Locality and simultaneous elements of reality

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1 Locality and simultaneous elements of reality G. Nisticò and A. Sestito Citation: AIP Conf. Proc. 1508, 487 (2012); doi: / View online: View Table of Contents: Published by the American Institute of Physics. Additional information on AIP Conf. Proc. Journal Homepage: Journal Information: Top downloads: Information for Authors:

2 Locality and Simultaneous Elements of Reality G. Nisticò and A. Sestito 1 Dipartimento di Matematica, Università della Calabria, Cosenza, Italy and INFN - gruppo collegato di Cosenza, Cosenza, Italy Abstract. We show that the extension of quantum correlations stemming from a strict" interpretation of the criterion of reality raises the failure of Hardy s non-locality theorem. Then, by suggesting an ideal experiment, we prove that such an extension, though strictly smaller than the one derived by Einstein, Podolsky and Rosen and usually adopted, allows for the assignment of simultaneous objective values of two non-commuting observables. Keywords: Quantum Machanics, Locality, Criterion of Reality PACS: Ca, Db, Ud 1. INTRODUCTION In [1] Einstein, Podolsky and Rosen (EPR) describe a physical situation able to attain simultaneous knowledge about two non-commuting quantities of a system on the basis of measurements made on another system that had previously interacted with it". The following statements are crucial for their argument: (R) Criterion of Reality. If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. (L) Principle of Locality. Let R 1 and R 2 be two space-time regions which are separated space-like. The reality in R 2 is unaffected by operations performed in R 1. Since its appearance, two different interpretations of the criterion (R) are specified, a wide interpretation, used and maintained in particular by EPR themselves, and a strict one, maintained by Bohr. They are responsible of two different extensions of the validity of quantum correlations; in the present work, after presenting the necessary formalism, such extensions, (EQC) and (seqc), are explicitly found within that formalism (section 2). The main task of the non-locality theorems is to prove an inconsistency between quantum mechanics (QM) and (EQC): it s really so? In section 3 we prove the failure of Hardy s theorem when we replace (EQC) by (seqc). While EPR argument in [1] can be used to infer the simultaneous knowledge of incompatible properties when (EQC) is adopted, this no longer holds if (seqc) replace (EQC); on the other hand, (EQC) conflict with locality and (seqc) is consistent with 1 Angela Sestito s work was supported by the European Commission, European Social Fund and by the Calabria Region, Regional Operative Program (ROP) Calabria ESF 2007/ IV Axis Human Capital - Operative Objective M2- Action D.5 Quantum Theory: Reconsideration of Foundations 6 AIP Conf. Proc. 1508, (2012); doi: / American Institute of Physics /$

3 it. Does (seqc) allow for simultaneous knowledge of non commuting observables? In the concluding section 4 we affirmatively answer such a question by showing how the simultaneous knowledge can be attained by means of an ideal experiment involving spin particles. 2. EXTENSIONS OF QUANTUM CORRELATIONS According to standard quantum theory, every two-value observable A is represented by a self-adjoint operator  of the Hilbert space H associated with the physical system, with purely discrete spectrum σ(â)={1, 1}; every pure state of the system is represented by a state vector ψ H, with ψ = 1. The probability of obtaining the outcome 1 by measuring A in the state ψ is p ψ (A,1)= ψ 1 2 (1 + Â)ψ. Our argument requires the formal introduction of further terms. Given a state vector ψ, bysupport of ψ we mean any concrete non-empty set S (ψ) of individual physical systems (specimens) whose quantum state is ψ. Fixed a support S (ψ) we introduce: A, the concrete non-empty set of specimens in S (ψ) which actually undergo a measurement of A; A + (resp., A ) is the subset of A for which the outcome 1 (resp. -1) has been obtained. We assume that A = A + A. A, the set of the specimens in S (ψ) which objectively possess a value of the observable A, without being measured (for instance as a consequence of (R)); A + (resp., A ) is the subset of A of specimens which possess objective value 1 (resp., -1); we assume that A + A = A holds. We define A = A A, A + = A + A +, A = A A. two mappings a : A {1, 1} and a : A {1, 1}: a(x)= { 1, if x A+, 1, if x A ; a(x)= { 1, if x A+, 1, if x A. The following statements relate the formalism of standard quantum mechanics with the physical concepts so far introduced. Let A and B be two-value observables; then i) ψ, S (ψ) exists such that A /0, B /0; ii) [A,B]=0 implies that, for all ψ, S (ψ) exists such that A B /0; (1) iii) [A,B] 0 implies A B = /0 for all S (ψ). The conditions of locality and reality lead to further implications for correlated observables, A and B, in the case that they are separated. Two observables A and B are separated, written A B, if their measurements require operations confined in space-like separated regions, R A and R B respectively; as a consequence of (L), [A,B]=0, hence S (ψ) exists such that A B /0. The quantum correlation A B in the state ψ is defined as follows: A B if and only if x A B, then a(x) =1 b(x) =1, if and only if A + B B + S (ψ), if and only if B A A S (ψ), if and only if (a(x)+ 1)(b(x) 1)=0 x A B. 488

4 Now, by (L) the act of actually performing the measurement of A does not affect the reality in R B ; hence (R) could be applied to extend the validity of the quantum correlation. One possibility is to adopt the following strict interpretation. Strict Interpretation. To ascribe reality to B the measurement of A, whose outcome would allow for the prediction, must be actually performed. As a consequence, if an actual measurement of A yields the outcome 1, i.e. if x A + then x B and b(x)=1, i.e. x B + ; hence, A B and A B imply A+ B + B and the correlation, besides holding for all A B, also holds for all x A +. Analogously, if an actual measurement of B yields the outcome 1, i.e. if x B, then x A and a(x) = 1. Therefore it follows that B A A and that the correlation (a(x) = 1) (b(x)=1) also holds for every x B. Hence, if the strict interpretation of (R) is adopted, an extension for A B follows: (seqc) (a(x)+1)(b(x) 1)=0, x (A + B ) (A B) (2) The quantum correlation A B, i.e. A B and B A, in the state ψ means that the correlation (a(x)=1) (b(x)=1) holds for all x A B for all S (ψ). In this case, from (seqc) we can deduce that a(x)=b(x), x A B, S (ψ). The following wide interpretation of (R) is at the basis of EPR argument in [1]. Wide Interpretation. For ascribing reality to B it is sufficient the possibility of performing the measurement of A whose outcome would allow for the prediction, with certainty, of the outcome of a measurement of B. As a consequence of adopting the wide interpretation the correlation A B extends to (EQC) (a(x)+1)(b(x) 1)=0, x A + B. (3) As a consequence, for the quantum correlation A B, we deduce A + = B +, B = A and A = S (ψ), S (ψ). 3. (EQC), (SEQC) AND HARDY S THEOREM The scheme of Hardy s theorem involves four 2-value observables A α, B α, A β, B β, chosen in such a way that (i) A α B α, A α B β, A β B α A β B β ; (ii) [ Aˆ α, A ˆβ ] 0 and [ Bˆ α, B ˆβ ] 0. The choice of the state vector ψ is constrained by a non-vanishing probability of obtaining (1, 1) as pair of outcomes of a measurement of A α and B β ; this implies that a support S 0 (ψ) of ψ exists such that A+ α B β /0, i.e. S 0 (ψ) and x 0 S 0 (ψ) exist such that a α (x 0 )=1 and b β (x 0 )= 1. (4) Furthermore, ψ is chosen so that, according to quantum theory, the correlations A α B α, B α A β, A β B β hold, i.e. so that for any S (ψ) the following statements (5.i)- 489

5 (5.iii) hold. i) (a α (x)+1)(b α (x) 1) =0, x A α B α ii) (b α (y)+1)(a β (y) 1) =0, y B α A β iii) (a β (z)+1)(b β (z) 1) =0, z A β B β. (5) Under these conditions, if (EQC) is assumed to hold, the correlations A α B α, B α A β and A β A β can be extended by making use of (3). As a result, the following equations hold for any S (ψ). i) (a α (x)+1)(b α (x) 1)=0, x A+ α B α ii) (b α (x)+1)(a β (x) 1)=0, x B+ α A β (6) iii) (a β (x)+1)(b β (x) 1)=0, x A β + Bβ. By using elementary algebra we can deduce that the following equation holds: (a α (x)+1)(b β (x) 1)=0 x A α + B β S (ψ). (7) Thus, (4) is contradicted, because by choosing S (ψ) =S 0 (ψ) we have (a α (x 0 )+ 1)(b β (x 0 ) 1)= 4 and x 0 A α + B β A α + B β. Now we show that no contradiction arises if we replace (EQC) by (seqc). The extension of correlations (5) obtained by applying (seqc) is expressed by i) (a α (x)+1)(b α (x) 1) =0, x A+ α B α (A α B α ) X ii) (b α (y)+1)(a β (y) 1) =0, y A β Bα + (A β B α ) Y iii) (a β (z)+1)(b β (z) 1) =0, z A β + Bβ (Aβ B β ) Z. (8) These extensions no longer imply (7). Indeed, equation (a α (x) 1)(b β (x) 1)=0 can be derived from (8) if all three equations therein hold for the same specimen x, i.e. if x X Y Z. But this last set is empty whatever support S (ψ) of ψ is chosen. 4. (SEQC) AND SIMULTANEOUS ELEMENTS OF REALITY While EPR s argument can be used to infer simultaneous elements of reality for noncommuting properties when (EQC) is adopted, this no longer holds if (seqc) replaces (EQC) [6]. For this reason we provide an ideal experiment enabling to ascribe simultaneous reality to two incompatible observables when the strict interpretation of the criterion of reality is adopted. The observables involved are 0-1 observables, i.e. having 0 and 1 as possible outcomes, represented in the theory by projection operators. In such a case statements (2) and (3), involved in our argument, while derived for two-value observables, turn out to be valid for 0-1 observables. The physical system consists of two separated and non-interacting particles, I and II. Particle I is a spin-5/2 particle localized in a region R I and described in the Hilbert space 490

6 H I ; particle II is a spin-3/2 particle localized in a region R II and described in the Hilbert space H II ; therefore, H I H II is the Hilbert space describing the entire system. We adopt Heisenberg s picture; in the notation for operators, the suffix I (resp. II) denotes an operator of H I (resp., H II ). By A j i i, i = I,II, we denote the projection operator of H i representing the event the spin component in the z-direction is j i ", with j I = 5 2, 3 2, 1 2, 1 2, 3 2, 5 2 and j II = 3 2, 1 2, 1 2, 3 2 ;by j i i we denote their respective eigenvectors relative to the eigenvalue 1. One of two non-commuting observables, E = E I 1 II =(A 1 I + A2 I + A3 I ) 1 II or G = G I 1 II =(B 1 I + B2 I + B3 I ) 1 II, can be measured on system I, where B i I = ψi 1 ψi 1 with i = 1,2,3, and ψ1 1 = 2 1( 5 2 I 3 2 I I 3 2 I), ψ1 2 = 1 2 I and ψ1 3 = 5 2 I. Now we consider projection operators T = 1 I (A 1 II + A2 II ) and Y = 1 I (A 1 II + A3 II ); they turn out to represent commuting properties of particle II, hence a support S 0 (ψ) exists such that T Y /0; furthermore, their measurements require operations confined in the region R II, so that they are separated from, hence commuting (with), both E and G. Let the system be prepared in the entangled state represented by ψ = 3 4 ( 5 2 I I) 1 2 II I 3 2 II ( 1 2 I I) II I 1 2 II. In the state ψ, the following conditions hold: i) E T, i.e. e(x)=t(x) x E T, S (ψ); ii) G Y, i.e. g(z)=y(z) z G Y, S (ψ). (seqc) incorporates the following extensions: i) e(x)=t(x) x E T = X, S (ψ); ii) g(z)=y(z) z G Y = Z, S (ψ). As a consequence, the objective values of both E and G are inferred, in spite of their incompatibility, by actual measurements of T and Y. We notice that in the present ideal experiment, the act of ascertaining the value of E does not affect the value of the other observable G, contrary to what happens in EPR experiment, because the measurement of T and Y are performed in region R II, which is space-like separated from R I. REFERENCES 1. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 (1935) G. Nisticò, A. Sestito, Found. Phys. 41, n. 7, (2011) J.S. Bell, Physics, 1 (1964) L. Hardy, Phys. Rev. Lett., 71 (1993) D.M. Greenberger, M.A. Horne, A. Shimony and A. Zeilinger, Am. J. Phys., 58, (1990). 6. A. Sestito, submitted. 491