Excerpt from an unpublished note I wrote in April, 1997 with some earlier calculations and observations concerning the relationship between energy and effect size in MMI generators. My present level of modeling enables more extensive and precise calculations of effect size versus energy, including the ability to model the performance of virtually every type of entropy source and generator type. © July, 2021 Scott A. Wilber

[Page 1]

With single photon, signal is just 1 or 0 with equal probability. Since both paths are indistinguishable, this is a quantum superposition. Therefore, the collapse of the wave function at the “1” or “0” detector results in no difference in entropy versus the “0” or “1” detector.

The goal is to obtain a detectable signal from a system while maintaining entropy, E = 1.0 – ε, ε is vanishingly small. If the designation of the “1” or “0” detector is randomly chosen before [Page 2] detection, then the entropy may be maintained at exactly 1.0, ε → 0.

Test correlation coefficient of the two detector systems.

Check correlation of the two pairs of detectors: analog or binary.

[Page 3] Three primary factors obscure the detection of intention on a physical device:

- The largest is the result of a very small effect superimposed on a large number of “natural” events. This is illustrated by the random signal caused by thermal noise in a resistor. (see Fig 1)

Only those bits which are taken during the time when the perturbation potential crosses the threshold of the detector (comparator), i.e., 0 in fig 1, may be “switched” to the opposite state. All other bits, such as “b”, may not be switched. In a real system the perturbation amplitude is much smaller than the noise [Page 4] amplitude so that the percentage of total bits which may be switched is small.

- The transfer function of the noise generation & detection circuitry spreads out the energy of the effect, with the result that more power is required to produce a given perturbation amplitude.

- The type of mechanism used to produce the random output will determine the amount of energy needed to perturb the output a given amount.

A perfect PBS [Polarization Beam Splitter] with a single photon input will produce two possible output states which are exactly equal from an energy or entropic point of view. Therefore, the amount of energy needed to produce either particular output is exactly 0. Any system which produces a simple quantum superposition can provide such an output.

[Page 6] A thermal noise source, which is primarily not quantum mechanical in nature, requires a finite, calculable amount of energy to produce a given perturbation.

[Page 8] Optical or nuclear sources may ultimately be better detectors, but they are much more complex and expensive than thermal or shot noise sources.

Optical sources:

[Page 9] Nuclear Source [Americium-241]:

[Page 11]

ES = fraction of bits switched/switchable bits (effect size)

ES = (Total high bits, ie, in intended direction - ½ total bits)/(½ total bits)

ES = (2 Ibits – Tbits)/Tbits = 2 Sbits/Tbits

Ibits = number of bits in the intended direction.

Tbits = total number of bits in sample.

Sbits = number of bits switched to intended direction.

Given Nrms and VMs, what fraction of the Vn < VMs ? VMs = max voltage of the [mentally-produced] sig.

1 – 2(1 – CDF) = 2 CDF – 1

Sbits = 2 CDF – 1

w/100% efficiency, given Normal (μ = 0, σ = Nrms) eval[uated] @ VMs.