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Given a stream of random binary numbers(*)

Is there any way to differentiate if they came from a Truly Random or from a formula/algorithm ? how?

if there is no way to decide this, then, I can't find any basis, to keep denying that behind the "truly random" of quantum mechanics can be a hidden algorithm.

I know I am talking about the posibility of a "hidden variables" theory, but I can't find any other explanation.

(*) Is known that is possible to create computable normal numbers from which first is possible to extract an infinite random binary stream, second, it tell us that a finite logic expression (again a binary stream) can contain a rational, and even an irrational number, so there is no big difference within a bit stream and the measurement of a random outcome, (and even less if we consider the accuracy) I say this, because the argument that obtaining bits is not the same as answer the complex question that can be made to quantum experiments, I think that "random" results can be perfectly read from a random stream of bits

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No, there is always a formula you can fit to any sequence of numbers –  user1708 Apr 4 '11 at 16:08
    
@kakemonsteret: Give me a formula which will give only prime numbers exactly ;-) –  user1355 Apr 4 '11 at 16:11
    
@sb1 en.wikipedia.org/wiki/Formula_for_primes –  user1708 Apr 4 '11 at 16:14
    
@Luboš Motl funny, it was intentionally duplicated by a moderator request –  HDE Apr 4 '11 at 16:17
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I think the OP is asking whether there is any way to know if some sequence of numbers are truly random or pseudo random. If not, then is there any proof that the randomness of QT is not pseudo randomness. –  user1355 Apr 4 '11 at 16:21
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This question involving "randomness" and Quantum Mechanics introduces some subtleties. Firstly we have the definition of "randomness" to consider. It turns out that there are various ways to define this term: the two I shall consider here are (the digit sequence of a) normal number (mentioned in the Question) and Martin-Lof Randomness.

As explained in the Wikipedia article a "Normal Number" is "Finite State Machine Random". So it looks random to a FSM. However such numbers can be computable by a Turing Machine (as the link shows that Turing proved).

A Martin-Lof random sequence is based on the familiar notion of incompressibility and cannot be computed by a Turing Machine. Phrased alternatively because it is an infinite sequence it will not have a finite compression onto a Turing Machine - so there can be no program for it (which has to be finite).

The logical link between the two is that every Martin-Lof random sequence is normal (but not conversely - as shown by Turing).

Also note that every finite sequence can be generated by an algorithm (and also it will be the solution of a polynomial). The reason why the random sequences work is that although the first N digits can be replicated via a program P, a different program (in general) is required for the first N+1 digits of the sequence ie P will fail to "predict" N+1. To get the entire sequence requires an infinite series of programs so we are back where we started with the random sequence.

Now that we have some definitions available we can examine connection to Physics. The problem in a experimental based theory is that we only ever have a finite amount of data. Thus the claim that a given series is "random" is strictly not empirically provable.

So this puts the Copenhagen claim that "QM sequences are random" into an awkward semi-scientific status. Such claims cannot be proved experimentally, yet Copenhagen asserts this. So where is the proof? Indeed what constitutes a proof? Also we have seen several definitions of randomness (there are more) - so which type of randomness does QM have exactly?

To return to specific points in the OP question:

Is there any way to differentiate if they came from a Truly Random or from a formula/algorithm ? how?

No, because one only ever has a finite amount of stream data to analyse, which can always be explained algorithmically (as discussed above).

if there is no way to decide this, then, I can't find any basis, to keep denying that behind the "truly random" of quantum mechanics can be a hidden algorithm.

The "truly random" of quantum mechanics might be Martin-Lof randomness, for which there is no algorithm; however if it is really normal number randomness then there might be an algorithm. I suspect that most physicists take the view that QM is as "random as it gets", hence would prefer the Martin-Lof option.

I know I am talking about the possibility of a "hidden variables" theory, but I can't find any other explanation.

The link between algorithmic underlying structure and "hidden variables" would appear to be close. If its a non-algorithmic type of randomness then we have to decide whether it belongs to one of "oracle classes" associated with Martin-Lof randomness, and what that would mean in terms of "hidden variables".

Some readers might recall that in his book "The Emperor's New Mind" in 1989 Roger Penrose proposed that aspects of quantum mechanics were "non-computable". Although that argument was not formulated as I have here, it is consistent (I believe) with the idea of Martin-Lof randomness too.

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+1 for an exceptionally good answer for questions like this. IMHO even the previous question by HDE was not all that non-sense and surely deserved good answers than an abrupt close! Science should be practiced in a free and open mind and not like fanaticism. –  user1355 Apr 5 '11 at 3:14
    
Thanks Roy Simpson, I've just found a paper supporting (aparently) what you have said. arxiv.org/abs/1004.1521 –  HDE Apr 13 '11 at 1:03
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No, you cannot prove that a string of numbers is random, and no, this is not a problem for quantum mechanics.

Suppose you are observing integers between 0 and 9 and you observe the 12-number sequence

4 6 7 1 9 4 0 8 5 2 8 8

Is it random? It looks like it, but there's always a possibility it was generated by the following algorithm:

1) print 4 2) print 6 3) print 7 4) print 1 ... 12) print 8 13) go back to 1)

You can test whether this was the correct algorithm by getting the next digit. Say you get "3" when the previous algorithm said you should get "4". That's no problem, you can amend the previous algorithm to add printing "3" at the end of it before it repeats.

So there's an algorithm for any sequence of numbers, but it's vacuous. The point isn't so much that quantum measurements are random. According to the Copenhagen interpretation they are, but what matters is whether they can be predicted, not whether they can be post-dicted.

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That's a periodic result, the question is about normal numbers, and algorithms capable to compute irrational numbers with no period. Surely are some irrational numbers that cannot be computed, but to say that quantum experiment results could came from that exactly kind of non computable random, is strange, and nothing but an easy path, of course we could stick to Bell's theorem and not make this kind of questions, but that's (at least) boring.. It's better to have an answer to gain a deeper undestanding –  HDE Apr 4 '11 at 18:23
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An individual measurement from a quantum mechanical wave function describing the system is random within the probability distribution described by the whole function. A sufficient number of measurements will show the probability distribution and not randomness.

Suppose we have a Psi that has expectation +1 and -1. Randomness means 50% of the time the measurement will find 1 and 50% -1. One can make an algorithm to mimic this. The only way the algorithm could have a connection with measurements is if, by some method, the experimenter could control it, and get either 1 or -1 at will. Immediately one gets to extra interactions, outside the solution Psi, by the word "control".

There could be a million algorithms. The existence of algorithms is not sufficient to demand the existence of hidden variables.

Hidden variables would be necessary if the data did not agree with the 50/50 calculated expectation value for this Psi and data in general.

If hidden variables exist, there will be a mathematical description for them, which you could call an algorithm, that might even give 50/50 in this simple case, and something spectacularly different for scattering, for example, that had necessitated their supposition.

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