Geometric interpretations of the exponential of entropy Question:
Might there be a natural geometric interpretation of the exponential of entropy in Classical and Quantum Information theory? This question occurred to me recently via a geometric inequality concerning the exponential of the Shannon entropy.
Original motivation:
The weighted AM-GM inequality states that if $\{a_i\}_{i=1}^n,\{\lambda_i\}_{i=1}^n \in \mathbb{R}_+^n$ and $\sum_{i=1}^n \lambda_i = 1$, then:
\begin{equation}
\prod_{i=1}^n a_i^{\lambda_i} \leq \sum_{i=1}^n \lambda_i \cdot a_i \tag{1}
\end{equation}
As an application, we find that if $H(\vec{p})$ denotes the Shannon entropy of a discrete probability distribution $\vec{p} = \{p_i\}_{i=1}^n$ and $r_p^2 = \lVert \vec{p} \rVert^2 $ is the $l_2$ norm of $\vec{p}$ then:
\begin{equation}
e^{H(\vec{p})} \geq \frac{1}{r_p^2} \tag{2}
\end{equation}
This result follows from the observation that if $a_i = p_i$ and $\lambda_i = p_i$,
\begin{equation}
e^{-H(\vec{p})} = e^{\sum_i p_i \ln p_i} = \prod_{i=1}^n p_i^{p_i} \tag{3}
\end{equation}
\begin{equation}
\sum_{i=1}^n p_i^2 = \lVert \vec{p} \rVert^2 \tag{4}
\end{equation}
and using (1), we may deduce (2) where equality is obtained when the Shannon entropy is maximised by the uniform distribution i.e. $\forall i, p_i = \frac{1}{n}$.
A remark on appropriate geometric embeddings:
If we consider that the Shannon entropy measures the quantity of hidden information in a stochastic system at the state $\vec{p} \in [0,1]^n$, we may define the level sets $\mathcal{L}_q$ in terms of the typical probability $q \in (0,1)$:
\begin{equation}
\mathcal{L}_q = \{\vec{p} \in [0,1]^n: e^{H(\vec{p})} = e^{- \ln q} \} \tag{5}
\end{equation}
which allows us to define an equivalence relation over states $\vec{p} \in [0,1]^n$. Such a model is appropriate for events which may have $n$ distinct outcomes.
Now, we'll note that $e^{H(\vec{p})}$ has a natural interpretation as a measure of hidden information while $e^{-H(\vec{p})}$ may be interpreted as the typical probability of the state $\vec{p}$. Given (5), a natural relation between these measures may be found using the Hyperbolic identities:
\begin{equation}
\cosh^2(-\ln q) - \sinh^2(-\ln q) = 1 \tag{6}
\end{equation}
\begin{equation}
\cosh(-\ln q) - \sinh(-\ln q) = q \tag{7}
\end{equation}
where $2 \cdot \cosh(-\ln q)$ is the sum of these two measures and $2 \cdot \sinh(-\ln q)$ may be understood as their difference. This suggests that the level sets $\mathcal{L}_q$ have a natural Hyperbolic embedding in terms of Hyperbolic functions.
References:

*

*Olivier Rioul. This is IT: A Primer on Shannon’s Entropy and Information. Séminaire Poincaré. 2018.


*David J.C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press 2003.


*John C. Baez, Tobias Fritz, Tom Leinster. A Characterization of Entropy in Terms of Information Loss. Arxiv. 2011.
 A: If I may broaden the question somewhat to include other interpretations of the exponential of entropy, it is commonly used in ecology to measure how many different species there are in a community. In that context one considers the generalized diversity index (or Hill number)
$${}^q\!D=\left ( {\sum_{i} p_i^q} \right )^{1/(1-q)},$$
with the exponential of entropy being the $q\rightarrow 1$ limit,$^\ast$
$$\lim_{q\rightarrow 1} {}^q\!D=\exp\left(-\sum_{i} p_i\ln p_i\right).$$
The quantity ${}^1\!D=e^S$ is called the Shannon diversity.
Different values of $q$ correspond to different weights given to different species $i=1,2,\ldots$, dependent on their relative abundance $p_i$.
Increasing the value of $q$ increases the weight given to the most abundant species, thereby reducing the effective number of species, hence
$${}^q\!D\geq {}^{q'}\!D\;\;\text{if}\;\;q<q'$$
The inequality in the OP is the special case $q=1$, $q'=2$.

$\ast$

$$\lim_{q\rightarrow 1}=\left ( {\sum_{i} p_i^q} \right )^{1/(1-q)}=\lim_{q\rightarrow 1}\exp\left( -(1-q)^{-1}\left(1-{\sum_{i} p_i^q}\right) \right )=\lim_{q\rightarrow 1}\exp\left( \sum_i p_i\frac{1-p_i^{q-1}}{q-1} \right )$$
$$=\exp\left(-\sum_i p_i\ln p_i\right).$$
I have used $\sum_i p_i=1$ and the two identies $\lim_{x\rightarrow 0}(1-xa)^{1/x}=e^{-a}$ and $\lim_{x\rightarrow 0}(a^x-1)/x=\ln a$.

A: Not sure what you consider to be "geometric" in this context, but the exponential of the Shannon entropy is simply a measure of the "effective size of support" (ESS) of the underlying probability distribution.  In classical mechanics, it is a measure of the hyper-volume of the reasonably probable part of phase space, and is independent of the choice of canonical phase space coordinates.  In high energy physics, this is also closely related to the notion of the "emittance" of a particle beam in an accelerator or storage ring.  In quantum mechanics, the exponential of the von Neumann entropy is, roughly speaking, equal to the number of accessible microstates, and is independent of the choice of basis.  The logarithm is taken to (i) tame otherwise huge numbers, and more fundamentally, (ii) make the quantity additive over independent sub-systems.
A: With apologies for promoting my own work, there's a whole book on the
mathematics of the exponentials of various entropies:

Tom Leinster, Entropy and Diversity: The Axiomatic
Approach. Cambridge University Press, 2021.

You can download a free copy by clicking, although persons of taste will
naturally want to grace their bookshelves with the bound work.
The direct answer to your literal question is that I don't know of a
compelling geometric interpretation of the exponential of entropy. But the
spirit of your question is more open, so I'll explain (1) a non-geometric
interpretation of the exponential of entropy, and (2) a geometric
interpretation of the exponential of maximum entropy.
Diversity as the exponential of entropy
As Carlo Beenakker says, the exponential of entropy (Shannon or more
generally Rényi) has long been used by ecologists to quantify biological
diversity. One takes a community with $n$ species and writes $\mathbf{p} =
(p_1, \ldots, p_n)$ for their relative abundances, so that $\sum p_i =
1$. Then $D_q(\mathbf{p})$, the exponential of the Rényi entropy of
$\mathbf{p}$ of order $q \in [0, \infty]$, is a measure of the
diversity of the community, or "effective number of species" in the
community.
Ecologists call $D_q$ the Hill number of order $q$, after
the ecologist Mark Hill, who introduced them in 1973 (acknowledging the
prior work of Rényi). There is a precise mathematical sense in which the
Hill numbers are the only well-behaved measures of diversity, at least if one
is modelling an ecological community in this crude way. That's Theorem
7.4.3 of my book. I won't talk about that here.
Explicitly, for $q \in [0, \infty]$
$$
D_q(\mathbf{p}) = \biggl( \sum_{i:\,p_i \neq 0} p_i^q \biggr)^{1/(1 - q)}
$$
($q \neq 1, \infty$). The two exceptional cases are defined by taking limits
in $q$, which gives
$$
D_1(\mathbf{p}) = \prod_{i:\, p_i \neq 0} p_i^{-p_i}
$$
(the exponential of Shannon entropy) and
$$
D_\infty(\mathbf{p}) = 1/\max_{i:\, p_i \neq 0} p_i.
$$
Rather than picking one $q$ to work with, it's best to consider all of
them. So, given an ecological community and its abundance distribution
$\mathbf{p}$, we graph $D_q(\mathbf{p})$ against $q$. This is called the
diversity profile of the community, and is quite informative. As Carlo
says, different values of the parameter $q$ tell you different things about
the community. Specifically, low values of $q$ pay close attention to
rare species, and high values of $q$ ignore them.
For example, here's the diversity profile for the global community of great
apes:

(from Figure 4.3 of my book). What does it tell us? At least two things:

*

*The value at $q = 0$ is $8$, because there are $8$ species of great ape
present on Earth. $D_0$ measures only presence or absence, so that a
nearly extinct species contributes as much as a common one.


*The graph drops very quickly to $1$ — or rather, imperceptibly more
than $1$. This is because 99.9% of ape individuals are of a single
species (humans, of course: we "outcompeted" the rest, to put it diplomatically). It's only the very
smallest values of $q$ that are affected by extremely rare
species. Non-small $q$s barely notice such rare species, so from their
point of view, there is essentially only $1$ species. That's why
$D_q(\mathbf{p}) \approx 1$ for most $q$.
Maximum diversity as a geometric invariant
A major drawback of the Hill numbers is that they pay no attention to how
similar or dissimilar the species may be. "Diversity" should depend on the
degree of variation between the species, not just their
abundances. Christina Cobbold and I found a natural generalization of the Hill
numbers that factors this in — similarity-sensitive diversity
measures.
I won't give the definition (see that last link or Chapter 6 of the book),
but mathematically, this is basically a definition of the entropy or
diversity of a probability distribution on a metric space. (As before,
entropy is the log of diversity.) When all the distances are $\infty$, it
reduces to the Rényi entropies/Hill numbers.
And there's some serious geometric content here.
Let's think about
maximum diversity. Given a list of species of known similarities to one
another — or mathematically, given a metric
space — one can ask what the maximum possible value of the diversity
is, maximizing over all possible species distributions $\mathbf{p}$. In other words,
what's the value of
$$
\sup_{\mathbf{p}} D_q(\mathbf{p}),
$$
where $D_q$ now denotes the similarity-sensitive (or metric-sensitive)
diversity? Diversity is not usually maximized by the uniform distribution
(e.g. see Example 6.3.1 in the book), so the question is not trivial.
In principle, the answer depends on $q$. But magically, it doesn't! Mark
Meckes and I proved this. So
$$
D_{\text{max}}(X) := \sup_{\mathbf{p}} D_q(\mathbf{p})
$$
is a well-defined real invariant of finite metric spaces $X$, independent
of the choice of $q \in [0, \infty]$.
All this can be extended to compact metric spaces, as Emily Roff and I
showed. So every compact metric space
has a maximum diversity, which is a nonnegative real number.
What on earth is this invariant? There's a lot we don't yet know, but we do
know that maximum diversity is closely related to some classical geometric
invariants.
For instance, when $X \subseteq \mathbb{R}^n$ is compact,
$$
\text{Vol}(X)
=
n! \omega_n
\lim_{t \to \infty}
\frac{D_{\text{max}}(tX)}{t^n},
$$
where $\omega_n$ is the volume of the unit $n$-ball and $tX$ is $X$ scaled
by a factor of $t$. This is Proposition 9.7 of my paper with
Roff and follows from work of Juan
Antonio Barceló and Tony Carbery. In short: maximum diversity determines
volume.
Another example: Mark Meckes showed that the Minkowski dimension of a
compact space $X \subseteq \mathbb{R}^n$ is given by
$$
\dim_{\text{Mink}}(X)
=
\lim_{t \to \infty}
\frac{D_{\text{max}}(tX)}{\log t}
$$
(Theorem 7.1 here). So, maximum diversity
determines Minkowski dimension too.
There's much more to say about the geometric aspects of maximum
diversity. Maximum diversity is closely related to another recent invariant of
metric spaces, magnitude. Mark
and I wrote a survey paper on the more
geometric and analytic aspects of magnitude, and you can find more on all
this in Chapter 6 of my book.
Postscript
Although diversity is closely related to entropy, the diversity viewpoint really opens up new mathematical questions that you don't see from a purely information-theoretic standpoint. The mathematics of diversity is a rich, fertile and underexplored area, waiting for mathematicians to come along and explore it.
A: With apologies also for promoting my own work, a colleague (Tim Hosgood) and I have recently published an article in Entropy, where we give a geometric interpretation for the exponentiated entropy of empirical distributions, namely as the length $L$ of a certain rectangle. What makes this rectangle interesting is that its width $W$ and area $A$ (from which you can calculate its length as $L=\frac{A}{W}$ and hence entropy as $H=\log(L)$) are additively and multiplicatively homomorphic in the original distribution.
It's well known that the entropy of a product of distributions is the sum of their entropies, so the exponentiated entropy (sometimes called the perplexity) of a product is the product of the exponentiated entropies. Thus one says that exponentiated entropy is multiplicatively homomorphic in the distribution. What we're saying in this short note is that if you use the whole rectangle (its area and width) instead of just its length, you can get not only a multiplicative but also an additive homomorphism; that is you get a rig homomorphism from empirical distributions to rectangles.
