August 26, :27 WSPC/INSTRUCTION FILE ws-ijmpcbellcomputerref 2 CHSH AND A SIMPLE VIOLATING PROGRAM BASED ON LOCALITY.

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1 International Journal of Modern Physics C c World Scientific Publishing Company CHSH AND A SIMPLE VIOLATING PROGRAM BASED ON LOCALITY. HAN GEURDES CvdLijnstraat 164 The Hague, 2593NN, Netherlands han.geurdes@gmail.com Received 19 August 2013 In the present paper a simple computer code shows CHSH can be violated with strict LHV. Subject Classification: PACS 3.65 Keywords: CHSH & computer code violating the CHSH. 1. Introduction. Einstein never fully agreed with the foundational nature of quantum mechanics [1]. He wanted to explain a.o. Heisenberg s uncertainty and the correlation between particles with additional local hidden variables unknown to physics. In 1964, John Bell [2] reformulated Einstein s idea of local hidden variables (LHVs) into a form that allowed experimental research. Bell s probabilistic nature of the argument turned experimentation into checking of violation of inequalities [3], [4]. Especially strong is the claim that it is impossible with local means to violate the CHSH inequality even if the history of the system may be included [6]. We demonstrate in the present paper that a simple code exists such that this violation is possible. 2. Computer code Preliminaries. CHSH no-go conclusions are tested with the attempt to construct a computer program that shows a small but significant violation of the CHSH contrast. In the construction of the code use is made of simple LHV principles. The program employs the following expression of the CHSH contrast CHSH = 2 + 2N(= 1, 1) N(1, 1) 2N(= 1, 2) N(1, 2) 2N(= 2, 1) N(2, 1) 2N(= 2, 2) N(2, 2) (1) 1

2 2 Han Geurdes Here we have CHSH = Ê(1, 1) Ê(1, 2) Ê(2, 1) Ê(2, 2) and Ê(a, b) = N(= a, b) N( a, b) N(a, b) In the ideal experiment N(a, b) = N(= a, b) + N( a, b). Let us start the explanation of the code with the following remarks. The response in one measurement at A and B is denoted with S A and S B. At first instance let us take S B (b) = 1 and (P-a.s. = P almost sure) a = 1 S A (a) = 1, P a.s. a = 2 S A (a) = 1, P a.s. This leads to N(= 1, 1) = 0, N(= 2, 1) = N(2, 1), N(= 1, 2) = 0 and also N(= 2, 2) = N(2, 2). Hence, CHSH = 2 with obvious local means. In order to have sufficient random outcomes the source Σ sends out randomly selected numbers, r i { 1, 1} per trial i such that (S A, S B ) i r i (S A, S B ). For the construction of the algorithm this random multiplier r i is unimportant. (2) 2.2. Slack to violate. The general concept of the the program is to have a split somewhere in the trial number series and change strategy. In a half number of the total number of trials, the focus lies mainly on getting the CHSH = 2. In the other half, the second in the example program, a compensation for rigidity i.e. the slack in the algorithm is introduced. In the program we randomize 4N = setting pairs. The randomization is performed on N(1, 1) = N, N(1, 2) = N, N(2, 1) = N, N(2, 2) = N generated setting pairs (a, b). We have N = Subsequently we may note that the source Σ in Gill s supermartingale treatment [6] is allowed to have complete knowledge of the past. That implies, if we are in the i-th trial, i = 2,..., 4N, the source Σ is allowed to use N i 1 (= a, b), i.e. the count of equal spin measurements up until and including the i 1 trial for setting pair (a, b) {(1, 1), (1, 2), (2, 1), (2, 2)}. Note that we can also use the information N i 1 (= a, b) at the i-th trial in A and B but the explanation is conceptually somewhat more easy when the information is projected in the source Σ. In Σ the factor p = N i 1(= 1, 2) + 1 N i 1 (= 2, 2) + 1 is employed. In the computer simulation dp = 0.01 and the scan starts at p = 0.4 and ends at p = 0.8. In the code p is denoted P grens. We have (N max = 4N) for the first N max /2 number of trials, per selected p, that ɛ = p. For the remaining trials ɛ = 2. In case t i = 1 then S(b) is redefined. In the computer code we have replaced t i = 1 by a false Boolean. In the program code we have blselect for Boolean. The (3)

3 CHSH and violation with LHV computer program. 3 redefinition works as follows. Suppose ( ) i < Nmax 2 here i < then ɛ = p, if not then ɛ = 2. Under the condition that p from (3) such that p > p we, for i = 2,..., N max, see the following ( ) if η 3 = (Nmax/4) N i 1(= 1,2) (N max/4) < ɛ then t i = 1 else ( ) t i = 1 The first line in the rule (*) is there to minimize the N i 1 (= 1, 2) overshoot in relation to the N i 1 (= 2, 2) when ɛ < 1. If p p in trial i we have t i = 1, i.e. the Boolean is false. This part of the complete code in the appendix is given for reference:... If (n(1, 2) + 1) / (n(2, 2) + 1) > P grens T hen If eta3 < eps T hen blselect = T rue t i = 1 blselect = F alse t i = 1 blselect = F alse t i = 1... The n(a, b) represent the N j (= a, b) in the j-the trial, η 3 and ɛ are obviously translated in the code. When η 3 < ɛ then in case ɛ = p we can conclude that apparently N i 1 (= 1, 2) is close to N max /4. Considering the change in S B (b) response it follows that if t i = 1 then S B (b) = 1 for b {1, 2} else S B (2) = 1 & S B (1) = 1 In that scan, viewed over a number of experiments, i.e. different (a, b) distributions, the best violation found so far is CHSH = The worst value was CHSH = Measured in standard deviations the CHSH violation result is, seen over experiments, in m 40, p-scans per experiment, roughly 2 standard deviations away from 2. In Figure-1, a series of results with the computer program, given in the code section below, is presented. The computer code is an essential part of the paper. In order to demonstrate that the CHSH violation with the code can be improved, the following adaptation was introduced. In the first place we inspected the N max /4 approximation of pair (2, 2) with the use of (Nmax/4) Ni 1(= 2,2) (N max/4). Secondly, the code

4 4 Han Geurdes presented in (*) was adapted to if η 3 = if η 2 = t i = 1 else t i = 1 endif else t i = 1 endif ( (Nmax/4) N i 1(= 1,2) ( (N max/4) (Nmax/4) N i 1(= 2,2) (N max/4) ) < ɛ then ) < ɛ 1.5 then ( ) The use of endif is to aid the readability of the pseudocode. Of course, the rule if t i = 1 then S B (b) = 1 for b {1, 2} else S B (2) = 1 & S B (1) = 1 follows rule (**) in the same way as in the case of rule (*). As an addition to the present computation we studied N max = Step size is dp = 10 4 and the scan was from p = to In Figure-3 we can observe a relatively steep fall in CHSH, (1). This computation is only an indication to see a possible increase along the lines of this algorithm. The result of the more fine grained scan with p from 0.78 to 0.8 with dp = is presented in Figure-2. The number of computations increases dramatically with smaller dp and the rule in (**). That is why this type of fine grained scanning is run after possible identification of interesting sections in a more coarse first p scan. A run with N max = 10 6 and p from to with dp = and (*) replaced with (**) is given in Figure Scavenging ahead Algorithm A separate tactics shows that the code can even be improved to the order CHSH = The principle of this separate piece of coding is based on the fact that the scanning on p can be extended with a scanning to search for optimal scavenging violating photon pair equivalents. Let us note that the overshoot of photon pair equivalents is expressed in N(= 1, 2) + N(= 2, 2) ζ = ) (4) ( Nmax 4 The Napier logarithm of ζ will not be positve nonzero when N(= 1, 2) + N(= 2, 2) ( N max ) 4 but will be positive otherwise. The optimization scavenging of photon equivalent pair measurements will take place after log(ζ) > 0 occurred. The B measurement instrument will, in case b = 2 scan ahead two setting possibilities

5 CHSH and violation with LHV computer program. 5 in over eight possible array values of S b i.e. the tripples in the set T T = {(1, 1, 1), ( 1, 1, 1), (1, 1, 1), (1, 1, 1), (1, 1, 1), ( 1, 1, 1), (1, 1, 1), ( 1, 1, 1)} (5) The history is condensed in 8 array values at the i-th trial log(ζ j ) for j = 1, 2,.., 8. Before the (i + 3)-th trial starts an (S B,i, S B,(i+1), S B,(i+2) ) triple is selected from T under the condition b = 2. Now because all N k (= 1, 2) and N k (= 2, 2) data, k < i + 3, is known to B on the (i + 3)-th trial, the eightfold array with each possiblity for (S B,i, S B,(i+1), S B,(i+2) ) T is also known to B. If, for b = 2 only, the triple (S B,i, S B,(i+1), S B,(i+2) ) is used that has a maximal log(ζ j ) for j = 1, 2,.., 8 then the CHSH can reach the order of 2.04 with the present computer code. The drawback of this method is that the B waits two photon equivalent pairs before providing the maximal log(ζ) triple to be archived as measurement. The priniciple in Gill s method that B immediately produces the S B (b) appears violated on first sight in the linear not- parallel running computer program that we had to employ. However, the change from log(ζ) 0 to log(ζ) > 0 occurs somewhere around i = in our example case of N max = If we interpret the scanning for the optimal p parameter as simple parallel processing, the scavenging for optimal residual photon pair equivalents after log(ζ) > 0 can run parallel for the residual i = trials while the first i = is running for a maximal violating p. So with in principle classical computing running parallel the first trials run for maximal violating p while the is running parallel classically with the previous scavenging algorithm. In the optimum situation the maximal violating sequence of S B (b) is presented and archived for registration exactly as Gill prescribes in [6]. The variations discussed in the next section work however, without the scavenging residual algorithm disussed above. Results are provided in Figure Conclusion & Discussion In the present paper it was demonstrated that there is a numerical loophole in the no-go conclusions of CHSH. This conclusion is based on a simple computer code that uses a one-parameter optimization scan. The physical picture that can be imagined behind the optimization is the following. Nature uses parallel computing to find the optimal violating p parameter and then presents the optimal plus and minus outcomes in A and B. The code does not meet the Randi challenge requirement because the random distributed pairs of settings must be known first in order to find the maximal violating p parameter. The reason is that the results are tested with one thread non parallel computing. However, as one can clearly verify, the code only uses local principles to arrive at the result and shows that CHSH is not a reliable criterium irrespective of the necessary scanning to find the optimal parameter. If CHSH was waterproof and the supermartingale proof of Gill [6] was valid then this simple computer model would be impossible. This means no program

6 6 Han Geurdes based on local principles would give CHSH < 2. Additional arguments against Gill s supermartingale no-go can be found in [7]. The reader should note that mathematically CHSH 2. This is a strict claim for all LHV models. Finding with a computer simulation LHV models that numerically show CHSH > 2, simply refutes the for all part of the CHSH no-go claim and therewith rejects CHSH as a reliable means to claim no-go for LHV. The program is lagrely mechanical (with a slight probability to deviate: i.e. P-a.s). This concurs with previous work of the author [5] although some believe that the argument in [5] is still incomplete a. Because the results in the present computer simulation is simply based on countings as in an ordinary experiment the violation with LHV principles shows that the strict theoretical claim CHSH 2 can at its least be questioned because the probability density, ρ(λ) 0 that Bell uses gives dλ ρ(λ) = 1. Critics of λ Λ the violation conclusion can only avoid abandoning CHSH by increasing the γ in CHSH γ from 2 to 2 + δγ without a physics theory to support the δγ > 0 increase. We also note that the computer program is based on simple counting principles such as are found in experiment. The program is mechanistic. The usual statistical explaining away of the present result clearly ignores that. As an important final remark we would like to stress that the rejection of the use of history in the scanning for optimal parameter must be founded in physical theory not in statistical concepts. For after all, if we just gamble on the optimal parameter, there is a nonzero but small probability to get it right for just one step of a scan. The same argument goes for the optimal scavenfing algorithm introduced for the residual or final trials. The results of this algorithm are provided in Figure-4. In Figure-5 the numerical proof is given for an increase in amount of violation without scavenging algorithm. The parallel computing option takes away all the scanning effort but maintains classical local hidden variables. The no-scavenging present deepened algorithm arrives at approximately 320 to 340 photon pairs overshoot in case CHSH = to CHSH = violation. In numbers: we have roughly N(= 1, 2) + N(= 2, 2) (N max /4) 320 in N max = 10 5 trials and maintain N(= 1, 1) close to 0 while N(= 2, 1) is very close to N max /4. Figure-5 shows the minimum that was attained here. Moreover, Figure-6 shows with a different configuration of setting distributed over N max trials that CHSH = can be attained independently of that distribution. Another interesting observation is that introducing the following snippet in equation (6) into the code presented in the section below, the order N(= 1, 2) > N(= 2, 2) was changed in N(= 1, 2) < N(= 2, 2) in the violation CHSH > 2. This implies that the hidden structure that is expressed in the program code enables to redirect violating photon pair equivalents from N(= 1, 2) to N(= 2, 2) and adds to the a The counter arguments to [5] will be dealt with elsewhere.

7 CHSH and violation with LHV computer program. 7 power of LHV explanation with the code given in the appendix. If blselect T hen Sb = 1 If b = 1 T hen Sb = 1 If b = 2 T hen If ((n(1, 2) + n(2, 2))/(Nmax/4) ((Nmax/4)/Ncnt) 0) T hen Sb = 1 Sb = 1 (6) In addition it is noted that the history of the countings, i.e. employing N i 1 (= a, b) at trial i are only employed to scan for the proper parameters in the program. We claim that if parallel computing in nature occurs then the scanning-with-history such as employed in the present program is not essential. This is also true for a parallel scavenging ahead algorithm. Moreover, De Raedt, Jin and Michielsen employed the possibility of memory, i.e. the use of history, in neutron interferometry with good result [8]. In our program the history is employed more strategically to find a parameter and less in a fundamental sense such as in [8]. In passing we note that Gill s supermartingale argument [6] against CHSH is only refuted in a weak sense. The code does not produce CHSH > However, we showed that with more fine grained scanning and a change in rule from (*) to (**) the amount of violating the CHSH 2 can be improved. We employed N(= 2, 2) in rule (**) and a smaller dp in a scan of p from 0.78 to The differences in (a, b) distributions does not affect the outcome of the violation and when parallel computing is supposed instead of scanning, the order effect in the scanning, i.e. in one distribution the optimal p is found at p 1 while for another it is found at p 2 and p 1 p 1 is no longer of any importance to the optimization of the S B (b) outcome. We finally note that several kinds of improvements can be considered to get a better violation in one thread non-parallel computing to restrict the scanning effort. Parallel computing takes away the objection to know the (a, b) ditribution beforehand. Note that for e.g. the initial p = 0.4 the (a, b) distribution is unknown and random. Only when single thread computing is needed, the p = 0.41 (when e.g. dp = 0.01), runs through an known sequence of (a, b). If however, p = 0.4 runs in one thread and p = 0.41 parallel in another, the (a, b) is new and random to both threads. Note that, as already claimed, the use of history N k (= a, b) and k i 1 at the i-th trial, cannot be rejected on physical grounds. For k = 2, 3,..., N max we

8 8 Han Geurdes have N k (= a, b) N max /4. Parallel processing of different types of history would also be the solution to avoid having to use the history of measurement results such as in Gill s supermartingale restriction [6]. Most likely, if there are LHVs they express themselves in parallel processing for, both, optimal parameter scanning p and history N k (= a, b), for k = 2, 3,...N max. Appendix A. VBA Code The code was written in VBA for excel and is presented in the subsection below. Option Explicit Sub BellAgainM odoutdoors() Dim Sa, Sb As Integer Dim eta1, eta2, eta3, eps As Double Dim i, j, k, l As Long Dim a, b As Integer Dim n(2, 2), m(2, 2) As Long Dim CHSH As Double Dim Ncnt As Long Dim blselect As Boolean Const N max As Long = Dim P grens, dp, P max As Double If MsgBox( Rerandomize Settings?, vby esno) = vby es T hen Call Settings(1, 1, 1, N max/4) Call Settings(1, 2, (N max/4) + 1, N max/2) Call Settings(2, 1, (N max/2) + 1, 3 N max/4) Call Settings(2, 2, (3 Nmax/4) + 1, Nmax) Call Rerandomize(N max) Call Rerandomize(N max) Call Rerandomize(N max) P grens = 0.4 P max = 0.8 dp = 0.01 P grens = P grens dp j = 0 If P max < P grens T hen GoT o eind Do Until P grens > P max n(1, 1) = 0 : n(1, 2) = 0 : n(2, 1) = 0 : n(2, 2) = 0 m(1, 1) = 0 : m(1, 2) = 0 : m(2, 1) = 0 : m(2, 2) = 0 Ncnt = 0

9 CHSH and violation with LHV computer program. 9 Do Until Ncnt > Nmax Ncnt = Ncnt + 1 a = AcceptSetting(N cnt, 1) b = AcceptSetting(N cnt, 2) m(a, b) = m(a, b) + 1 eta3 = ((Nmax / 4) n(1, 2)) / (Nmax / 4) If Ncnt < (Nmax / 2) T hen eps = P grens eps = 2 If (n(1, 2) + 1) / (n(2, 2) + 1) > P grens T hen If eta3 < eps T hen blselect = T rue blselect = F alse blselect = F alse If blselect T hen Sb = 1 If b = 2 T hen Sb = 1 Sb = 1 Sa = f nresponse(a) If Sa = Sb T hen n(a, b) = n(a, b) + 1 Loop CHSH = ((2 n(1, 1) / m(1, 1)) 1) ((2 n(1, 2) / m(1, 2)) 1) ((2 n(2, 1) / m(2, 1)) 1) ((2 n(2, 2) / m(2, 2)) 1)

10 10 Han Geurdes W orksheets( Sheet1 ).Activate j = j + 1 Cells(j, 1) = CHSH Cells(j, 2) = P grens Cells(1, 3) = n(1, 1) : Cells(1, 4) = n(1, 2) : Cells(2, 3) = n(2, 1) : Cells(2, 4) = n(2, 2) : Cells(2, 5) = CHSH P grens = P grens + dp Loop eind : MsgBox Ok End Sub F unction f nresponse(a) As Integer Dim r As Double r = randomly If a = 2 T hen If r < T hen f nresponse = 1 fnresponse = 1 If r < T hen f nresponse = 1 fnresponse = 1 End F unction F unction AcceptSetting(k, i) As Integer W orksheets( Sheet4 ).Activate AcceptSetting = Cells(k, i) End F unction Sub Settings(x, y, p, q) Dim i As Long W orksheets( Sheet4 ).Activate F or i = p T o q Cells(i, 1) = x Cells(i, 2) = y Next End Sub

11 CHSH and violation with LHV computer program. 11 Sub Rerandomize(N max) Dim r As Double Dim i, j, k, n, m As Long r = randomly F or i = 1 T o Nmax k = 1 + Int(r (N max 1)) n = Cells(k, 1) m = Cells(k, 2) Cells(k, 1) = Cells(i, 1) Cells(k, 2) = Cells(i, 2) Cells(i, 1) = n Cells(i, 2) = m r = randomly Next End Sub F unction randomly() As Double Dim T istr As String Dim CtrStri As String Dim Nloop As Long Dim i, j, k As Long Dim r As Double T istr = T ime CtrStri = Right(T istr, 2) N loop = V al(ctrstri) + 1 F or i = 1 T o Nloop r = Rnd Next r = Rnd randomly = r End F unction

12 12 Han Geurdes References 1. A.Einstein,B.Podolsky and N.Rosen Phys. Rev. 47(10) (1935). 2. J.S. Bell,Physics (1964). 3. G.Weihs,T.Jennewein, C. Simon, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett (1998). 4. A.Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett (1982). 5. J.F. Geurdes, Optical Engineering. 51(12) (2012). 6. R. Gill, Time, Finite Statistics and Bell s fifth position, p188, Foundations of Probability and Physics, Växjö Sweden, (2002). 7. J.F. Geurdes, Bell s fifth position is not supported by supermartingale statistics, arxiv: H. De Raedt, F. Jin and K. Michielsen AIP Conf. Proc. 1508, (2012).

13 CHSH and violation with LHV computer program. 13-1,98-1, ,99-1, ,005-2,01 Series1 Series2 Series3-2,015-2,02-2,025 Fig. 1. Three series of trials where p ran from 0.60 to 0.89 and dp = 0.01.

14 August 26, :27 WSPC/INSTRUCTION FILE 14 Han Geurdes -2,016-2,018-2,02-2,022 Series1-2,024-2,026-2,028 Fig. 2. One series of trials where p ran from 0.78 to 0.80 with dp = and rule (*) is replaced by rule (**).

15 CHSH and violation with LHV computer program. 15-1,985-1, , ,005-2,01 Series1-2,015-2,02-2,025-2,03 Fig. 3. One run with N max = 10 6 where p ran from to with dp = and rule (*) is replaced by rule (**).

16 16 Han Geurdes -2,032-2, ,036-2,038-2,04-2,042-2,044-2,046-2,02708 Series1-2,0271-2, , ,02716 Fig. 4. One series of trials where p ran from 0.78 to 0.80 with dp = and rule (*) is replaced by rule (**) together with applying the scavenging algorithm for the final based on the change in sign of log(ζ), defined in (4) described in subsection-2.3.

17 CHSH and violation with LHV computer program. 17-2, ,0271-2, , , , ,026-2,0265-2,027-2,0275 Series1-2,0272-2,02722 Fig. 5. One run with N max = 10 5 where p ran from to with dp = and rule (*) is replaced by rule (**) and ɛ = 2 in the Ncnt 0.5 N max branch replaced with ɛ = 1. -1,99-1, ,005-2,01 Series1-2,015-2,02-2,025-2,03 Fig. 6. One run with N max = 10 5 where p ran from 0.20 to 0.84 with dp = 0.01 and rule (*) is employed on a different distribution of (a, b) settings over trials than in the previous figures.