# Uncertainty principle and Cramer-Rao bound - is there relation ?

Just out of curiosity.

The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound. Saying that we cannot measure something with certain accuracy. However looking closer they have completely different setup.

Question May be nevertheless there is some relation ?

I googled a little, there are many papers with these two words inside, but may be too many ...

If someone can give self-contained comment it would be interesting, (well, any comment is welcome).

Let me comment on "uncertainty principle" and "Cramer-Rao bound".

http://en.wikipedia.org/wiki/Uncertainty_principle

Uncertainty principle in "quantum mechanics" - actually is a fact about matrices. We know that if two matrices commute, then they have joint eigenbasis (assuming smth). So we might ask if they do not commute - what happens ? How to measure the deviation from the fact that they do not commute ? The basic example is A, B: [A,B]=1. e.g. A = x, B = d/dx. The answer goes as follows: take any vector v and decompose it in eigenbasis for A, the "uncertainty principle" says that if vector is "localized" in eigenbasis for A, then it should be unlocalized in eigenbasis for B.

In the basic example A= x, B = d/dx, the two eigenbasis related by the Fourier Transform and so we get that if function is localized in "delta-function" basis - e.g. it is just localized in common sense, then its FT will be delocalized.

Cramer-Rao bound. (Statistics)

http://en.wikipedia.org/wiki/Cramer-Rao

It basically says the following - one have a sample of some random variable and we want estimate some parameter - then we cannot kill randomness completely, e.g. there will always be certain inaccuracy in our estimation of parameter. The Cramer-Rao inequality provides quantitative bound.

So you see setups are completely different - one is about matrices, another is in probabilty theory.

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Great question. As far as I can tell, the uncertainty principle is a complex version of the Cramer-Rao inequality. Or the Cramer-Rao inequality is a real version of the uncertainty principle. Both give a lower bound for the product of the second moment of position and the second moment of momentum (which is called Fisher information in information theory). I believe the proofs are also quite similar (integration by parts and Cauchy-Schwarz), but I've never investigated this carefully enough. – Deane Yang Dec 27 '11 at 4:22
@Deane Thanks for the comment ! Reading it comes to my mind that if we take normal distribution in CramerRao setup we somehow can arrive to uncertainty relation. And it should not be that much complicated... – Alexander Chervov Dec 27 '11 at 4:35
One of my friends had immediately expressed the same reaction when I had told him about the Cramer-Rao bound :) Wonder what connection it might have? Or is it that the uncertainty principle is a concrete example of Cramer-Rao bound? that would be intuitive, I guess. What say @Alexander Chervov and @Deane Yang? – NikBels Dec 27 '11 at 6:58
@Nikhil. Thank You for the comment, so if question came not only to my mind, makes me think it was not so bad to post it to MO. I do not have clear idea on the questions you ask - at the beginning I thought connection might not exists at all or something not very clear exist. But after comment it seems it might be not that difficult. So let us wait may be some one generate some more idea or idea will come to our mind by itself:) – Alexander Chervov Dec 27 '11 at 7:21
@Nikhil. the problem is that the setup is quite different, the words "accuracy of measurement" are the same, but have different meanings in the two setups. Quite completely different ! – Alexander Chervov Dec 27 '11 at 7:24

Not so much a connection as a lurking variable, but both Cramér-Rao and the Uncertainty Principle follow from the Cauchy-Schwartz inequality.

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Thanks, and also Deane Yang already mentioned Cauchy-Schwartz in a comment... – Bjørn Kjos-Hanssen Dec 27 '11 at 9:27

I can't answer the actual question, but one of the first places I'd look for an answer is Science from Fisher information: a unification by B. Roy Frieden. He seems to be saying the Cramér–Rao lower bound explains just about everything in science.

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Sounds attractive :) – Alexander Chervov Dec 27 '11 at 11:31
I would be wary of Frieden's book - see a scathing review by Cosma Shalizi: cscs.umich.edu/~crshalizi/reviews/physics-from-fisher-info – Marcin Kotowski Dec 27 '11 at 16:00
I found the book to be quite unenlightening. – Deane Yang Dec 27 '11 at 16:42
@Dean, Marcin Thanks for comments ! – Alexander Chervov Dec 28 '11 at 7:04
I don't actually (yet) have an opinion of that book. But it seems to address the topic of this question. – Michael Hardy Dec 28 '11 at 18:23