Simulation of Bell violations in Walker system
Discussing poster at emQM17 with R. Brady (Cambridge)
Abstract
We describe results from an implementation of a Monte-Carlo simulation of Bell-CHSH type correlations with hydrodynamic walkers as suggested by [Vervoort2017]. We observe the formation of pairs of walkers strongly anti-correlated in position and velocity under various random initial conditions. With a non-relativistic representation of the walkers, i.e. one where the hydrodynamic waves propagate faster than the walkers, as in real life walkers, we observe numerical S values of CHSH correlations above 2, violating the Bell limit as an explicitly non-local system. We observe Bell violations up to the Tsirelson limit of 2sqrt(2), but not violating it, under fine-tuned observation and post selection conditions. We report various such runs in the 2 < S <= 2.82 range under the non-separable assumptions. However when we numerically enforce programmatic separability of the walkers, a numerically enforced locality, then we lose the violations and recover, as predicted if locality holds, a classical S <= 2.
Conceptual framework
M(x,y)= sum(sigma1*sigma2*P(sigma1,sigma2)).
We do the correlation measure for four vectorial values a,a’,b and b’ following the rules of the standard Bell game and we calculate the scalar value of the CHSH form:
S = M(a,b)+M(a,b’)+M(a’,b)-M(a’,b’)
Figure 1: Anti-correlated walkers in red dots seen with their path as filmed from above. Bath dimensions in our simulation: 1.2m*0.6m. Pin measures at position a and b in blue. Sigma1 and sigma2 as UP and DOWN outcomes of the measure. Source of anticorrelated walkers S from [VERVOORT 2017].
Source of AntiCorrelated Walkers
The original paper [Vervoort2017] called for the creation of two walkers, each one with opposite velocities, and thus perfectly anti-correlated by construction. This construction, in theory, does not require wave mediation for the walkers would get created with perfectly anti-correlated values at birth and propagated by inertia. Here we choose velocities randomly (although narrowly) as a more general starting point for the creation of pairs. Anti-correlation emerges as a rather generic outcome for two field mediated walkers. Walker to walker interaction, via wakes, creates perfect anti-correlation in our simulations.
Below you can see pictures of walkers going in opposite directions. Bear in mind that these walkers had random initial conditions. The differences at birth quickly get wave mediated and a high degree of anti-correlation appear even numerically as in figure 3 where we show the velocities as vx and vy (2D). Take a moment to read through the raw data if you can.
Below you can see pictures of walkers going in opposite directions. Bear in mind that these walkers had random initial conditions. The differences at birth quickly get wave mediated and a high degree of anti-correlation appear even numerically as in figure 3 where we show the velocities as vx and vy (2D). Take a moment to read through the raw data if you can.
Figure 2: anti-correlation, the walkers get initiated with different and random velocities.
Figure 3: anti-correlation, random initial velocities.
We characterize our source as a statistical mix of up-down and down-up combinations. The following data shows 4 different runs and P(-+)=P(+-)=1/2. We abuse the language to talk about psi, for we have a statistical ensemble here, and still need to show S-Value >2 to talk about entanglement and superposition.PSI =1/sqrt2* (|-+>+|+->)
Figure 4: psi =1/sqrt2* (|-+>+|+->)
Importance of Anechoid Cavity
It should be noted that anticorrelated walkers do not seem to appear generically in real walker systems as they do in the simulation. In private communications, both Couder's team (Fort) and MIT's team (Bush), confirm that they haven't seen them yet for we haven't actively looked for them either. The walkers in the presence of cavity noise integrate it chaotically [ChaosBook] and default to confined walkers such as the 2 walker Oroborous. This rationalization can be tested in software by artificially removing all wall effects (just not coding them will do). Removing all wall effects from the walkers requires more care. 
Figure 5: 2 walker Oroborous
Bell-CHSH Methods
Walker particle assumptions
We operate in the so-called stroboscopic approximation where we only consider the horizontal point dynamics of a walker.
Walker field assumptions
Each impact of each walker creates a standing wave centered at said point of impact. We represent these wavelets with standing Bessel profiles. The walkers offer a non-Lorentzian ontology. The waves, both travelling and standing, propagating faster than the walkers. Given the finite nature of the velocities, we can choose to simulate separability in walkers or not, knowing real life walkers by default do not meet ideal separability criteria.
Figure 6: A 2D Bessel function of order 0
Pin wave assumptions
The pins are represented by Static Bessel functions which do not decay in time. We set the wavelength at Faraday and the amplitude on the order of 1mm, in order to roughly mimic the hydrodynamic range and as a first approximation of a mirror image of the walker field on the pins. We mimic separability of measure to measured e.g. the last minute insertion of the pins of [Vervoort2017]. We can turn it on or off and it proves a necessary condition for the observation of the Bell violations. In effect we only allow the pins to interact with walkers that are close to them and without cross-talk in both channels, mimicking the photonic Bell conditions of in-flight setting of the polarizers as done in [AA1982].
Force computation
We use a a-dimensional numerical representation in the hydrodynamic walker domain [Bush2014] acceleration = VISCOUS_force + FIELD_force
field_force comes from the gradient of the wake.
Time Decay
Pin waves do not decay in time (static). Walker waves do decay in time.
Walker Separability (outcome independence), OI
We do not include waves coming from another walker if the walkers are separated by a large distance in a simplified implementation of causal relativism. We code for that separability as a boolean. It should be noted that our Bell results, Bell violation vs no Bell violation, were not sensitive to this OI. While the Bell status did not depend on it, the correlated outcome statistics were.
Parameter separability (parameter independence), PI
When calculating the field force we can choose to not include waves coming from the other PIN and account only for the pin present in the arm. The Bell violations were dependent on the value of this parameter.
Artificial separability and real walker systems.
These two separability conditions OI (Walker-Walker) and PI (Pin-Walker), probably do not exist in the real walker systems simply because the waves vastly outpace the walkers.
Violation of Bell inequality w/ non separability (real walker).
Without Parameters Indepence (PI=false) we let the left pin influence the right walker and vice versa. We obtain violations of Bell
Figure 8: Bell violation (S-VALUE>2) under non-separability (ENFORCE_PI set to false)
Loss of Bell violation with separability (software walker).
With Parameters Indepence (PI=true) we lose the violations and recover Clauser separability.
Figure 9: No-Bell violation
Emergence of the Tsirelson limit in post-selection
When we observe violations and when we narrow the admission window on the back walls to +-50mm (Y_BLOWUP) then we observe various violations at 2.80+/-0.05. In other words we observe Bell violations at the Tsirelson limit 2*sqrt2= 2.82. We have not seen violations above the Tsirelson limit. We have not specifically looked for them either.
Figure 10: Tsirelson limit emergence (2.82)
For anyone interested, which should mean about 2 people, you already have my paper version of this blog/poster entry, if you read this far and want to know more I can send you the 26 pages of it.
Bibliography
[Vervoort2017] Louis Vervoort, Are Hidden-Variable Theories for Pilot-Wave Systems Possible ? https://arxiv.org/abs/1701.08194
[ChaosBook.org] Chaos Book by Predrag Cvitanovic et al. http://chaosbook.org/
[AA1982] Alain Aspect 2006 review. https://arxiv.org/pdf/quant-ph/0402001.pdf
[Bush2014] http://math.mit.edu/~bush/?p=3113
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