How much information is required to completely describe a given volume in spacetime?

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    $\begingroup$ Depends on what's inside the volume. The limit is given by the Bekenstein bound. $\endgroup$ – psitae Jan 25 '19 at 16:42

There exists a limit on the maximum amount of entropy (or information) that can be contained in a physical system with a given volume and total energy, which is termed as Bekenstein bound. Though the second law of thermodynamics proposes that the entropy of an isolated system grows to a maximum value, this limit fills that knowledge gap. One can view this as an extension of the second law, limiting the growth of entropy beyond a maximum.

If a system of total energy $E$, including rest masses, is enclosed in a sphere of radius $R$, then the entropy $S$ of the system is bounded by $$S \leq \lambda RE$$ where $λ > 0$ is a constant (the value $λ = 2π$ is often proposed).

Here is a rigorous derivation using the general mathematical framework concerning quantum information of infinite systems.

References: Comments on Berkenstein bound

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