# Formulation of Heisenberg's uncertainty in terms of Shannon entropy

Heisenberg's uncertainty principle $$\Delta x \Delta p \geq h$$ asserts a fundamental limit to the accuracy versus uncertainty with which the values for the two canonically conjugate variables $$\Delta x$$ (deviation of position) and $$\Delta p$$ (deviation of momentum) can be predicted from initial conditions.

Question: From an information theory point of view, how may we mathematically interpret and formulate this notion of uncertainty in terms of the Shannon entropy?

• Take a look here or here. Dec 18, 2021 at 17:24
• That's not what the uncertainty relation states. Wikipedia is not a reliable source on physics. Dec 18, 2021 at 23:06
• At the very least, it’s references are mostly reliable. It’s most constructive to recommend better sources instead of making faces, @Jan Lalinky … Dec 18, 2021 at 23:34

Meanwhile I found to a solid answer.

The uncertainty principle of Heisenberg, as any on a Fourier transformation based uncertainty principle, is a consequence of an entropic uncertainty principle, but not vice versa: $$H_{a}+H_{b}\geq \log(e/2)$$ where $$H_{a}$$ and $$H_{b}$$ be Shannon information entropies (Hirschman entropies) of the Fourier transform conjugates: $$f(a)=\int _{-\infty }^{\infty }g(b)\ e^{2\pi iab}\,db$$ $$g(b)=\int _{-\infty }^{\infty }f(a)\ e^{-2\pi iab}\,da$$ meaning: $$H_{a}=\int _{-\infty }^{\infty }f(a)\log(f(a))\,da$$ $$H_{b}=\int _{-\infty }^{\infty }g(b)\log(g(b))\,db$$

References:

• DeWitt, B. S.; Graham, N. (1973), The Many-Worlds Interpretation of Quantum Mechanics, Princeton: Princeton University Press, pp. 52–53, ISBN 0-691-08126-3
• Beckner, W. (1975), "Inequalities in Fourier analysis", Annals of Mathematics, 102 (6): 159–182
• Bialynicki-Birula, I.; Mycielski, J. (1975), "Uncertainty Relations for Information Entropy in Wave Mechanics", Communications in Mathematical Physics, 44 (2): 129–132
• Hirschman, I. I., Jr. (1957), "A note on entropy", American Journal of Mathematics, 79 (1): 152–156
• Everett, Hugh (1957). On the foundations of quantum mechanics (PhD thesis)

I like the references and the notion of Fourier defined entropies that I never saw, although I was triggered from a similar definition of Heisenberg relations starting from Legendre transforms some time ago.

Even without multi-world, I would like to add yet another Wikipedia link that we were browsing a few days ago and a perspective from the H-Theorem in there you see that the H-theorem has a basis on the Shannon entropy where the Shannon entropy is a kind of discrete precursor of the Boltzmann entropy.

Let's say that Shannon entropy in essence it may be well defined as combinatorial entropy but to the extent of physics it would be only well defined purely in a kind of equilibrium and decoherent state or approximations thereof, e.g local equilibrium approx or linear response to name two.

If you imagine by extreme that the local state is some shape of fractal in the thermodynamics variable space, or discrete jumps if you think of it in the Kolmogorov perspective, it will essentially not be possible to define properly an entropy as a whole, for sure not as a standard function with a standard minimum as a differentiable function would do, maybe you can define an entropy per Kolmogorov scale, or you may end up with some weird looking entropy, I imagine here some shape of Weierstrass/fractal alike function, where the definition of minimum would be tricky, and it may be well defined only globally and not locally. From a physics standpoint is questionable that entropy may make sense at all, in a system which is wildly not at equilibrium.

On the opposite end of the spectrum a coherent quantum system typically would have entropy 0 being one state, and it may become even a finite entropy in another eigenfunction representation.

Therefore there is definitely some shape of extra "information" jump from Shannon entropy for a finite set of states, to a countable set (like eigenfunctions or coherent states which is actually still countable but eventually "larger"), to a fully "not necessarily" countable one (i.e. decoherent, plus equilibrium, plus entanglement).

Comparing The number operator representation (which would be logical to use for understanding Shannon entropy) with the coherent states where the number of states is not fixed, let's say a gas of photons which you can consider gapless (i.e a system there is no effort or energy gap to create an extra photon), there are also some considerations with the density matrix, but this does not seem to create troubles given both sets are still countable.

Therefore coherence will give you the minimum possible entropy, i.e the equal sign in the Heisenberg relation which I would expect that is dual to a minimum in your Hirschman entropies.