• Glyn Adgie and Tim van Beek, Increasing the signal-to-noise ratio with more noise. [...]

int getRandomNumber()
{
     return 4;  // chosen by fair dice roll.
                    // guaranteed to be random.
}

But the best quote I know in this context is from numerical recipes (an author cites a system administrator from memory):

“We guarantee that each number is random individually, but we don’t guarantee that more than one of them is random.”

Electrical Engineers designing analog to digital converters have known about this effect for decades. They call it dither. It’s random noise that improves the accuracy of finite-bucket assignment of a sampled analog signal to a digital value. This classic paper from 1964 shows that the optimum is white noise with RMS 1/3 of the difference between the smallest assignment values (least significant bit or LSB):

• L. Schuchman, Dither signals and their effect on quantization noise, IEEE Trans. Commun. Technol. (1964), vol. COM-12, 162 -165.

I replied:

I think dither is a bit different from stochastic resonance: as the blog entry explains, stochastic resonance happens when you have a dynamical system with two stable equilibria and a small oscillatory force that would push the system back and forth between these equilibria if it were big enough. Then adding noise boosts the chance that the system actually makes it over the hump, especially if the noise has some power at the same frequency as the oscillatory force.

However, it’s cool that dither is yet another way that noise can help. Thanks! – I’ll post a comment about this on my blog. I wonder if there’s some precise mathematical relationship.

1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1

They look good, don’t they? But watch out: I deliberately put one in myself, which was not generated randomly!

Or, if you need lots of random numbers, you can buy their product Quantis, which produces up to 16 megabits per second. You can read about it here:

• Random number generation using quantum physics, Id Quantique White Paper, version 3.0, April 2010.

It uses a half-silvered mirror to split a beam of light, and single-photon detectors that can tell which way each photon went.

But there’s a catch: while ideally 50% of the photons would go through and 50% would get reflected, Id Quantique can only guarantee that the number is somewhere between 45% and 55%. So, they need to use an algorithm called an ‘unbiasing technique’ to take the somewhat random but still biased string of bits and try to purify it to maximal randomness.

This problem shows up with any physical method of generating random digits. The first unbiasing technique was developed by von Neumann:

• John von Neumann, Various techniques used in connection with random digits, Applied Mathematics Series 12 (1951), 36-38.

The output of the unbiasing technique needs to have fewer bits (per second) than the input. But even apart from this, I suspect that all the difficulties in defining a random sequence produced using an algorithm—the “sin” von Neumann was talking about—reappear when trying to define what counts as a valid unbiasing technique, unless for some reason you know the input string is identically independently distributed… or something like that: something you probably can’t actually know about a realistic, imperfect physical system.

It is striking that in the cricket’s cercal mechanoreceptors (at least at the level of single neurons) the optimal signal-to-noise ratio was achieved with a noise intensity 25x that of the signal.

Even more intriguing to me was an abstract referenced from Tim’s second URL: “Ubiquitous crossmodal Stochastic Resonance in humans: auditory noise facilitates tactile, visual and proprioceptive sensations.” I realize we are now far afield from climate physics but the apparent pervasiveness of this phenomenon is eye opening.

From a practical point of view a random number generator is good if it is random enough for the application that consumes the numbers it produces, in the sense that the application does not produce any artifacts that depend on the generator used. Since no random number generator is truly random, there is no objective mathematical criterion to classify generators; in practice, users have specified a list of statistical tests that a “good” generator should pass – on the average – while “bad” generators do not – on the average (see the references for details).

From a pure mathematical point of view this means that we know some ways in which the “bad” generators fail, while we did not figure out how to unmask the “good” ones. So the bad ones are actually those generators that we happen to know more about!