A = CH), the C -algebra composed of all compact operators on a Hilbert space H, the B 1 ) is called quantum measurement theory or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics Also, when A is commutative that is, when A is characterized by C 0 Ω), the C -algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space Ω cf [12, 14])), the B 2 ) is called classical measurement theory Also, note cf [12]) that, when A = CH), i) A = TrH) =trace class), = BH), = TrH) ie, pre-dual space), thus, TrH) ρ, T )BH) = Tr ρt ) ρ TrH), T BH)) H Also, when A = C 0 Ω), ii) A = the space of all signed measures on Ω, = L Ω, ν) BL 2 Ω, ν))), = L 1 Ω, ν), where ν is some measure on Ω, thus, L 1 Ω,ν) ρ, T )L Ω,ν) = Ω ρω)t ω)νdω) ρ L1 Ω, ν), T L Ω, ν)) cf [12]) The measure ν is usually omitted in this paper, that is, L Ω, ν) and L 1 Ω, ν) is respectively written by L Ω) and L 1 Ω) Let [A, ] BH) be a basic structure Let A BH)) be a C -algebra, and let A be the dual Banach space of A That is, A = {ρ ρ is a continuous linear functional on A }, and the norm ρ A is defined by sup{ ρf ) F A such that F A = F BH) ) 1} Define the mixed state ρ A ) such that ρ A = 1 and ρf ) 0 for all F A such that F 0 And define the mixed state space S m A ) such that S m A )={ρ A ρ is a mixed state} 2

3 A mixed state ρ S m A )) is called a pure state if it satisfies that ρ = θρ θ)ρ 2 for some ρ 1, ρ 2 S m A ) and 0 < θ < 1 implies ρ = ρ 1 = ρ 2 Put S p A )={ρ S m A ) ρ is a pure state}, which is called a state space It is well known cf [12]) that S p CH) ) = { u u ie, the Dirac notation) u H = 1}, and S p C 0 Ω) ) = {δ ω0 δ ω0 is a point measure at ω 0 Ω}, where Ω fω)δ ω 0 dω) = fω 0 ) f C 0 Ω)) The latter implies that S p C 0 Ω) ) can be also identified with Ω called a spectrum space or simply spectrum) such as S p C 0 Ω) ) state space) δ ω ω Ω spectrum) For instance, in the above ii) we must clarify the meaning of the value of F ω 0 ) for F L Ω, ν) and ω 0 Ω An element F ) is said to be essentially continuous at ρ 0 S p A )), if there uniquely exists a complex number α such that C) if ρ, ρ = 1) converges to ρ 0 S p A )) in the sense of weak topology of A, that is, ρg) ρ 0 G) G A )), then ρf ) converges to α And the value of ρ 0 F ) is defined by the α According to the noted idea cf [4]), an observable O :=X, F, F ) in is defined as follows: i) [σ-field] X is a set, F 2 X PX)), the power set of X) is a σ-field of X, that is, Ξ 1, Ξ 2, F Ξ n F, Ξ F X \ Ξ F ii) [Countable additivity] F is a mapping from F to satisfying: a): for every Ξ F, F Ξ) is a nonnegative element in such that 0 F Ξ) I, b): F ) = 0 and F X) = I, where 0 and I is the 0-element and the identity in respectively c): for any countable decomposition {Ξ 1, Ξ 2,, Ξ n, } of Ξ ie, Ξ, Ξ n F n = 1, 2, 3, ), Ξ n = Ξ, Ξ i Ξ j = i j) ), it holds that F Ξ) = F Ξ n) in the sense of weak topology in 13 Axiom 1 [Measurement] and Axiom 2 [Causality] Measurement theory B) is composed of two axioms ie, Axioms 1 and 2) as follows With any system S, a basic structure [A, ] BH) can be associated in which the measurement theory B) of that system can be formulated A state of the system S is represented by an element ρ S p A )) and an observable is represented by an observable O :=X, F, F ) in Also, the measurement of the observable O for the system S with the state ρ is denoted by M O, S [ρ] ) or more precisely, M O :=X, F, F ), S [ρ] ) ) An observer can obtain a measured value x X) by the measurement M O, S [ρ] ) The Axiom 1 presented below is a kind of mathematical generalization of Born s probabilistic interpretation of quantum mechanics And thus, it is a statement without reality ow we can present Axiom 1 in the W -algebraic formulation as follows Axiom 1 [ Measurement ] The probability that a measured value x X) obtained by the measurement M O :=X, F, F ), S [ρ] ) belongs to a set Ξ F) is given by ρf Ξ)) if F Ξ) is essentially continuous at ρ S p A )) ext, we explain Axiom 2 Let [A 1, 1 ] BH1) and [A 2, 2 ] BH2) be basic structures A continuous linear operator Φ 1,2 : 2 with weak topology) 1 with weak topology) is called a Markov operator, if it satisfies that i): Φ 1,2 F 2 ) 0 for any non-negative element F 2 in 2, ii): Φ 1,2 I 2 ) = I 1, where I k is the identity in k, k = 1, 2) In addition to the above i) and ii), in this paper we assume that Φ 1,2 A 2 ) A 1 and sup{ Φ 1,2 F 2 ) A1 F 2 A 2 such that F 2 A2 1} = 1 It is clear that the dual operator Φ 1,2 : A 1 A 2 satisfies that Φ 1,2S m A 1)) S m A 2) If it holds that Φ 1,2S p A 1)) S p A 2), the Φ 1,2 is said to be deterministic If it is not deterministic, it is said to 3

4 be non-deterministic or decoherence Also, note that, for any observable O 2 :=X, F, F 2 ) in 2, the X, F, Φ 1,2 F 2 ) is an observable in 1 ow Axiom 2 is presented as follows: Axiom 2 [Causality] Let t 1 t 2 The causality is represented by a Markov operator Φ t1,t 2 : t2 t1 14 The linguistic interpretation = the manual to use Axioms 1 and 2) In the above, Axioms 1 and 2 are kinds of spells, ie, incantation, magic words, metaphysical statements), and thus, it is nonsense to verify them experimentally Therefore, what we should do is not to understand but to use After learning Axioms 1 and 2 by rote, we have to improve how to use them through trial and error We can do well even if we do not know the linguistic interpretation However, it is better to know the linguistic interpretation = the manual to use Axioms 1 and 2), if we would like to make progress quantum language early The essence of the manual is as follows: D) Only one measurement is permitted And thus, the state after a measurement is meaningless since it can not be measured any longer Thus, the collapse of the wavefunction is prohibited cf [9]) We are not concerned with anything after measurement That is, any statement including the phrase after the measurement is wrong Also, the causality should be assumed only in the side of system, however, a state never moves Thus, the Heisenberg picture should be adopted, and thus, the Schrödinger picture should be prohibited and so on For details, see [10] 15 Simultaneous measurement, parallel measurement Definition 1 i): Let [A, ] BH) be a basic structure Consider observables O k = X k, F k, F k ) k = 1, 2,, K) in Let K k=1 X k, K k=1f k ) be the product measurable space An observable O = K k=1 O k = K k=1 X k, K k=1f k, F ) in is called the simultaneous observable of O k k = 1, 2,, K), if it holds that K k=1 F k Ξ k ) = F K k=1 Ξ k ) Ξ k F k ) 2) Also, the measurement M O, S [ρ0 ]) is called a simultaneous measurement of measurements M O k, S [ρ0 ]) k = 1, 2,, K) ote that a simultaneous observable O = K k=1 X k, K k=1, F ) in always exists if observables O = K k=1 X k, K k=1f k, F ) commute, ie, F k Ξ k )F l Ξ l ) = F l Ξ l )F k Ξ k ) Ξ k F l, Ξ l F k, k l) 3) ii): Let [A k, k ] BHk ) k = 1, 2,, K) be basic structures, and let [ K k=1 A k, K k=1 k] K k=1 BH be the k) tensor basic structure cf [10]) Consider measurements M k O k = X k, F k, F k ), S [ρk ]) k = 1, 2,, K) in k Let K k=1 O k = K k=1 X k, K k=1 F k, K k=1 F k) be the parallel observable in a tensor W -algebra K k=1 k And let K k=1 ρ k S p K k=1 A k) )) Then, the measurement M K k=1 K k k=1 O k = K k=1 X k, K k=1 F k, K k=1 F k), S [ K k=1 ρ k] ) is called a parallel measurement of M k O k = X k, F k, F k ), S [ρk ]) k = 1, 2,, K) ote that the parallel measurement always exists uniquely 2 Bell s inequality in quantum language 21 Our view about Bell s inequality In this paper, I assert that Bell s inequality should be studied in the framework of quantum theory ie, quantum theory with the linguistic Copenhagen interpretation) Let us start from the following definition, which is a slight modification of the simultaneous observable in Definition 1 4

5 Definition 2 [Combined observable] Let [A, ] BH) be a basic structure Put X = { 1, 1} Consider four observables: O 13 = X 2, PX 2 ), F 13 ), O 14 = X 2, PX 2 ), F 14 ), O 23 = X 2, PX 2 ), F 23 ), O 24 = X 2, PX 2 ), F 24 ) in The four observables are said to be combinable if there exists an observable O = X 4, PX 4 ), F ) in such that F 13 {x 1, x 3 )}) = F {x 1 } X {x 3 } X), F 14 {x 1, x 4 )}) = F {x 1 } X X {x 4 }) F 23 {x 2, x 3 )}) = F X {x 2 } {x 3 } X), F 24 {x 2, x 4 )}) = F X {x 2 } X {x 4 }) 4) for any x 1, x 2, x 3, x 4 ) X 4 The observable O is said to be a combined observable of O ij i = 1, 2, j = 3, 4) ote that the O is regarded as a kind of simultaneous observable of O ij i = 1, 2, j = 3, 4) Also, the measurement M O = X 4, PX 4 ), F ), S [ρ0 ]) is called the conbined measurement of M O 13, S [ρ0 ]), M O 14, S [ρ0 ]), M O 23, S [ρ0 ]) and M O 24, S [ρ0 ]) The following theorem is all of our insistence concerning Bell s inequality Theorem 3 [Bell s inequality in quantum language] Let [A, ] BH) be a basic structure Put X = { 1, 1} Fix the pure state ρ 0 S p A ) ) And consider the four measurements M O 13 = X 2, PX 2 ), F 13 ), S [ρ0 ]), M O 14 = X 2, PX 2 ), F 14 ), S [ρ0 ]), M O 23 = X 2, PX 2 ), F 23 ), S [ρ0 ]) and M O 24 = X 2, PX 2 ), F 24 ), S [ρ0 ]) Or equivalently, consider the parallel measurement i=1,2,j=3,4 M O ij = X 2, PX 2 ), F ij ), S [ρ0 ]) Define four correlation functions i = 1, 2, j = 3, 4), R ij = u v ρ 0 F ij {u, v)})) u,v) X X Assume that four observables O 13 = X 2, PX 2 ), F 13 ), O 14 = X 2, PX 2 ), F 14 ), O 23 = X 2, PX 2 ), F 23 ) and O 24 = X 2, PX 2 ), F 24 ) are combinable, that is, we have the observable O = X 4, PX 4 ), F ) in such that it satisfies 4) Then we have the combined measurement M O = X 4, PX 4 ), F ), S [ρ0 ]) of M O 13, S [ρ0 ]), M O 14, S [ρ0 ]), M O 23, S [ρ0 ]) and M O 24, S [ρ0 ]) And further, we have Bell s inequality in quantum language as follows R 13 R 14 + R 23 + R ) Proof Clearly we see, i = 1, 2, j = 3, 4, R ij = x i x j ρ 0 F {x 1, x 2, x 3, x 4 )})) 6) x 1,x 2,x 3,x 4) X X X X for example, R13 = x 1,x 2,x 3,x 4 ) X X X X x 1 x 3 ρ 0 F {x 1, x 2, x 3, x 4 )})) ) Therefore, we see that R 13 R 14 + R 23 + R 24 [ ] = x 1 x 3 x 1 x 4 + x 2 x 3 + x 2 x 4 ρ 0 F {x 1, x 2, x 3, x 4 )})) = x 1,x 2,x 3,x 4 ) X X X X x 1,x 2,x 3,x 4) X X X X This completes the proof [ ] x 3 x 4 + x 3 + x 4 ρ 0 F {x 1, x 2, x 3, x 4 )})) 2 As the corollary of this theorem, we have the followings: 5

6 Corollary 4 Consider the parallel measurement i=1,2,j=3,4 M O ij = X 2, PX 2 ), F ij ), S [ρ0 ]) as in Theorem 3 Let x = x 1 13, x 2 13), x 1 14, x 2 14), x 1 23, x 2 23), x 1 24, x 2 24) ) X 2 { 1, 1} 8 ) i,j=1,2 be a measured value of the parallel measurement i=1,2,j=3,4 M O ij = X 2, PX 2 ), F ij ), S [ρ0 ]) Let be sufficienly large natural number Consider -parallel measurement [ i=1,2,j=2,3 M O ij := X 2, PX 2 ), F ij ), S [ρ0 ]) ] Let {x n } be the measured value That is, {x n } = x 1,1 x 1,2 13, x2, , x2,2 13 Here, note that the law of large numbers says: ), x1,1 14, x2,1 14 ), x1,1 23, x2,1 23 ), x1,1 24, x2,1 24 )) ), x1,2 14, x2,2 14 ), x1,2 23, x2,2 23 ), x1,2 24, x2,2 24 )) x 1, 13, x2, 13 ), x1, 14, x2, 14 ), x1, 23, x2, 23 R ij 1 Then, it holds, by the formula 5), that 13 x2,n x2,n 14 ) ), x1, 24, x2, 24 ) ij x2,n ij i = 1, 2, j = 3, 4) + which is also called Bell s inequality in quantum language 23 x2,n x2,n 24 X 8 ) 2, 7) Remark 5 [The conventional Bell s inequality cf [13])] The mathematical Bell s inequality is as follows: Let Θ, B, P ) be a probability space Let f 1, f 2, f 3, f 4 ) : Θ X 4 { 1, 1} 4 ) be a measurable functions Define the correlation functions R ij i = 1, 2, j = 3, 4) by Θ f iθ)f j θ)p dθ) Then, the following mathematical Bell s inequality holds: This is easily proved as follows the left-hand side of the above 8) R 13 R 14 + R 23 R ) Θ f 3 θ) + f 4 θ) P dθ) + f 3 θ) f 4 θ) P dθ) 2 Θ 3 Bell s inequality is violated in classical systems as well as quantum systems In the previous section, we show that E 1 ) Under the combinable condition cf Definition 1), Bell s inequality 5) or, 7)) holds in both classical systems and quantum systems 6

7 Or, equivalently, E 2 ) If Bell s inequality 5) or 7)) is violated, then the combined observable does not exist, and thus, we cannot obtain the measured value by the measurement of the combined observable) This makes us expect that F) Bell s inequality 5) or 7)) is violated in classical systems as well as quantum systems without the combined condition This F) was already shown in my previous paper [7] However, I got a lot of questions concerning F) from the readers Thus, in this section, we again explain the F) precisely For this, three steps [Step:I] [Step:III]) are prepared in what follows [Step: I] Put X = { 1, 1} Define complex numbers a k = α k + β k 1) k = 1, 2, 3, 4) such that a k = 1 Define the probability space X 2, PX 2 ), ν aia j ) such that i = 1, 2, j = 3, 4) ν ai a j {1, 1)})= ν ai a j { 1, 1)})= 1 α i α j β i β j )/4 ν aia j { 1, 1)})= ν aia j {1, 1)})= 1 + α i α j + β i β j )/4 9) The correlation Ra i, a j ) i = 1, 2, j = 3, 4) is defined as follows: Ra i, a j ) ow we have the following problem: x 1,x 2 ) X X x 1 x 2 ν ai a j {x 1, x 2 )}) = α i α j β i β j 10) G) Find a measurement M O aia j [A, ] BH) such that := X 2, PX 2 ), F aia j ), S [ρ0 ]) i = 1, 2, j = 3, 4) in a basic structure ν ai a j Ξ) = ρ 0 F ai a j Ξ)) Ξ PX 2 )) 11) and [Step: II] F a1a 3 {x 1 } X) = F a1a 4 {x 1 } X) F a1a 3 X {x 3 }) = F a2a 3 X {x 3 }) F a2a 3 {x 2 } X) = F a2a 4 {x 2 } X) F a1a 4 X {x 4 }) = F a2a 4 X {x 4 }) x k X { 1, 1}), k = 1, 2, 3, 4) Let us answer this problem G) in the two cases ie, classical case and quantum case), that is, i):the case of quantum systems: [A = BC 2 C 2 )] ii):the case of classical systems: [A = C 0 Ω Ω)] i):the case of quantum system: [A = BC 2 ) BC 2 ) = BC 2 C 2 )] Put e 1 = [ ] 1, e 0 2 = [ ] 0 1 C 2 ) 7

8 For each a k k = 1, 2, 3, 4), define the observable O ak X, PX), G ak ) in BC 2 ) such that Then, we have four observable: and further, G ak {1}) = 1 2 [ ] 1 āk, G a k 1 ak { 1}) = 1 [ ] 1 āk 2 a k 1 Ô ai = X, PX), G ai I), Ô aj = X, PX), I G aj ) i = 1, 2, j = 3, 4) 12) O aia j = X 2, PX 2 ), F aia j := G ai G aj ) i = 1, 2, j = 3, 4) 13) in BC 2 C 2 ), where it should be noted that F ai a j is separated by G ai and G aj Further define the singlet state ρ 0 = ψ s ψ s S p BC 2 C 2 ) ) ), where ψ s = e 1 e 2 e 2 e 1 )/ 2 Thus we have the measurement M BC2 C 2 )O aia j, S [ρ0 ]) in BC 2 C 2 ) i = 1, 2, j = 3, 4) The followings are clear: for each x 1, x 2 ) X 2 { 1, 1} 2 ), ρ 0 F ai a j {x 1, x 2 )})) = ψ s, G ai {x 1 }) G aj {x 2 }))ψ s = ν ai a j {x 1, x 2 )}) i = 1, 2, j = 3, 4) 14) For example, we easily see: ρ 0 F ai b j {1, 1)})) = ψ s, G ai {1}) G aj {1}))ψ s = 1 [ ] [ 1 8 e āi 1 āj 1 e 2 e 2 e 1 ), a i 1 a j 1 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] = 1 āi 1 āj 1 0 ), ) a i 1 a j = 1 [ ] [ ] [ ] [ ] [ ] ] ] [ ] [āj [āi 1 8 ), ) a i ] )e 1 e 2 e 2 e 1 ) = aā j ā i a j ) = 1 α i α j β i β j )/4 = ν ai a j {1, 1)}) Therefore, the measurement M BC2 C 2 )O aia j, S [ρ0 ]) satisfies the condition G) a j [ ] 0 1 [ ] 1 ) 0 ii):the case of classical systems: [A = C 0 Ω) C 0 Ω) = C 0 Ω Ω)] Put ω 0 = ω 0, ω 0 )) Ω Ω ρ 0 = δ ω0 S p C 0 Ω Ω) ), ie, the point measure at ω 0 ) ) Define the observable O ai a j := X 2, PX 2 ), F ai a j ) in L Ω Ω) such that [F ai a j {x 1, x 2 )})]ω) = ν ai a j {x 1, x 2 )}) x 1, x 2 ) X 2, i = 1, 2, j = 3, 4, ω Ω Ω) 15) Thus, we have four observables O ai a j = X 2, PX 2 ), F ai a j ) i = 1, 2, j = 3, 4) 16) in L Ω Ω) though the variables are not separable cf the formula 13) ) Then, it is clear that the measurement M L Ω Ω)O ai a j, S [δω0 ]) satisfies the condition G) [Step: III] 8

9 As defined by 9), consider four complex numbers a k = α k + β k 1; k = 1, 2, 3, 4) such that ak = 1 Thus we have four observables O a1a 3 := X 2, PX 2 ), F a1a 3 ), O a1a 4 := X 2, PX 2 ), F a1a 4 ), O a2a 3 := X 2, PX 2 ), F a2a 3 ), O a2a 4 := X 2, PX 2 ), F a2a 4 ), in the W -algebra Thus, we have the parallel measurement i=1,2,j=3,4 M O ai a j := X 2, PX 2 ), F ai a j ), S [ρ0]) in i=1,2,j=3,4 Thus, putting we see, by 10), that a 1 = 1, a 2 = 1, a 3 = 1 + 1, a 4 = 1 1, 2 2 Ra 1, a 3 ) Ra 1, a 4 ) + Ra 2, a 3 ) + Ra 2, a 4 ) = ) Further, assume that the measured value is x X 8 ) That is, x = x 1 13, x 2 13), x 1 14, x 2 14), x 1 23, x 2 23), x 1 24, x 2 24) ) X 2 { 1, 1} 8 ) i,j=1,2 Let be sufficiently large natural number Consider -parallel measurement [ i=1,2,j=3,4 M O ai a j := X 2, PX 2 ), F ai a j ), S [ρ0 ]) ] Assume that its measured value is {x n } That is, {x n } = x 1,1 x 1,2 13, x2, , x2,2 13 ), x1,1 14, x2,1 14 ), x1,1 23, x2,1 23 ), x1,1 24, x2,1 24 )) ), x1,2 14, x2,2 14 ), x1,2 23, x2,2 23 ), x1,2 24, x2,2 24 )) x 1, 13, x2, 13 ), x1, 14, x2, 14 ), x1, 23, x2, 23 Then, the law of large numbers says that This and the formula 17) say that 13 x2,n 13 Ra i, a j ) 1 14 x2,n 14 ) ), x1, 24, x2, 24 ) ij x2,n ij i = 1, 2, j = 3, 4) + 23 x2,n 23 + X 2) { 1, 1} 8 ) i=1,2,j=3,4 24 x2,n ) Therefore, Bell s inequality 5) or 7)) is violated in classical systems as well as quantum systems 4 Conclusions In this paper, contrary to Bell s spirit which inherits Einstein s spirit), we try to discuss Bell s inequality in the framework of quantum theory with the linguistic Copenhagen interpretation of quantum mechanics And we show Theorem 3 Bell s inequality in quantum language), which says the statement E 2 ), that is, H) E 2 )): If Bell s inequality 5) or 7)) is violated, then the combined observable does not exist, and thus, we cannot obtain the measured value 9

10 Also, recall that Bell s original argument based on Einstein s spirit) says, roughly speaking, that I) : If the mathematical Bell s inequality 8) is violated, then hidden variables do not exist which is rather philosophical It should be note that the I) is a statement in Einstein s spirit, on the other hand, the H) is a statement in scientific theory ie, quantum theory with the linguistic Copenhagen interpretation) Therefore, we assert that our H) is a scientific representation of the philosophical I) If so, we can, for the first time, understand Bell s inequality in science We hope that our proposal will be examined from various points of view 1 References [1] Bell, JS On the Einstein-Podolosky-Rosen Paradox, Physics 1, ) [2] Bohr, Can quantum-mechanical description of physical reality be considered complete?, Phys Rev 48) ) [3] Born, M Zur Quantenmechanik der Stoßprozesse Vorläufige Mitteilung), Z Phys 37) [4] Davies, E B Quantum Theory of Open Systems, Academic Press, 1976 [5] Einstein, A, Podolosky, B and Rosen, Can quantum-mechanical description of reality be considered completely? Phys Rev 47) ) [6] Ishikawa,S, Mathematical Foundations of Measurement Theory, Keio University Press Inc 335pages, 2006 http: //wwwkeio-upcojp/kup/mfomt/) [7] Ishikawa,S, A ew Interpretation of Quantum Mechanics, Journal of quantum information science, Vol 1, o 2, 35-42, 2011 doi: /jqis [8] Ishikawa,S, Quantum Mechanics and the Philosophy of Language: Reconsideration of traditional philosophies, Journal of quantum information science, Vol 2, o 1, 2-9, 2012 doi: /jqis [9] Ishikawa,S, Linguistic interpretation of quantum mechanics ; Projection Postulate, Journal of quantum information science, Vol 5, o4, , 2015, DOI: /jqis PaperInformationaspx?PaperID=62464%20) [10] Ishikawa,S, Linguistic interpretation of quantum mechanics: Quantum language Version 2, Research Report Department of mathematics, Keio university), KSTS-RR-16/001, 2016, 426 pages) This preprint is a draft of my book Linguistic Interpretation of Quantum Mechanics Towards World-Description in Quantum Language - Shiho-Shuppan Publisher,2016) [11] Ishikawa,S, Final solution to mind-body problem by quantum language, Journal of quantum information science, Vol 7, o2, 48-56, 2017, DOI: /jqis aspx?paperid=76391) [12] Sakai, S, C -algebras and W -algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete Band 60), Springer- Verlag, 1971) [13] Selleri, F Die Debatte um die Quantentheorie, Friedr Vieweg&Sohn Verlagsgesellscvhaft MBH, Braunschweig 1983) [14] Yosida, K, Functional Analysis, Springer-Verlag, 6th edition, For the further information of quantum language, see my home page: ishikawa/indexehtml 10