# How does quantum entropy scales with the size of the sample?

Suppose i have a 3D bulk of physical matter with no black holes enclosed in a sphere of radius $R$. What is the scaling law of all quantum entropy in function of $R$?

If the scaling is not $R^2$, in what sense can we argue that the holographic principle holds in our universe?

Under what scenarios quantum entropy of non-black hole matter will scale with the area of the enclosed volume?

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Non-black-hole matter – more precisely one in which the curvature may be considered a small perturbation everywhere – always has a volume-extensive entropy that scales like $R^3$. It follows from the locality. In various condensed-matter systems, one may develop environments with various long-distance correlations that make the entropy non-extensive and may replace $R^3$ by $R^n$ with rather general fractional powers.
The scaling of the entropy as $R^2$ applies to the maximum entropy that can be squeezed into a given region, and the maximum entropy of a localized/squeezed system is the black hole entropy. The most fundamental laws of physics are those that can describe the most general state or conditions i.e. the highest entropy. And those fundamental laws see the $R^2$ upper bound on the entropy which is the entropy counting supporting the holographic nature of the Universe. Whether some lower-entropy (non-gravitational) systems with a volume-extensive entropy exist is irrelevant and doesn't create any contradiction; instead, this fact known for centuries is what makes the holographic principle non-trivial and powerful.