Entropic Cone Closure: Geometry & Applications

• The closure of the entropic cone is the topological closure of entropy vectors from discrete random variables, forming a convex but non-polyhedral set for n ≥ 4.
• It is analyzed by decomposing into tight and modular parts, with quasi-uniform random vectors revealing boundary features beyond classical Shannon inequalities.
• Computational and geometric methods, including support enumeration and information geometry, help explore its unknown boundaries and applications in network coding and quantum contexts.

Searching arXiv for recent and foundational papers on closure of the entropic cone and related boundary characterizations. The closure of the entropic cone is the topological closure of the set of entropy vectors generated by discrete random variables. For $n$ variables, the entropy vector records the joint entropies of all nonempty subsets, and the corresponding entropy region is denoted in the literature by symbols such as $\Gamma_n^*$, $\mathcal{T}_n^*$, or $\mathcal{E}_N$. Its closure, written for example as $\overline{\Gamma_n^*}$ or $\overline{\mathcal{T}_n^*}$, is also called the almost entropic region or entropic cone. In the classical setting this closure is a convex cone, and it is the natural object of study because the set of entropic vectors itself need not a priori be closed; indeed, whether the full region of entropy vectors is closed is described as a major open problem in information theory (Thakor et al., 2022, Gadouleau, 2013).

1. Definitions, outer bounds, and basic geometry

For a set of $n$ discrete random variables $X_{[n]}$, the entropy vector $h$ consists of the joint entropies $h_A$ of all nonempty subsets $\Gamma_n^*$0. The entropy region $\Gamma_n^*$1 is the set of all such vectors, while its closure $\Gamma_n^*$2 is the almost entropic region. An equivalent notation widely used for the region of entropic vectors is $\Gamma_n^*$3, with closure $\Gamma_n^*$4 (Thakor et al., 2022, Liu et al., 2015).

A universal outer bound is the polymatroidal cone, denoted $\Gamma_n^*$5 in one notation and $\Gamma_n^*$6 in another. It consists of all vectors satisfying the basic Shannon-type inequalities, including nonnegativity and submodularity: $\Gamma_n^*$7 and

$\Gamma_n^*$8

For $\Gamma_n^*$9, the closure of the entropy region coincides with this Shannon outer bound: $\mathcal{T}_n^*$0 For $\mathcal{T}_n^*$1, the inclusion is strict: the almost entropic region is a proper subset of the Shannon cone (Thakor et al., 2022).

The closure is convex, but for four or more variables it is not polyhedral. Multiple sources in the literature emphasize that non-Shannon inequalities are required for $\mathcal{T}_n^*$2, and that the boundary cannot be captured by finitely many Shannon-type inequalities alone (Liu et al., 2015, Matveev et al., 2019). This distinction between the Shannon cone and the closure of the actual entropy region is the central geometric obstruction in the subject.

2. Boundary geometry for three random variables

The three-variable case is exceptional because the closure is fully described by Shannon-type inequalities, yet its boundary already exhibits the distinction between entropic and almost entropic phenomena. The proper faces of the Shannon region form the boundary, and for $\mathcal{T}_n^*$3 there are eight extreme rays. All but one correspond to entropy vectors; the exceptional ray $\mathcal{T}_n^*$4 contains non-entropic vectors. In one explicit description, $\mathcal{T}_n^*$5 is proportional to $\mathcal{T}_n^*$6, corresponding to random variables that are pairwise independent and each a deterministic function of the other two (Tiwari et al., 2020).

Faces containing $\mathcal{T}_n^*$7 are precisely the faces that can contain non-entropic boundary points. This produces a detailed face stratification: one-dimensional, two-dimensional, and higher-dimensional faces containing $\mathcal{T}_n^*$8 become the natural testing ground for sharp inner and outer bounds (Tiwari et al., 2020).

Several boundary characterizations are known exactly in low-dimensional faces. On the face $\mathcal{T}_n^*$9, an entropy vector is entropic if and only if

$\mathcal{E}_N$0

On the face $\mathcal{E}_N$1, an entropy vector is entropic if and only if

$\mathcal{E}_N$2

The same line of work also establishes new outer bounds on larger faces. For $\mathcal{E}_N$3, entropicity implies that $\mathcal{E}_N$4 and $\mathcal{E}_N$5 have the same marginal distribution, hence $\mathcal{E}_N$6. For $\mathcal{E}_N$7, entropicity implies

$\mathcal{E}_N$8

Explicit constructions further show that previously used inner bounds obtained from lower-dimensional faces by addition are not tight on faces such as $\mathcal{E}_N$9 and $\overline{\Gamma_n^*}$0 (Tiwari et al., 2020).

This three-variable analysis is significant because it shows that even where the closure is completely known as a cone, the entropic points on its boundary are already highly structured and not exhausted by simple conic or additive recipes.

3. Tight and modular decomposition of the closure

A structural reduction of the closure is obtained by decomposing polymatroids into tight and modular parts. Let $\overline{\Gamma_n^*}$1 denote the cone of all polymatroidal rank functions, let $\overline{\Gamma_n^*}$2 be the cone of modular functions, and let $\overline{\Gamma_n^*}$3 be the tight cone consisting of all $\overline{\Gamma_n^*}$4 such that

$\overline{\Gamma_n^*}$5

Then the closure of the entropy region admits the direct-sum decomposition

$\overline{\Gamma_n^*}$6

Equivalently, every point in the closure has a unique decomposition into a tight part and a modular part (Matúš et al., 2013).

For a given $\overline{\Gamma_n^*}$7, these components are given explicitly by

$\overline{\Gamma_n^*}$8

and

$\overline{\Gamma_n^*}$9

This decomposition reduces the study of the closure to the tight part, where the nontrivial geometry is concentrated (Matúš et al., 2013).

A particularly useful consequence is the relative-interior statement

$\overline{\mathcal{T}_n^*}$0

Thus, the relative interior of the tight reduction is exhausted by entropic points; non-entropic closure effects occur only at the boundary. For four variables, this reduction interfaces naturally with the Ingleton inequality and with symmetrized three-dimensional sections used for visualization and computation (Matúš et al., 2013).

This suggests that much of the difficulty of the closure problem lies in the fine boundary geometry of the tight component rather than in the modular directions.

4. Quasi-uniform random vectors and the almost entropic boundary

Quasi-uniform random vectors occupy a central position in the characterization of the closure. A random vector is quasi-uniform when its probability mass function assigns either $\overline{\mathcal{T}_n^*}$1 or a constant positive value to elements of its support; equivalently, every nonempty marginal $\overline{\mathcal{T}_n^*}$2 is uniformly distributed on its support. If $\overline{\mathcal{T}_n^*}$3 denotes the set of entropy vectors achievable by quasi-uniform random vectors, then a fundamental equivalence is

$\overline{\mathcal{T}_n^*}$4

Hence the convex closure of quasi-uniform entropy vectors coincides with the almost entropic region itself (Thakor et al., 2022).

For quasi-uniform entropy vectors, a key structural condition is

$\overline{\mathcal{T}_n^*}$5

for every nonempty $\overline{\mathcal{T}_n^*}$6. This arithmetic form makes quasi-uniformity particularly effective for boundary constructions (Thakor et al., 2022).

Using this framework, the boundary of the almost entropic region for three random variables can be probed more sharply than with traditional additive inner bounds. On the four-dimensional face

$\overline{\mathcal{T}_n^*}$7

a standard inner-bound construction uses vectors of the form

$\overline{\mathcal{T}_n^*}$8

with the restriction $\overline{\mathcal{T}_n^*}$9 for some integer $n$0. The quasi-uniform approach produces entropy vectors in $n$1 outside this inner bound. A concrete example is

$n$2

which is realized by an explicit quasi-uniform distribution, while $n$3 is not of the form $n$4 for $n$5 (Thakor et al., 2022).

Analogous looseness occurs on the five-dimensional face

$n$6

These constructions show that the boundary contains entropic points that cannot be reached by addition and convexity from lower-dimensional subfaces. In communication-theoretic terms, the one-to-one correspondence between quasi-uniform codes and quasi-uniform random vectors makes these results directly relevant to the design of network codes and to converse arguments in network information theory (Thakor et al., 2022).

5. Computational and geometric methods for the unknown boundary

For four or more variables, the closure is not fully characterized, so much of the literature develops methods for mapping its unknown boundary. One prominent approach is support enumeration. The region of entropic vectors is explored by enumerating non-isomorphic supports of joint probability mass functions, using group actions on set partitions and canonical representatives. In this setting the “Snakes and Ladders” algorithm is used to construct a transversal of support orbits efficiently, after which numerical optimization over the nonzero probabilities yields inner bounds in the unknown part of the region (Liu et al., 2015).

This approach is especially informative in the four-variable case, where the Shannon outer bound $n$7 is decomposed into the Ingleton inner bound $n$8 and six cones corresponding to violation of one of the six Ingleton inequalities. The same work reports that among 75 nonisomorphic 4-atom supports, only one can assign probabilities so as to violate Ingleton (Liu et al., 2015).

Information geometry supplies a complementary analytic description. Conditional independence constraints define e-autoparallel submanifolds in exponential coordinates, and equality in a Shannon inequality corresponds to an affine condition in $n$9-coordinates. In this way, faces of the Shannon cone and some Ingleton-violating families become geometric objects in a coordinate system adapted to statistical structure (Liu et al., 2015).

A distinct asymptotic framework is tropical probability theory. There the entropic cone is studied through quasi-linear sequences of diagrams, equipped with the asymptotic entropy distance

$X_{[n]}$0

The space of tropical diagrams forms a complete closed convex cone, and homogeneous diagrams are dense in it. Within this framework, a dimension-reduction theorem for four variables states that the boundary function $X_{[n]}$1 describing the maximal value of the 15th entropy coordinate in terms of the first 14 satisfies

$X_{[n]}$2

Thus only 11 of the 15 coordinates are essential for that boundary description (Matveev et al., 2019).

Taken together, these methods indicate that the unknown boundary is accessible neither by purely combinatorial enumeration nor by purely inequality-based reasoning alone; rather, it is approached through an overview of support geometry, information geometry, and asymptotic cone methods.

6. Related cones, applications, and extensions

The closure of the entropic cone is closely connected to several adjacent cones defined by operational or structural constraints. In Bell locality and Kochen-Specker noncontextuality scenarios, the joint Shannon entropies of local or noncontextual models define a convex cone whose non-trivial facets are tight entropic Bell or contextuality inequalities. The experimentally relevant object is often a projection of the full entropy cone onto observable marginals, and Fourier-Motzkin elimination is used to derive the corresponding facet inequalities (Chaves et al., 2012).

In compatibility scenarios, the entropic no-disturbance cone is defined by an intersection of context-wise cones,

$X_{[n]}$3

with the hierarchy

$X_{[n]}$4

This construction parallels the probabilistic no-disturbance or nonsignaling picture and uses the convexity of the closure of entropic vectors in an essential way (Jia et al., 2018).

Another dual perspective is provided by the monotonicity cone of linear entropic formulas under local operations. For classical and quantum systems, this cone is polyhedral; its facets are derived from strong subadditivity and lower-set constraints, and the paper introducing it proves that all linear entropic monotones are implied by strong subadditivity. In the classical case, non-Shannon inequalities valid for the entropy cone do not produce new monotonic linear formulas under local operations (Alhejji et al., 2018). This clarifies that the dual geometry relevant for monotonicity is narrower than the full geometry of the closure.

Extensions also appear in the theory of closure operators. There, Shannon entropy is defined as a polymatroidal relaxation, closure entropies are shown to be dense, and the work emphasizes that density of attainable entropy values does not settle the closure question for the associated entropic sets (Gadouleau, 2013). A plausible implication is that scalar entropy observables can be dense even when the full vector geometry remains unresolved.

Quantum variants complicate the classical conical picture. In the $X_{[n]}$5-partite quantum setting, homogeneous inequalities such as strong subadditivity constrain the closure of the set of entropy vectors, but non-homogeneous inequalities near the apex impose strictly positive lower bounds under certain saturation conditions. In particular, the set $X_{[n]}$6 of realizable quantum entropy vectors is shown not to be a cone for $X_{[n]}$7, because arbitrary down-scaling can fail even though integer up-scaling is always possible (Christandl et al., 2023). This does not alter the classical definition of the almost entropic region, but it shows that “entropy cone” terminology becomes more delicate outside the classical Shannon framework.

The closure of the entropic cone therefore serves simultaneously as a geometric object, an asymptotic limit, and a unifying language across network coding, contextuality, monotonicity, and combinatorial closure theory. Its most robustly established features are convexity, the Shannon outer approximation, the non-polyhedral nature of the four-variable and higher-variable case, and the special role of quasi-uniform constructions. Its least understood aspect remains the precise boundary for $X_{[n]}$8, where explicit constructions, asymptotic reductions, and geometric parametrizations continue to supply partial but increasingly sharp descriptions (Thakor et al., 2022, Liu et al., 2015).