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I was looking at a website that claims to generate random numbers from observation of quantum effects. This lead me to question how we know that the numbers are truly random.

When we observe a probability wave and it collapses in one place into a particle, how do we know that the location of the particle is really random?

Do we have any evidence of the randomness, or is it just that no one can predict the location right now?

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  • $\begingroup$ "When we observe a probability wave" - ??? $\endgroup$ – Alfred Centauri Oct 14 '17 at 23:33
  • $\begingroup$ Which website? Some existing protocols are based on violations of Bell inequalities in ways that are not fully covered by the existing answers, but it'd be good to know if the example you're looking at is in that class before writing that up. $\endgroup$ – Emilio Pisanty Oct 15 '17 at 11:39
  • $\begingroup$ @EmilioPisanty The site that I am speaking directly of is qrng.anu.edu.au $\endgroup$ – user3465829 Oct 16 '17 at 0:16
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There are two main views. The first view relates to the Copenhagen interpretation of Quantum Mechanics. According to this interpretation, a particle does not have a specific path, but travels like a wave. Upon detection, the wave function collapses and the particle appears at a random point on the screen (according to the probability defined by the wave function).

The second view relates to the "Pilot Waves" theory. It states that a particle has a definite trajectory that ends up with a dot on the screen. However, the trajectory depends on the emission parameters, as the particle is emitted by the source at a certain angle, with a certain phase, etc. These parameters are random, so the result is exactly the same.

In the Copenhagen interpretation, the trajectory is unknown, because a definite trajectory does not exist. In the Pilot Waves theoty, the trajectory is definite, but cannot be known, because it depends on the random parameters of the emission.

In other words, whether we don't know the trajectory, because it doesn't exist, or we don't know it, because it exists, but we can't ever know it, the result is exactly the same. Whether randomness is at the end of the path or at the beginning, the result is unpredictable anyway.

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Given a sequence of numbers, whatever their origin, how can we assert whether they are random or not? This is an important question in many fields of computing science. Often such sequences are actually only pseudo-random, in the sense that they are produced by a deterministic algorithm, but one which is, naively speaking, chaotic enough to emulate a true uniformly distributed random sequence. In order to evaluate the quality of the randomness, statisticians have long developed a batterie of tests, one of the best known one being the so-called Diehard Test. One of the most important feature of such test suites is to verify that there are no correlations between the number in the sequence. I refer you to the Wikipedia page and the references therein if you are interested in the nitty-gritty details.

I guess, you see where I am going: has anybody thought of applying the Diehard Test, or some equivalent, to a sequence of quantum measurements. This has not been done for position as the precision of measurement is not enough and would get in the way of the actual test. But this has been done with photons, for example as reported in [1]. I quote the authors to explain the principle of their device.

Here we present an optical Quantum Random Number Generator (QRNG), whose randomness is based on the very principles of quantum physics. The compact setup consists of a light source with stabilized intensity attenuated to the single photon level and one single photon detector. The detection events are counted during a sampling time interval $\tau_s$ and are interpreted as ’0’ for an even number of counts, whereas an odd reading corresponds to ’1’. According to fundamental laws of quantum optics the probability distribution of the number of photons in a sampling interval should follow a Poissonian distribution with mean $\mu$ for a constant intensity light source [19], fully analogous to radioactive sources for low $\mu$. This fact would cause a considerable bias between the num- ber of ’0’s and ’1’s in the random bit sequence. However, as we demonstrate below, dead time effects of the photomultiplier together with the read-out electronics allow to eliminate the bias even for very fast generation of random bits.

Then, the author passed their QRNG through a battery of statistical tests, including a version of Diehard, and they all passed with flying colours. Therefore in that sense, which, again, is the widely acknowledge standard of randomness, their device produces true random numbers. Such a device is particularly useful for cryptography, as a final note.

[1] Harald Fürst, Henning Weier, Sebastian Nauerth, Davide G. Marangon, Christian Kurtsiefer, and Harald Weinfurter. High speed optical quantum random number generation. Opt. Express, 18(12):13029–13037, Jun 2010.

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So the question is what you mean by random.

In a very superficial sense of the term, one which has to do with excluding the possibility of determinism and therefore asking, "is there a way to understand the system as having an initial state which forced it to come to this conclusion," the answer is a qualified "no": there are hidden-variable interpretations like the pilot-wave theory which interpret quantum mechanics as a deterministic theory containing unknowable global information.

The qualification here is the word "global": using some thought experiments (my favorite is a game called Betrayal) one can prove that there are quantum effects which cannot be understood in terms of classical local information: in Betrayal's case, a game which cannot be won by any classical algorithm more than 3/4 of the time is won by a quantum algorithm 100% of the time. The proof contains some interesting insight: it is possible to describe classical information, and classical probability, in a way where we take the three players and ask them to give us their individual hypothetical answers and then we choose randomly which questions we asked afterwards. In quantum mechanics there is no good way to do this; you cannot extract an entire quantum state from a single measurement in its full generality, particularly not as it may correlate with other states in other parts of the space.

In a deeper sense randomness is our way of reasoning about information that we do not know, and the point of these "random numbers" for example is that nobody else is likely to have any knowledge of any bit of them. And this answer is then a "yes" no matter how you slice it: whether there is some unknowable information which makes everything deterministic, it is known that we cannot (not just do not) know it. Those 3 players of Betrayal cannot have any sort of local knowledge which allows them to beat this game 100% of the time, assuming that we are able to choose the 4 types of experiment evenly without their knowledge. That remaining 25% gap between what classical and quantum players can achieve corresponds to a real information since it lets you win a game, but it's a real information which cannot be accounted for by any classical information that they have available to them.

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