The Effect of `Probability Skew' in Bell-test Experiments
Dale Hodgson



 Novel Sensor Scheduling Scheme for Intruder 

	Tracking in Energy Efficient Sensor Networks 
    Raghuram Bharadwaj D.^1,
        Prabuchandran K.J.^1,
        and Shalabh Bhatnagar^1
^1 Authors are with the Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India


    December 30, 2023
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We consider typical experiments that use Bell-inequalities to test local-realist theories of quantum mechanics and gain insight into how certain results can be obtained. We see that results against local-realism arise from some `quantum skew' of the correlation between entangled qubit pairs. Furthermore, we find some conditions necessary for a conclusion against local-realism. Finally we show that the problem of `no-signalling' that arises in these experiments cannot be reduced to arbitrary experimental accuracy.





§ INTRODUCTION

The testing of Bell-inequalities is the favoured method of investigating the relationship between local-realism and quantum mechanics. Significant results have been achieved in this area, but further work is still ongoing.

Realism is the assumption that the physical universe exists (has some `element of reality') independent of any measurement or observation. Locality is the assumption that physical influences cannot travel faster than the speed of light — the cornerstone of special relativity. In their 1935 paper <cit.>, Einstein, Podolsky and Rosen noted the disparity arising between the assumptions of local-realism and the behaviour of entangled quantum objects. John Bell's later work on that idea <cit.> lead to experimentally testable results, now known as Bell-inequalities, that can test whether certain quantum mechanical systems obey the ideas of local-realism.

A popular type of Bell-test experiment involves two entangled qubits being measured under space-like separation sufficient to exclude relativistic interaction and noting the correlation of the two sets of results. The correlations become a test value which, under the assumption of local-realism, has a theoretical bound. Hence such experiments can verify or contradict any local-realist theory for the behaviour of quantum objects.

The testable theoretical bound arises from consideration of a `correlation coefficient'. If a is a measurement examining the state of the first qubit, and likewise b for the second, then the correlation coefficient for the joint outcome ab is

    E_ab    = probability of correlation - probability of anti-correlation 
       realised experimentally as: n(corr)_ab - n(anti-corr)_ab/n(ab trials).


Bell's work shows that if some `hidden variable' that could not be directly measured exists (the assumption of reality), then the inequality:

    E_ac - E_ba - E_bc⩽ 1

should hold (in a set-up with perfect anti-correlation).

The later work of Clauser, Horne, Shimony and Holt <cit.> generalised this to allow for non-perfect anti-correlation in what is now known as the CHSH inequality. With two measurements prepared for each qubit, chosen such that measurement 1 is precisely the complement of measurement 0 (measures of spin in perpendicular directions).

    E_00+E_01+E_10-E_11⩽ 2


Recent work by Hensen et al. <cit.> <cit.> have shown statistically significant violation of the CHSH-inequality, suggesting to us that there is no local-realist theory that completely describes quantum mechanics.



§ A BELL-TEST EXPERIMENT USING CHSH

A general Bell-test experiment produces results with eight independent variables: N the total number of trials, a, b, c the number of trials with setting configurations 00, 01, 10, respectively, and the number of correlated results under each setting (n_00 , n_01 , n_10, n_11). (See Table 1.)



Giving test value: 

    S= E_00 + E_01+E_10-E_11 = 2 ( n_00/a + n_01/b + n_10/c - n_11/d - 1 ).


Local-realism predicts S ⩽ 2, so an experiment disproving local-realist explanations of quantum entanglement will violate the CHSH-inequality — showing S > 2.



§ THE EFFECT OF `QUANTUM PROBABILITY SKEW'


Suppose that the probabilities of correlation under each experimental setting are equal (to P say). Then S=2  2P -1; in which case S>2 ⇔ P > 1. Hence uniformity of these correlations will never violate the CHSH-inequality — the nature of quantum mechanics lends some `unnatural skew' to these results.

Similarly, we may examine Bell's original inequality (<ref>). Let n_ac denote the number of correlated results under setting ac, N_ac the total number of trials under setting ac, n_ba the number of correlated results under measurement setting ba Then correlation values take the form E_ac=n_ac- N_ac-n_ac/N_ac=2 n_ac/N_ac-1 and Bell's inequality asserts that a local-realist theory implies the following inequality:

    n_ac/N_ac - n_ba/N_ba - n_bc/N_bc⩽ 0  .

Again, if these three fractions (each equal to the probability of correlation under a given setting) are all equal, then the inequality will always be satisfied.



§ APPROXIMATING THIS EXPERIMENT FOR LARGE NUMBER OF TRIALS


Since the experimental settings are determined randomly, we would expect that the number of trials under each setting would tend to a uniform distribution for suitable large N. Supposing this is the case, let a=b=c=N/4, then we get 

    S=8/N n_a + n_b + n_c - n_d - N/4 .


Let us denote the range of correlation counts σ.

    σ := n_a,n_b,n_c,n_d=


Where n_max = maxn_a,n_b,n_c,n_d and n_min = minn_a,n_b,n_c,n_d. Let us only consider σ⩾ 1 (for if = 0, then we will always get S<2). Furthermore, we must have that n_a+n_b+n_c+n_d ⩽ N; consequently, it is clear that we must have n_min⩽N/4.

Let us denote (n_a+n_b+n_c-n_d) by S. Then we will observe violation of the CHSH inequality (S>2) S  > N/2. Note also that S is bounded above and below by S  _max = 3 n_max-n_min=2n_min+3σ and S  _min=3n_min-n_max=2n_min-σ.

Violation of the CHSH inequality is only possible if S  _max > N/2, that is to say: 2n_min+3σ>N/2 is necessary (but not sufficient) for violation. If we desire violation of a certain magnitude δ, then we may say that S  _max > N/2+δ⇔ 2n_min+3σ>N/2+δ is necessary. From this we may see that for a violation of magnitude δ, the `quantum skew' (range of correlation counts) σ must be strictly greater than δ/3:


    [ σ >N/6+δ/3-2/3n_min                 2*}         2*⇒σ > δ3 .;           n_min⩽N/4                                         ]


If we desire S>2+ Δ, then we are in fact asking that S  > N/2+N Δ/8, so δ = N Δ/8; hence σ > N Δ/24 is necessary. 



§ THE NO-SIGNALLING PROBLEM


A crucial factor in these experiments is that the two measurements must be made independent of one another. Khrennikov's analysis of the first Hensen experiments <cit.> noted that the independence of the measurements could be verified by analysing marginal probabilities available from the data.  Bednorz <cit.> also reviewed recent experiments in a similar way.

That is to say, if the measurements are truly independent, then we should observe:

    p(corr|s_1=0)     = p(corr | s_1 = 0 ∧ s_2=0)
        = p(corr | s_1 = 0 ∧ s_2 = 1)  ,

and similarly for other settings.

In the experimental set-up we are considering, all such probability differences look like

    p(corr|0s_b)-p(corr|00)=n_a  +  n_b/a +  b-n_a/a=an_b  -  bn_a/a(a  +  b).


The family of all these equations can be described by α n_β-β n_α/α (α + β) and β n_α - α n_β/β (α + β) where α , β∈a,b,c,d and α≠β.

We can say that we have achieved experimental results with a good no-signalling accuracy if all these probability differences are small. All such probabilities will be `small' (to some ε), if

    α n_β - β n_α/α + β < ε·minα , β.


In the case of uniform experimental test settings considered above (a=b=c=N/4), this condition becomes: n_α-n_β < ε N/2 ∀α , β. This is satisfied if 

    < ε N/2 .


So σ < ε N/2 is necessary to achieve results with a good no-signalling accuracy. Since σ⩾ 1 is necessary for a CHSH-inequality violation, N > 2/ε trials must be performed if an experiment violating the CHSH-inequality is to achieve no-signalling tolerance of accuracy ε.

Combining this with our knowledge that σ>N Δ/24 is necessary for violation of the CHSH inequality to magnitude Δ, we conclude that 

    ε > Δ/12

is a limit on our no-signalling tolerance — independent of the number of trials we perform.



§ ACKNOWLEDGEMENTS

With thanks to my PhD supervisor Dr Vladimir Kisil for putting me on to this topic and his advice while researching and writing.



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