In this note we discuss the effect of the unpolarized state in the
spin-correlation measurement of the $^1S_0$ two-proton state produced in
12C(d,2He) reaction at the KVI, Groningen. We show that in the presence of the
unpolarized state the maximal violation of the CHSH-Bell inequality is lower
than the classical limit if the purity of the state is less than $ \sim
\verb+70%+$. In particular, for the KVI experiment the violation of the
CHSH-Bell inequality should be corrected by a factor $\sim\verb+10%+$ from the
pure $^1S_0$ state.Comment: 6 pages, to appear in J. Phys. 

Bell J S

Bennett C H

Clauser J

Guhne O

Hamieh S

Polachic C Rangacharyulu C van den Berg A Harakeh M de Huu M Wortche H Heyse J Baumer C Frekers D Brooke J

S Hamieh

Werner R F

English

arXiv

Abstract: In this note we discuss the effect of the unpolarized state in the spin-correlation measurement of the $^1S_0$ two-proton state produced in 12C(d,2He) reaction at the KVI, Groningen. We show that in the presence of the unpolarized state the maximal violation of the CHSH-Bell inequality is lower than the classical limit if the purity of the state is less than $ sim verb+70%+$. In particular, for the KVI experiment the violation of the CHSH-Bell inequality should be corrected by a factor $simverb+10%+$ from the pure $^1S_0$ state.

Hamieh, S.

University of Groningen Digital Archive

Effect of the unpolarized spin state in spin-correlation measurement of two protons produced in the 12C(d,2He) reaction

Crossref

Effect of the unpolarized spin state in spin-correlation measurement of two protons produced in the
                    12

In this paper we discuss the effect of the unpolarized state in the spin-correlation measurement of the S-1(0) two-proton state produced in the C-12(d, He-2) reaction at the Kernfysisch Versneller Instituut (KVI), Groningen. We show that in the presence of the unpolarized state the maximal violation of the Clauser, Home, Shimony and Holth (CHSH)-Bell inequality is lower than the classical limit if the purity of the state is less than similar to70%. In particular, for the KVI experiment the violation of the CHSH-Bell inequality should be corrected by a factor of similar to10% from the pure S-1(0) state.</p

Hamieh, S

University of Groningen

University of GroningenEffect of the unpolarized spin state in spin-correlation measurement of two protons producedin the 12C(d,2He) reactionHamieh, SPublished in:Journal of Physics A-Mathematical and GeneralDOI:10.1088/0305-4470/37/7/018IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2004Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Hamieh, S. (2004). Effect of the unpolarized spin state in spin-correlation measurement of two protonsproduced in the 12C(d,2He) reaction. Journal of Physics A-Mathematical and General, 37(7), 2777-2780.[PII S0305-4470(04)71131-2]. https://doi.org/10.1088/0305-4470/37/7/018CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Effect of the unpolarized spin state inspin-correlation measurement of two protonsproduced in the 12C(d,2He)reactionDecember 23, 2003S. HamiehKernfysisch Versneller Instituut,Zernikelaan 25, 9747 AA Groningen, The Netherlands.AbstractIn this note we discuss the effect of the unpolarized state in thespin-correlation measurement of the 1S0 two-proton state produced in12C(d,2He) reaction at the KVI, Groningen. We show that in the pres-ence of the unpolarized state the maximal violation of the CHSH-Bellinequality is lower than the classical limit if the purity of the state is lessthan ∼ 70%. In particular, for the KVI experiment the violation of theCHSH-Bell inequality should be corrected by a factor ∼ 10% from the pure1S0 state.11 IntroductionIn an experiment performed at the Kernfysisch Versneller Instituut (KVI),Groningen [5] with the goal to test Bell inequality violation in Nuclear Physics(perhaps to be applied in quantum information physics), the experimental group,by bombarding a 12C target with 170 MeV d, was able to generate a singlet-spin, two-proton state coupled to unpolarized state with  10% contribution. Inthis paper we will analyze the experimental results of this experiment and wewill show that the eect of the unpolarized state is important and could not beneglected.2 CHSH inequalities and entanglement in a mixedensembleBell-type inequalities relating averages of four random dichotomic variables a,a0 and b, b0 representing measurements of spin in directions a^, a^0 and b^, b^0. TheClauser, Horne, Shimony and Holth (CHSH) [3] form of Bell-type inequalitiesfor spin 1=2 case could be written in this formjE(1; ′1; 2; ′2)j = jE(1; 2) + E(1; ′2) + E(′1; 2)− E(′1; ′2)j  2; (1)where i is the analyzer angular setting for the ith particles (i = 1, 2) andE(i; j) is the correlation function dened asE(i; j) =N++ + N−− −N+− −N−+Ntotal: (2)In quantum-theory language the CHSH operator corresponding to the CHSHinequality is represented by an operatorB = a^   ⊗ (b^ + b^0) + a^0 ⊗ (b^− b^0) ; (3)acting in Hilbert space HA ⊗HB in 2⊗2 dimension. The correlation function isgiven by the mean value of the operator a^⊗b^. For a pure state this correlationfunction could be easily computed, e:g: for singlet state we haveE(i; j) = −cos(i − j) : (4)For mixed state, however, the mean value should be averaged over the ensem-ble and therefore the CHSH inequality not a sucient condition to test thepresence of entanglement[7]. Dierent measures of the entanglement have beenproposed in the literature for mixed state1, e.g. entanglement of formation,distillation, relative entropy of entanglement, negativity, etc: : : Here we will usethe entanglement of formation as our measure of the entanglement.1Any measurement of the entanglement should not increase by local operation (e.g. unitarytransformation) and classical communication (e.g. phone calls.), known as LOCC2In a mixed ensemble any bipartite quantum state AB can be written as:AB =14I ⊗ I + A   ⊗ I + I ⊗P   +3∑i,j=1Diji ⊗ j : (5)i are the pauli matrices, I is the identity operator, A and P are vectors inR3. The Dij form a 3  3 matrix called the correlation matrix D. In thisrepresentation of the density matrix the mean value of the CHSH-Bell operatoris given by [4]hBi = a^ [D(b^ + b^0)]+ a^0 [D(b^− b^0)]: (6)Using the representation of the density matrix given in Eq. (5), we characterizeany bipartite quantum state AB by The entanglement measured by the \tangle",  , of the entanglement offormation [6] and dened by = maxf1 − 2 − 3 − 4; 0g ; (7)where the ’s are the square roots of the eigenvalues, in decreasing order,of the matrix, AB(y⊗y?ABy⊗y) and ?AB is the complex conjugationof AB in the computational basis fj+ +i; j+−i; j −+i; j − −ig. The maximum amount of the CHSH-Bell violation of the state AB [4]hBimax = 2√M(AB) : (8)M(AB) is the sum of the two larger eigenvalues of DDy 2. The purity of the state that measures how far the state is from pure stateSL = Tr(2AB) : (9)3 Analysis of the experimental data of the KVIexperimentThe spin state of the two protons produced in the 12C(d,2He) reaction at Ed =170 MeV at KVI [5] is a singlet state mixed with the unpolarized (randomcontamination) state with γ (0  γ  1) controlling the degree of mixing (thedetails of the experimental setup and analysis method of the (d,2He) reaction2In this case the directions bˆ and bˆ′ of the analyser setting are equal to cos(θ)cˆmax ±sin(θ)cˆ′max and the direction aˆ, aˆ′ are equal toDcˆmax||Dcˆmax|| ,Dcˆ′max||Dcˆ′max|| , respectively. cˆmax andcˆ′max are two unit (not unique) and mutually orthogonal vectors in R3 that maximize thefunction ||Dcˆ||2 + ||Dcˆ′||2 (see Ref. [4] for more detail).30.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.0maxKVI limitFigure 1: Plot of hBimaxWerner (dashed line) and  (solid line) versus the purity γ.The dotted line is the Bell limit, the circle is the KVI limit for γ  0:9.were described in detail in Ref. [5]). Given all that, we can write the densitymatrix of such state asW = (1− γ)I4 + γjΨ−ihΨ−j (10)which interpolates between the unpolarized state I=4 and singlet state jΨ−i =(j + −i − j − +i)=√(2). This class of states is called Werner states [7]. Thepurity of Werner states is a monotonic function of γ. Thus, in this paper weuse γ as our measure of purity. Also, for Werner state it is easy to prove usingthe condition noted above thathBimaxWerner = γhBimaxpure : (11)Note that, a violation of the modied Bell-inequality does not exclude anexplanation with a hidden variable theory. In Fig. 1 we plot hBimaxWerner andthe tangle  versus the purity γ. As we can see in this gure the Wernerstate does not violate the Bell inequality if its purity γ is less than 1=p2 70%. However, the entanglement is still non-zero in the Werner state untilγ > 1=3  33%. Therefore, some quantum correlation cannot be seen only4Table 1: Experimental data and quantum theory predictions for a pure singletstates (case 1) and mixed Werner states (case 2) for several violating cases ofthe CHSH-Bell inequality according to the denition given in Eq. 1.CHSH-Bell QM QM Exp. DataInequality case 1 case 2E(50; 0; 25; 75) 2.46 2.21 0.67 2.30E(60; 0; 30; 90) 2.60 2.34 1.21 2.42E(70; 0; 35; 105) 2.72 2.45 1.54 2.76E(80; 0; 40; 120) 2.80 2.52 2.11 2.86E(90; 0; 45; 135) 2.83 2.55 2.23 2.48E(100; 0; 50; 150) 2.79 2.51 2.39 2.87E(110; 0; 55; 165) 2.69 2.34 2.58 2.91E(120; 0; 60; 180) 2.50 2.25 2.75 2.952 1.26 0.85by measuring the violation of the Bell-type inequality because some of them(Werner states) are entangled but still do not violate Bell inequality. Note thatthere is a possible experimental measurement of the entanglement based on theentanglement witness [8] that we think to implement in the future experiment.In Tab. 1 we compare the quantum theory predictions assuming a puresinglet state (case 1) and mixed Werner states (case 2) for the spin of the twodetected protons for several violating cases of the CHSH-Bell inequality. Thevalue of 2 =∑i(Rith−Riexp∆Riexp)2 is given in the bottom of the table for both cases.We have found that 2Werner < 2Singlet as expected. However, we cannot judgethis result as evidence of the mixing of the singlet with the unpolarized statebecause the experimental data suer from large errors.4 ConclusionIn this paper we have discussed the eect of the unpolarized state in the spincorrelations measurement of the 1S0 two proton state produced in 12C(d,2He)reaction at KVI. We have shown that even a small coupling (less than 10%)of the pure singlet state with the unpolarized state changes dramatically theBell-violation value. After introducing the contribution of the unpolarized statewe have found a better 2. The experimental results are suering from a largestatistical error and therefore not conclusive for testing Bell’s inequality, butwith a modied experimental setup, measurements with signicantly improvedprecision will become feasible.5AcknowledgmentsThis work was performed as part of the research program of the Stichting voorFundamenteel Onderzoek der Materie (FOM) with nancial support from theNederlandse Organisatie voor Wetenschappelijk Onderzoek .References[1] S. Hamieh and H. Zaraket,J. Phys. A: Math. Gen. 36, L387-L391 (2003).[2] J.S. Bell, Physics 1, 195 (1964); Rev. Mod. Phys. 38, 447 (1966).[3] J. Clauser, M. Horne, A. Shimony, and R. Holt,Phys. Rev. Lett. 23, 880(1969).[4] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340(1995).[5] C. Polachic, C. Rangacharyulu, A. van den Berg, M. Harakeh, M. deHuu, H. Wortche, J. Heyse, C. Baumer, D. Frekers, and J. Brooke, quant-ph/0303136, submitted to Phys. Lett. A.[6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys.Rev. A 54, 3824 (1996).[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989).[8] O. Guhne, P. Hyllus, D. Brub, E. Ekert, M. Lewenstein, C. Macchiavello,and A. Sampera, Phys. Rev. A 66, 062305 (2002).6

ARTS repository - University of Groningen

University of GroningenEffect of the unpolarized spin state in spin-correlation measurement of two protons producedin the 12C(d,2He) reactionHamieh, SPublished in:Journal of Physics A-Mathematical and GeneralDOI:10.1088/0305-4470/37/7/018IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2004Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Hamieh, S. (2004). Effect of the unpolarized spin state in spin-correlation measurement of two protonsproduced in the 12C(d,2He) reaction. Journal of Physics A-Mathematical and General, 37(7), 2777-2780.Article PII S0305-4470(04)71131-2. https://doi.org/10.1088/0305-4470/37/7/018CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Effect of the unpolarized spin state inspin-correlation measurement of two protonsproduced in the 12C(d,2He)reactionDecember 23, 2003S. HamiehKernfysisch Versneller Instituut,Zernikelaan 25, 9747 AA Groningen, The Netherlands.AbstractIn this note we discuss the effect of the unpolarized state in thespin-correlation measurement of the 1S0 two-proton state produced in12C(d,2He) reaction at the KVI, Groningen. We show that in the pres-ence of the unpolarized state the maximal violation of the CHSH-Bellinequality is lower than the classical limit if the purity of the state is lessthan ∼ 70%. In particular, for the KVI experiment the violation of theCHSH-Bell inequality should be corrected by a factor ∼ 10% from the pure1S0 state.11 IntroductionIn an experiment performed at the Kernfysisch Versneller Instituut (KVI),Groningen [5] with the goal to test Bell inequality violation in Nuclear Physics(perhaps to be applied in quantum information physics), the experimental group,by bombarding a 12C target with 170 MeV d, was able to generate a singlet-spin, two-proton state coupled to unpolarized state with ∼ 10% contribution. Inthis paper we will analyze the experimental results of this experiment and wewill show that the effect of the unpolarized state is important and could not beneglected.2 CHSH inequalities and entanglement in a mixedensembleBell-type inequalities relating averages of four random dichotomic variables a,a′ and b, b′ representing measurements of spin in directions â, â′ and b̂, b̂′. TheClauser, Horne, Shimony and Holth (CHSH) [3] form of Bell-type inequalitiesfor spin 1/2 case could be written in this form|E(φ1, φ′1, φ2, φ′2)| = |E(φ1, φ2) + E(φ1, φ′2) + E(φ′1, φ2)− E(φ′1, φ′2)| ≤ 2, (1)where φi is the analyzer angular setting for the ith particles (i = 1, 2) andE(φi, φj) is the correlation function defined asE(φi, φj) =N++ + N−− −N+− −N−+Ntotal. (2)In quantum-theory language the CHSH operator corresponding to the CHSHinequality is represented by an operatorB = â · σ ⊗ (b̂ + b̂′)·σ + â′·σ ⊗ (b̂− b̂′)·σ , (3)acting in Hilbert space HA ⊗HB in 2⊗2 dimension. The correlation function isgiven by the mean value of the operator âσ⊗b̂σ. For a pure state this correlationfunction could be easily computed, e.g. for singlet state we haveE(φi, φj) = −cos(φi − φj) . (4)For mixed state, however, the mean value should be averaged over the ensem-ble and therefore the CHSH inequality not a sufficient condition to test thepresence of entanglement[7]. Different measures of the entanglement have beenproposed in the literature for mixed state1, e.g. entanglement of formation,distillation, relative entropy of entanglement, negativity, etc. . . Here we will usethe entanglement of formation as our measure of the entanglement.1Any measurement of the entanglement should not increase by local operation (e.g. unitarytransformation) and classical communication (e.g. phone calls.), known as LOCC2In a mixed ensemble any bipartite quantum state ρAB can be written as:ρAB =14I ⊗ I + A · σ ⊗ I + I ⊗P · σ +3∑i,j=1Dijσi ⊗ σj . (5)σi are the pauli matrices, I is the identity operator, A and P are vectors inR3. The Dij form a 3 × 3 matrix called the correlation matrix D. In thisrepresentation of the density matrix the mean value of the CHSH-Bell operatoris given by [4]〈B〉 = â ·[D(b̂ + b̂′)]+ â′ ·[D(b̂− b̂′)]. (6)Using the representation of the density matrix given in Eq. (5), we characterizeany bipartite quantum state ρAB by• The entanglement measured by the “tangle”, τ , of the entanglement offormation [6] and defined byτ = max{λ1 − λ2 − λ3 − λ4, 0} , (7)where the λ’s are the square roots of the eigenvalues, in decreasing order,of the matrix, ρAB(σy⊗σyρ?ABσy⊗σy) and ρ?AB is the complex conjugationof ρAB in the computational basis {|+ +〉, |+−〉, | −+〉, | − −〉}.• The maximum amount of the CHSH-Bell violation of the state ρAB [4]〈B〉max = 2√M(ρAB) . (8)M(ρAB) is the sum of the two larger eigenvalues of DD† 2.• The purity of the state that measures how far the state is from pure stateSL = Tr(ρ2AB) . (9)3 Analysis of the experimental data of the KVIexperimentThe spin state of the two protons produced in the 12C(d,2He) reaction at Ed =170 MeV at KVI [5] is a singlet state mixed with the unpolarized (randomcontamination) state with γ (0 ≤ γ ≤ 1) controlling the degree of mixing (thedetails of the experimental setup and analysis method of the (d,2He) reaction2In this case the directions b̂ and b̂′ of the analyser setting are equal to cos(θ)ĉmax ±sin(θ)ĉ′max and the direction â, â′ are equal toDĉmax||Dĉmax|| ,Dĉ′max||Dĉ′max|| , respectively. ĉmax andĉ′max are two unit (not unique) and mutually orthogonal vectors in R3 that maximize thefunction ||Dĉ||2 + ||Dĉ′||2 (see Ref. [4] for more detail).30.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.0maxKVI limitFigure 1: Plot of 〈B〉maxWerner (dashed line) and τ (solid line) versus the purity γ.The dotted line is the Bell limit, the circle is the KVI limit for γ ∼ 0.9.were described in detail in Ref. [5]). Given all that, we can write the densitymatrix of such state asρW = (1− γ)I4 + γ|Ψ−〉〈Ψ−| (10)which interpolates between the unpolarized state I/4 and singlet state |Ψ−〉 =(| + −〉 − | − +〉)/√(2). This class of states is called Werner states [7]. Thepurity of Werner states is a monotonic function of γ. Thus, in this paper weuse γ as our measure of purity. Also, for Werner state it is easy to prove usingthe condition noted above that〈B〉maxWerner = γ〈B〉maxpure . (11)Note that, a violation of the modified Bell-inequality does not exclude anexplanation with a hidden variable theory. In Fig. 1 we plot 〈B〉maxWerner andthe tangle τ versus the purity γ. As we can see in this figure the Wernerstate does not violate the Bell inequality if its purity γ is less than 1/√2 ∼70%. However, the entanglement is still non-zero in the Werner state untilγ > 1/3 ∼ 33%. Therefore, some quantum correlation cannot be seen only4Table 1: Experimental data and quantum theory predictions for a pure singletstates (case 1) and mixed Werner states (case 2) for several violating cases ofthe CHSH-Bell inequality according to the definition given in Eq. 1.CHSH-Bell QM QM Exp. DataInequality case 1 case 2E(50◦, 0◦, 25◦, 75◦) 2.46 2.21 0.67± 2.30E(60◦, 0◦, 30◦, 90◦) 2.60 2.34 1.21± 2.42E(70◦, 0◦, 35◦, 105◦) 2.72 2.45 1.54± 2.76E(80◦, 0◦, 40◦, 120◦) 2.80 2.52 2.11± 2.86E(90◦, 0◦, 45◦, 135◦) 2.83 2.55 2.23± 2.48E(100◦, 0◦, 50◦, 150◦) 2.79 2.51 2.39± 2.87E(110◦, 0◦, 55◦, 165◦) 2.69 2.34 2.58± 2.91E(120◦, 0◦, 60◦, 180◦) 2.50 2.25 2.75± 2.95χ2 1.26 0.85by measuring the violation of the Bell-type inequality because some of them(Werner states) are entangled but still do not violate Bell inequality. Note thatthere is a possible experimental measurement of the entanglement based on theentanglement witness [8] that we think to implement in the future experiment.In Tab. 1 we compare the quantum theory predictions assuming a puresinglet state (case 1) and mixed Werner states (case 2) for the spin of the twodetected protons for several violating cases of the CHSH-Bell inequality. Thevalue of χ2 =∑i(Rith−Riexp∆Riexp)2 is given in the bottom of the table for both cases.We have found that χ2Werner < χ2Singlet as expected. However, we cannot judgethis result as evidence of the mixing of the singlet with the unpolarized statebecause the experimental data suffer from large errors.4 ConclusionIn this paper we have discussed the effect of the unpolarized state in the spincorrelations measurement of the 1S0 two proton state produced in 12C(d,2He)reaction at KVI. We have shown that even a small coupling (less than 10%)of the pure singlet state with the unpolarized state changes dramatically theBell-violation value. After introducing the contribution of the unpolarized statewe have found a better χ2. The experimental results are suffering from a largestatistical error and therefore not conclusive for testing Bell’s inequality, butwith a modified experimental setup, measurements with significantly improvedprecision will become feasible.5AcknowledgmentsThis work was performed as part of the research program of the Stichting voorFundamenteel Onderzoek der Materie (FOM) with financial support from theNederlandse Organisatie voor Wetenschappelijk Onderzoek .References[1] S. Hamieh and H. Zaraket,J. Phys. A: Math. Gen. 36, L387-L391 (2003).[2] J.S. Bell, Physics 1, 195 (1964); Rev. Mod. Phys. 38, 447 (1966).[3] J. Clauser, M. Horne, A. Shimony, and R. Holt,Phys. Rev. Lett. 23, 880(1969).[4] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340(1995).[5] C. Polachic, C. Rangacharyulu, A. van den Berg, M. Harakeh, M. deHuu, H. Wortche, J. Heyse, C. Baumer, D. Frekers, and J. Brooke, quant-ph/0303136, submitted to Phys. Lett. A.[6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys.Rev. A 54, 3824 (1996).[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989).[8] O. Guhne, P. Hyllus, D. Brub, E. Ekert, M. Lewenstein, C. Macchiavello,and A. Sampera, Phys. Rev. A 66, 062305 (2002).6

University of GroningenEffect of the unpolarized spin state in spin-correlation measurement of two protons producedin the 12C(d,2He) reactionHamieh, SPublished in:Journal of Physics A-Mathematical and GeneralDOI:10.1088/0305-4470/37/7/018IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2004Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Hamieh, S. (2004). Effect of the unpolarized spin state in spin-correlation measurement of two protonsproduced in the 12C(d,2He) reaction. Journal of Physics A-Mathematical and General, 37(7), 2777-2780.[PII S0305-4470(04)71131-2]. https://doi.org/10.1088/0305-4470/37/7/018CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 11-12-2019Effect of the unpolarized spin state inspin-correlation measurement of two protonsproduced in the 12C(d,2He)reactionDecember 23, 2003S. HamiehKernfysisch Versneller Instituut,Zernikelaan 25, 9747 AA Groningen, The Netherlands.AbstractIn this note we discuss the effect of the unpolarized state in thespin-correlation measurement of the 1S0 two-proton state produced in12C(d,2He) reaction at the KVI, Groningen. We show that in the pres-ence of the unpolarized state the maximal violation of the CHSH-Bellinequality is lower than the classical limit if the purity of the state is lessthan ∼ 70%. In particular, for the KVI experiment the violation of theCHSH-Bell inequality should be corrected by a factor ∼ 10% from the pure1S0 state.11 IntroductionIn an experiment performed at the Kernfysisch Versneller Instituut (KVI),Groningen [5] with the goal to test Bell inequality violation in Nuclear Physics(perhaps to be applied in quantum information physics), the experimental group,by bombarding a 12C target with 170 MeV d, was able to generate a singlet-spin, two-proton state coupled to unpolarized state with  10% contribution. Inthis paper we will analyze the experimental results of this experiment and wewill show that the eect of the unpolarized state is important and could not beneglected.2 CHSH inequalities and entanglement in a mixedensembleBell-type inequalities relating averages of four random dichotomic variables a,a0 and b, b0 representing measurements of spin in directions a^, a^0 and b^, b^0. TheClauser, Horne, Shimony and Holth (CHSH) [3] form of Bell-type inequalitiesfor spin 1=2 case could be written in this formjE(1; ′1; 2; ′2)j = jE(1; 2) + E(1; ′2) + E(′1; 2)− E(′1; ′2)j  2; (1)where i is the analyzer angular setting for the ith particles (i = 1, 2) andE(i; j) is the correlation function dened asE(i; j) =N++ + N−− −N+− −N−+Ntotal: (2)In quantum-theory language the CHSH operator corresponding to the CHSHinequality is represented by an operatorB = a^   ⊗ (b^ + b^0) + a^0 ⊗ (b^− b^0) ; (3)acting in Hilbert space HA ⊗HB in 2⊗2 dimension. The correlation function isgiven by the mean value of the operator a^⊗b^. For a pure state this correlationfunction could be easily computed, e:g: for singlet state we haveE(i; j) = −cos(i − j) : (4)For mixed state, however, the mean value should be averaged over the ensem-ble and therefore the CHSH inequality not a sucient condition to test thepresence of entanglement[7]. Dierent measures of the entanglement have beenproposed in the literature for mixed state1, e.g. entanglement of formation,distillation, relative entropy of entanglement, negativity, etc: : : Here we will usethe entanglement of formation as our measure of the entanglement.1Any measurement of the entanglement should not increase by local operation (e.g. unitarytransformation) and classical communication (e.g. phone calls.), known as LOCC2In a mixed ensemble any bipartite quantum state AB can be written as:AB =14I ⊗ I + A   ⊗ I + I ⊗P   +3∑i,j=1Diji ⊗ j : (5)i are the pauli matrices, I is the identity operator, A and P are vectors inR3. The Dij form a 3  3 matrix called the correlation matrix D. In thisrepresentation of the density matrix the mean value of the CHSH-Bell operatoris given by [4]hBi = a^ [D(b^ + b^0)]+ a^0 [D(b^− b^0)]: (6)Using the representation of the density matrix given in Eq. (5), we characterizeany bipartite quantum state AB by The entanglement measured by the \tangle",  , of the entanglement offormation [6] and dened by = maxf1 − 2 − 3 − 4; 0g ; (7)where the ’s are the square roots of the eigenvalues, in decreasing order,of the matrix, AB(y⊗y?ABy⊗y) and ?AB is the complex conjugationof AB in the computational basis fj+ +i; j+−i; j −+i; j − −ig. The maximum amount of the CHSH-Bell violation of the state AB [4]hBimax = 2√M(AB) : (8)M(AB) is the sum of the two larger eigenvalues of DDy 2. The purity of the state that measures how far the state is from pure stateSL = Tr(2AB) : (9)3 Analysis of the experimental data of the KVIexperimentThe spin state of the two protons produced in the 12C(d,2He) reaction at Ed =170 MeV at KVI [5] is a singlet state mixed with the unpolarized (randomcontamination) state with γ (0  γ  1) controlling the degree of mixing (thedetails of the experimental setup and analysis method of the (d,2He) reaction2In this case the directions bˆ and bˆ′ of the analyser setting are equal to cos(θ)cˆmax ±sin(θ)cˆ′max and the direction aˆ, aˆ′ are equal toDcˆmax||Dcˆmax|| ,Dcˆ′max||Dcˆ′max|| , respectively. cˆmax andcˆ′max are two unit (not unique) and mutually orthogonal vectors in R3 that maximize thefunction ||Dcˆ||2 + ||Dcˆ′||2 (see Ref. [4] for more detail).30.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.0maxKVI limitFigure 1: Plot of hBimaxWerner (dashed line) and  (solid line) versus the purity γ.The dotted line is the Bell limit, the circle is the KVI limit for γ  0:9.were described in detail in Ref. [5]). Given all that, we can write the densitymatrix of such state asW = (1− γ)I4 + γjΨ−ihΨ−j (10)which interpolates between the unpolarized state I=4 and singlet state jΨ−i =(j + −i − j − +i)=√(2). This class of states is called Werner states [7]. Thepurity of Werner states is a monotonic function of γ. Thus, in this paper weuse γ as our measure of purity. Also, for Werner state it is easy to prove usingthe condition noted above thathBimaxWerner = γhBimaxpure : (11)Note that, a violation of the modied Bell-inequality does not exclude anexplanation with a hidden variable theory. In Fig. 1 we plot hBimaxWerner andthe tangle  versus the purity γ. As we can see in this gure the Wernerstate does not violate the Bell inequality if its purity γ is less than 1=p2 70%. However, the entanglement is still non-zero in the Werner state untilγ > 1=3  33%. Therefore, some quantum correlation cannot be seen only4Table 1: Experimental data and quantum theory predictions for a pure singletstates (case 1) and mixed Werner states (case 2) for several violating cases ofthe CHSH-Bell inequality according to the denition given in Eq. 1.CHSH-Bell QM QM Exp. DataInequality case 1 case 2E(50; 0; 25; 75) 2.46 2.21 0.67 2.30E(60; 0; 30; 90) 2.60 2.34 1.21 2.42E(70; 0; 35; 105) 2.72 2.45 1.54 2.76E(80; 0; 40; 120) 2.80 2.52 2.11 2.86E(90; 0; 45; 135) 2.83 2.55 2.23 2.48E(100; 0; 50; 150) 2.79 2.51 2.39 2.87E(110; 0; 55; 165) 2.69 2.34 2.58 2.91E(120; 0; 60; 180) 2.50 2.25 2.75 2.952 1.26 0.85by measuring the violation of the Bell-type inequality because some of them(Werner states) are entangled but still do not violate Bell inequality. Note thatthere is a possible experimental measurement of the entanglement based on theentanglement witness [8] that we think to implement in the future experiment.In Tab. 1 we compare the quantum theory predictions assuming a puresinglet state (case 1) and mixed Werner states (case 2) for the spin of the twodetected protons for several violating cases of the CHSH-Bell inequality. Thevalue of 2 =∑i(Rith−Riexp∆Riexp)2 is given in the bottom of the table for both cases.We have found that 2Werner < 2Singlet as expected. However, we cannot judgethis result as evidence of the mixing of the singlet with the unpolarized statebecause the experimental data suer from large errors.4 ConclusionIn this paper we have discussed the eect of the unpolarized state in the spincorrelations measurement of the 1S0 two proton state produced in 12C(d,2He)reaction at KVI. We have shown that even a small coupling (less than 10%)of the pure singlet state with the unpolarized state changes dramatically theBell-violation value. After introducing the contribution of the unpolarized statewe have found a better 2. The experimental results are suering from a largestatistical error and therefore not conclusive for testing Bell’s inequality, butwith a modied experimental setup, measurements with signicantly improvedprecision will become feasible.5AcknowledgmentsThis work was performed as part of the research program of the Stichting voorFundamenteel Onderzoek der Materie (FOM) with nancial support from theNederlandse Organisatie voor Wetenschappelijk Onderzoek .References[1] S. Hamieh and H. Zaraket,J. Phys. A: Math. Gen. 36, L387-L391 (2003).[2] J.S. Bell, Physics 1, 195 (1964); Rev. Mod. Phys. 38, 447 (1966).[3] J. Clauser, M. Horne, A. Shimony, and R. Holt,Phys. Rev. Lett. 23, 880(1969).[4] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340(1995).[5] C. Polachic, C. Rangacharyulu, A. van den Berg, M. Harakeh, M. deHuu, H. Wortche, J. Heyse, C. Baumer, D. Frekers, and J. Brooke, quant-ph/0303136, submitted to Phys. Lett. A.[6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys.Rev. A 54, 3824 (1996).[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989).[8] O. Guhne, P. Hyllus, D. Brub, E. Ekert, M. Lewenstein, C. Macchiavello,and A. Sampera, Phys. Rev. A 66, 062305 (2002).6

University of GroningenEffect of the unpolarized spin state in spin-correlation measurement of two protons producedin the 12C(d,2He) reactionHamieh, SPublished in:Journal of Physics A-Mathematical and GeneralDOI:10.1088/0305-4470/37/7/018IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2004Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Hamieh, S. (2004). Effect of the unpolarized spin state in spin-correlation measurement of two protonsproduced in the 12C(d,2He) reaction. Journal of Physics A-Mathematical and General, 37(7), 2777-2780.[PII S0305-4470(04)71131-2]. https://doi.org/10.1088/0305-4470/37/7/018CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Effect of the unpolarized spin state inspin-correlation measurement of two protonsproduced in the 12C(d,2He)reactionDecember 23, 2003S. HamiehKernfysisch Versneller Instituut,Zernikelaan 25, 9747 AA Groningen, The Netherlands.AbstractIn this note we discuss the effect of the unpolarized state in thespin-correlation measurement of the 1S0 two-proton state produced in12C(d,2He) reaction at the KVI, Groningen. We show that in the pres-ence of the unpolarized state the maximal violation of the CHSH-Bellinequality is lower than the classical limit if the purity of the state is lessthan ∼ 70%. In particular, for the KVI experiment the violation of theCHSH-Bell inequality should be corrected by a factor ∼ 10% from the pure1S0 state.11 IntroductionIn an experiment performed at the Kernfysisch Versneller Instituut (KVI),Groningen [5] with the goal to test Bell inequality violation in Nuclear Physics(perhaps to be applied in quantum information physics), the experimental group,by bombarding a 12C target with 170 MeV d, was able to generate a singlet-spin, two-proton state coupled to unpolarized state with ∼ 10% contribution. Inthis paper we will analyze the experimental results of this experiment and wewill show that the effect of the unpolarized state is important and could not beneglected.2 CHSH inequalities and entanglement in a mixedensembleBell-type inequalities relating averages of four random dichotomic variables a,a′ and b, b′ representing measurements of spin in directions â, â′ and b̂, b̂′. TheClauser, Horne, Shimony and Holth (CHSH) [3] form of Bell-type inequalitiesfor spin 1/2 case could be written in this form|E(φ1, φ′1, φ2, φ′2)| = |E(φ1, φ2) + E(φ1, φ′2) + E(φ′1, φ2)− E(φ′1, φ′2)| ≤ 2, (1)where φi is the analyzer angular setting for the ith particles (i = 1, 2) andE(φi, φj) is the correlation function defined asE(φi, φj) =N++ + N−− −N+− −N−+Ntotal. (2)In quantum-theory language the CHSH operator corresponding to the CHSHinequality is represented by an operatorB = â · σ ⊗ (b̂ + b̂′)·σ + â′·σ ⊗ (b̂− b̂′)·σ , (3)acting in Hilbert space HA ⊗HB in 2⊗2 dimension. The correlation function isgiven by the mean value of the operator âσ⊗b̂σ. For a pure state this correlationfunction could be easily computed, e.g. for singlet state we haveE(φi, φj) = −cos(φi − φj) . (4)For mixed state, however, the mean value should be averaged over the ensem-ble and therefore the CHSH inequality not a sufficient condition to test thepresence of entanglement[7]. Different measures of the entanglement have beenproposed in the literature for mixed state1, e.g. entanglement of formation,distillation, relative entropy of entanglement, negativity, etc. . . Here we will usethe entanglement of formation as our measure of the entanglement.1Any measurement of the entanglement should not increase by local operation (e.g. unitarytransformation) and classical communication (e.g. phone calls.), known as LOCC2In a mixed ensemble any bipartite quantum state ρAB can be written as:ρAB =14I ⊗ I + A · σ ⊗ I + I ⊗P · σ +3∑i,j=1Dijσi ⊗ σj . (5)σi are the pauli matrices, I is the identity operator, A and P are vectors inR3. The Dij form a 3 × 3 matrix called the correlation matrix D. In thisrepresentation of the density matrix the mean value of the CHSH-Bell operatoris given by [4]〈B〉 = â ·[D(b̂ + b̂′)]+ â′ ·[D(b̂− b̂′)]. (6)Using the representation of the density matrix given in Eq. (5), we characterizeany bipartite quantum state ρAB by• The entanglement measured by the “tangle”, τ , of the entanglement offormation [6] and defined byτ = max{λ1 − λ2 − λ3 − λ4, 0} , (7)where the λ’s are the square roots of the eigenvalues, in decreasing order,of the matrix, ρAB(σy⊗σyρ?ABσy⊗σy) and ρ?AB is the complex conjugationof ρAB in the computational basis {|+ +〉, |+−〉, | −+〉, | − −〉}.• The maximum amount of the CHSH-Bell violation of the state ρAB [4]〈B〉max = 2√M(ρAB) . (8)M(ρAB) is the sum of the two larger eigenvalues of DD† 2.• The purity of the state that measures how far the state is from pure stateSL = Tr(ρ2AB) . (9)3 Analysis of the experimental data of the KVIexperimentThe spin state of the two protons produced in the 12C(d,2He) reaction at Ed =170 MeV at KVI [5] is a singlet state mixed with the unpolarized (randomcontamination) state with γ (0 ≤ γ ≤ 1) controlling the degree of mixing (thedetails of the experimental setup and analysis method of the (d,2He) reaction2In this case the directions b̂ and b̂′ of the analyser setting are equal to cos(θ)ĉmax ±sin(θ)ĉ′max and the direction â, â′ are equal toDĉmax||Dĉmax|| ,Dĉ′max||Dĉ′max|| , respectively. ĉmax andĉ′max are two unit (not unique) and mutually orthogonal vectors in R3 that maximize thefunction ||Dĉ||2 + ||Dĉ′||2 (see Ref. [4] for more detail).30.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.0maxKVI limitFigure 1: Plot of 〈B〉maxWerner (dashed line) and τ (solid line) versus the purity γ.The dotted line is the Bell limit, the circle is the KVI limit for γ ∼ 0.9.were described in detail in Ref. [5]). Given all that, we can write the densitymatrix of such state asρW = (1− γ)I4 + γ|Ψ−〉〈Ψ−| (10)which interpolates between the unpolarized state I/4 and singlet state |Ψ−〉 =(| + −〉 − | − +〉)/√(2). This class of states is called Werner states [7]. Thepurity of Werner states is a monotonic function of γ. Thus, in this paper weuse γ as our measure of purity. Also, for Werner state it is easy to prove usingthe condition noted above that〈B〉maxWerner = γ〈B〉maxpure . (11)Note that, a violation of the modified Bell-inequality does not exclude anexplanation with a hidden variable theory. In Fig. 1 we plot 〈B〉maxWerner andthe tangle τ versus the purity γ. As we can see in this figure the Wernerstate does not violate the Bell inequality if its purity γ is less than 1/√2 ∼70%. However, the entanglement is still non-zero in the Werner state untilγ > 1/3 ∼ 33%. Therefore, some quantum correlation cannot be seen only4Table 1: Experimental data and quantum theory predictions for a pure singletstates (case 1) and mixed Werner states (case 2) for several violating cases ofthe CHSH-Bell inequality according to the definition given in Eq. 1.CHSH-Bell QM QM Exp. DataInequality case 1 case 2E(50◦, 0◦, 25◦, 75◦) 2.46 2.21 0.67± 2.30E(60◦, 0◦, 30◦, 90◦) 2.60 2.34 1.21± 2.42E(70◦, 0◦, 35◦, 105◦) 2.72 2.45 1.54± 2.76E(80◦, 0◦, 40◦, 120◦) 2.80 2.52 2.11± 2.86E(90◦, 0◦, 45◦, 135◦) 2.83 2.55 2.23± 2.48E(100◦, 0◦, 50◦, 150◦) 2.79 2.51 2.39± 2.87E(110◦, 0◦, 55◦, 165◦) 2.69 2.34 2.58± 2.91E(120◦, 0◦, 60◦, 180◦) 2.50 2.25 2.75± 2.95χ2 1.26 0.85by measuring the violation of the Bell-type inequality because some of them(Werner states) are entangled but still do not violate Bell inequality. Note thatthere is a possible experimental measurement of the entanglement based on theentanglement witness [8] that we think to implement in the future experiment.In Tab. 1 we compare the quantum theory predictions assuming a puresinglet state (case 1) and mixed Werner states (case 2) for the spin of the twodetected protons for several violating cases of the CHSH-Bell inequality. Thevalue of χ2 =∑i(Rith−Riexp∆Riexp)2 is given in the bottom of the table for both cases.We have found that χ2Werner < χ2Singlet as expected. However, we cannot judgethis result as evidence of the mixing of the singlet with the unpolarized statebecause the experimental data suffer from large errors.4 ConclusionIn this paper we have discussed the effect of the unpolarized state in the spincorrelations measurement of the 1S0 two proton state produced in 12C(d,2He)reaction at KVI. We have shown that even a small coupling (less than 10%)of the pure singlet state with the unpolarized state changes dramatically theBell-violation value. After introducing the contribution of the unpolarized statewe have found a better χ2. The experimental results are suffering from a largestatistical error and therefore not conclusive for testing Bell’s inequality, butwith a modified experimental setup, measurements with significantly improvedprecision will become feasible.5AcknowledgmentsThis work was performed as part of the research program of the Stichting voorFundamenteel Onderzoek der Materie (FOM) with financial support from theNederlandse Organisatie voor Wetenschappelijk Onderzoek .References[1] S. Hamieh and H. Zaraket,J. Phys. A: Math. Gen. 36, L387-L391 (2003).[2] J.S. Bell, Physics 1, 195 (1964); Rev. Mod. Phys. 38, 447 (1966).[3] J. Clauser, M. Horne, A. Shimony, and R. Holt,Phys. Rev. Lett. 23, 880(1969).[4] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340(1995).[5] C. Polachic, C. Rangacharyulu, A. van den Berg, M. Harakeh, M. deHuu, H. Wortche, J. Heyse, C. Baumer, D. Frekers, and J. Brooke, quant-ph/0303136, submitted to Phys. Lett. A.[6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys.Rev. A 54, 3824 (1996).[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989).[8] O. Guhne, P. Hyllus, D. Brub, E. Ekert, M. Lewenstein, C. Macchiavello,and A. Sampera, Phys. Rev. A 66, 062305 (2002).6

University of Groningen Research Database

University of Groningen
Effect of the unpolarized spin state in spin-correlation measurement of two protons produced
in the 12C(d,2He) reaction
Hamieh, S
Published in:
Journal of Physics A-Mathematical and General
DOI:
10.1088/0305-4470/37/7/018
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2004
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Hamieh, S. (2004). Effect of the unpolarized spin state in spin-correlation measurement of two protons
produced in the 12C(d,2He) reaction. Journal of Physics A-Mathematical and General, 37(7), 2777-2780.
[PII S0305-4470(04)71131-2]. DOI: 10.1088/0305-4470/37/7/018
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 10-02-2018
Effect of the unpolarized spin state in
spin-correlation measurement of two protons
produced in the 12C(d,2He)reaction
December 23, 2003
S. Hamieh
Kernfysisch Versneller Instituut,
Zernikelaan 25, 9747 AA Groningen, The Netherlands.
Abstract
In this note we discuss the effect of the unpolarized state in the
spin-correlation measurement of the 1S0 two-proton state produced in
12C(d,2He) reaction at the KVI, Groningen. We show that in the pres-
ence of the unpolarized state the maximal violation of the CHSH-Bell
inequality is lower than the classical limit if the purity of the state is less
than ∼ 70%. In particular, for the KVI experiment the violation of the
CHSH-Bell inequality should be corrected by a factor ∼ 10% from the pure
1S0 state.
1
1 Introduction
In an experiment performed at the Kernfysisch Versneller Instituut (KVI),
Groningen [5] with the goal to test Bell inequality violation in Nuclear Physics
(perhaps to be applied in quantum information physics), the experimental group,
by bombarding a 12C target with 170 MeV d, was able to generate a singlet-
spin, two-proton state coupled to unpolarized state with  10% contribution. In
this paper we will analyze the experimental results of this experiment and we
will show that the eect of the unpolarized state is important and could not be
neglected.
2 CHSH inequalities and entanglement in a mixed
ensemble
Bell-type inequalities relating averages of four random dichotomic variables a,
a0 and b, b0 representing measurements of spin in directions a^, a^0 and b^, b^0. The
Clauser, Horne, Shimony and Holth (CHSH) [3] form of Bell-type inequalities
for spin 1=2 case could be written in this form
jE(1; ′1; 2; 
′
2)j = jE(1; 2) + E(1; 
′
2) + E(
′
1; 2)− E(
′
1; 
′
2)j  2; (1)
where i is the analyzer angular setting for the ith particles (i = 1, 2) and
E(i; j) is the correlation function dened as
E(i; j) =
N++ + N−− −N+− −N−+
Ntotal
: (2)
In quantum-theory language the CHSH operator corresponding to the CHSH
inequality is represented by an operator
B = a^   ⊗ (b^ + b^0) + a^0 ⊗ (b^− b^0) ; (3)
acting in Hilbert space HA ⊗HB in 2⊗2 dimension. The correlation function is
given by the mean value of the operator a^⊗b^. For a pure state this correlation
function could be easily computed, e:g: for singlet state we have
E(i; j) = −cos(i − j) : (4)
For mixed state, however, the mean value should be averaged over the ensem-
ble and therefore the CHSH inequality not a sucient condition to test the
presence of entanglement[7]. Dierent measures of the entanglement have been
proposed in the literature for mixed state1, e.g. entanglement of formation,
distillation, relative entropy of entanglement, negativity, etc: : : Here we will use
the entanglement of formation as our measure of the entanglement.
1Any measurement of the entanglement should not increase by local operation (e.g. unitary
transformation) and classical communication (e.g. phone calls.), known as LOCC
2
In a mixed ensemble any bipartite quantum state AB can be written as:
AB =
1
4

I ⊗ I + A   ⊗ I + I ⊗P   +
3∑
i,j=1
Diji ⊗ j

 : (5)
i are the pauli matrices, I is the identity operator, A and P are vectors in
R3. The Dij form a 3  3 matrix called the correlation matrix D. In this
representation of the density matrix the mean value of the CHSH-Bell operator
is given by [4]
hBi = a^ 
[
D(b^ + b^0)
]
+ a^0 
[
D(b^− b^0)
]
: (6)
Using the representation of the density matrix given in Eq. (5), we characterize
any bipartite quantum state AB by
 The entanglement measured by the \tangle",  , of the entanglement of
formation [6] and dened by
 = maxf1 − 2 − 3 − 4; 0g ; (7)
where the ’s are the square roots of the eigenvalues, in decreasing order,
of the matrix, AB(y⊗y?ABy⊗y) and ?AB is the complex conjugation
of AB in the computational basis fj+ +i; j+−i; j −+i; j − −ig.
 The maximum amount of the CHSH-Bell violation of the state AB [4]
hBimax = 2
√
M(AB) : (8)
M(AB) is the sum of the two larger eigenvalues of DDy 2.
 The purity of the state that measures how far the state is from pure state
SL = Tr(2AB) : (9)
3 Analysis of the experimental data of the KVI
experiment
The spin state of the two protons produced in the 12C(d,2He) reaction at Ed =
170 MeV at KVI [5] is a singlet state mixed with the unpolarized (random
contamination) state with γ (0  γ  1) controlling the degree of mixing (the
details of the experimental setup and analysis method of the (d,2He) reaction
2In this case the directions bˆ and bˆ′ of the analyser setting are equal to cos(θ)cˆmax ±
sin(θ)cˆ′max and the direction aˆ, aˆ′ are equal to
Dcˆmax
||Dcˆmax|| ,
Dcˆ′max
||Dcˆ′max|| , respectively. cˆmax and
cˆ′max are two unit (not unique) and mutually orthogonal vectors in R3 that maximize the
function ||Dcˆ||2 + ||Dcˆ′||2 (see Ref. [4] for more detail).
3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
max
KVI limit
Figure 1: Plot of hBimaxWerner (dashed line) and  (solid line) versus the purity γ.
The dotted line is the Bell limit, the circle is the KVI limit for γ  0:9.
were described in detail in Ref. [5]). Given all that, we can write the density
matrix of such state as
W = (1− γ)I4 + γjΨ
−ihΨ−j (10)
which interpolates between the unpolarized state I=4 and singlet state jΨ−i =
(j + −i − j − +i)=√(2). This class of states is called Werner states [7]. The
purity of Werner states is a monotonic function of γ. Thus, in this paper we
use γ as our measure of purity. Also, for Werner state it is easy to prove using
the condition noted above that
hBimaxWerner = γhBimaxpure : (11)
Note that, a violation of the modied Bell-inequality does not exclude an
explanation with a hidden variable theory. In Fig. 1 we plot hBimaxWerner and
the tangle  versus the purity γ. As we can see in this gure the Werner
state does not violate the Bell inequality if its purity γ is less than 1=
p
2 
70%. However, the entanglement is still non-zero in the Werner state until
γ > 1=3  33%. Therefore, some quantum correlation cannot be seen only
4
Table 1: Experimental data and quantum theory predictions for a pure singlet
states (case 1) and mixed Werner states (case 2) for several violating cases of
the CHSH-Bell inequality according to the denition given in Eq. 1.
CHSH-Bell QM QM Exp. Data
Inequality case 1 case 2
E(50; 0; 25; 75) 2.46 2.21 0.67 2.30
E(60; 0; 30; 90) 2.60 2.34 1.21 2.42
E(70; 0; 35; 105) 2.72 2.45 1.54 2.76
E(80; 0; 40; 120) 2.80 2.52 2.11 2.86
E(90; 0; 45; 135) 2.83 2.55 2.23 2.48
E(100; 0; 50; 150) 2.79 2.51 2.39 2.87
E(110; 0; 55; 165) 2.69 2.34 2.58 2.91
E(120; 0; 60; 180) 2.50 2.25 2.75 2.95
2 1.26 0.85
by measuring the violation of the Bell-type inequality because some of them
(Werner states) are entangled but still do not violate Bell inequality. Note that
there is a possible experimental measurement of the entanglement based on the
entanglement witness [8] that we think to implement in the future experiment.
In Tab. 1 we compare the quantum theory predictions assuming a pure
singlet state (case 1) and mixed Werner states (case 2) for the spin of the two
detected protons for several violating cases of the CHSH-Bell inequality. The
value of 2 =
∑
i(
Rith−Riexp
∆Riexp
)2 is given in the bottom of the table for both cases.
We have found that 2Werner < 
2
Singlet as expected. However, we cannot judge
this result as evidence of the mixing of the singlet with the unpolarized state
because the experimental data suer from large errors.
4 Conclusion
In this paper we have discussed the eect of the unpolarized state in the spin
correlations measurement of the 1S0 two proton state produced in 12C(d,2He)
reaction at KVI. We have shown that even a small coupling (less than 10%)
of the pure singlet state with the unpolarized state changes dramatically the
Bell-violation value. After introducing the contribution of the unpolarized state
we have found a better 2. The experimental results are suering from a large
statistical error and therefore not conclusive for testing Bell’s inequality, but
with a modied experimental setup, measurements with signicantly improved
precision will become feasible.
5
Acknowledgments
This work was performed as part of the research program of the Stichting voor
Fundamenteel Onderzoek der Materie (FOM) with nancial support from the
Nederlandse Organisatie voor Wetenschappelijk Onderzoek .
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[3] J. Clauser, M. Horne, A. Shimony, and R. Holt,Phys. Rev. Lett. 23, 880
(1969).
[4] R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340
(1995).
[5] C. Polachic, C. Rangacharyulu, A. van den Berg, M. Harakeh, M. de
Huu, H. Wortche, J. Heyse, C. Baumer, D. Frekers, and J. Brooke, quant-
ph/0303136, submitted to Phys. Lett. A.
[6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys.
Rev. A 54, 3824 (1996).
[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989).
[8] O. Guhne, P. Hyllus, D. Brub, E. Ekert, M. Lewenstein, C. Macchiavello,
and A. Sampera, Phys. Rev. A 66, 062305 (2002).
6

In this paper we discuss the effect of the unpolarized state in the spin-correlation measurement of the S-1(0) two-proton state produced in the C-12(d, He-2) reaction at the Kernfysisch Versneller Instituut (KVI), Groningen. We show that in the presence of the unpolarized state the maximal violation of the Clauser, Home, Shimony and Holth (CHSH)-Bell inequality is lower than the classical limit if the purity of the state is less than similar to70%. In particular, for the KVI experiment the violation of the CHSH-Bell inequality should be corrected by a factor of similar to10% from the pure S-1(0) state

Effect of the unpolarized spin state in spin-correlation measurement of two protons produced in the 12C(d,2He) reaction

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