Entropohedron: Unified Entropy Geometry

• Entropohedron is a unifying geometric construct defined as a convex set representing entropy distributions and correlations in classical and quantum frameworks.
• It employs methodologies like entropy vector analysis, holographic bit thread formulations, and dynamical phase space studies to reveal fundamental entropic constraints.
• Its investigation bridges classical polymatroid theory, quantum entanglement, and chaotic dynamics, highlighting both universal properties and system-specific complexities.

The entropohedron is a unifying geometric concept that encodes the structure of entropy, correlations, and entanglement in mathematical physics, quantum information theory, and quantum gravity. It appears independently in several domains, each time as a central object characterizing the possible or allowed patterns of entropy distribution subject to intrinsic physical or mathematical constraints. The entropohedron is typically realized as a convex set—often a cone or polytope—in an appropriate entropy or flux space, with faces and extremal points corresponding to fundamental features of information structure or physical states.

1. Entropohedron in Classical and Quantum Entropy Regions

The earliest systematic appearance of the entropohedron is in the paper of Shannon entropy vectors of random variables. For a finite set $N$, the entropy region $\Gamma_N^*$ is the set of vectors of joint entropies $(H(X_I): I \subset N)$ taken over all distributions $(X_i)_{i\in N}$. Its closure, $\overline{\Gamma_N^*}$, often referred to as the "entropohedron," is a convex cone and forms the set of almost entropic polymatroids (Matúš et al., 2013). Every such vector is monotonic and submodular, connecting the entropy region to the theory of polymatroids.

The structure of the entropohedron in this context is non-polyhedral; it is not finitely characterized by linear inequalities due to the existence of non-Shannon-type information constraints. A key insight is that the overall entropy region can be decomposed into a direct sum of its modular and tight parts; the modular part is fully entropic and easy to describe, whereas the essential complexity—and hence the intricate geometry of the entropohedron—lies in the tight part.

For four variables, explicit computations and visualizations reveal that the entropohedron is a highly nontrivial low-dimensional convex body significantly smaller than the polyhedron suggested by all known inequalities. Computer experiments have demonstrated that its shape remains subtle and only partially understood, with notable results such as the refutation of the Four-Atom Conjecture via the construction of explicit entropy vectors with Ingleton scores lower than previously known (Matúš et al., 2013).

Notion	Mathematical Description	Context
Entropy region	$\overline{\Gamma_N^*}$ (closure of entropy vectors)	Random variables/polymatroids
Entropohedron (classical)	Set of almost entropic polymatroid rank functions	Information theory

2. The Entropohedron as a Flux Polytope in Holography

The entropohedron appears in holographic quantum gravity as the set of all possible assignments of entanglement density consistent with global entropy constraints (Headrick et al., 26 Oct 2025). This arises in the context of quantum bit thread formulations for the quantum extremal surface (QES) prescription of holographic entanglement entropy. Here, the entropohedron is a convex polytope, denoted $F_S$, in $\mathbb{R}^N$ for $N$ parties, defined as the set of entanglement distribution functions (EDFs):

$$F_S := \left\{ f : X \to \mathbb{R} \mid \forall A \subseteq X, \left| \sum_{x \in A} f(x) \right| \leq S(A) \right\}$$

where $S(A)$ is the von Neumann entropy for subset $A$. Each face of the entropohedron corresponds to a set of parties for which the entropy bound is saturated. This construction packages the total entropy structure of a quantum state and is tightly connected to allowed quantum bit thread flows: every point corresponds to a boundary flux vector of a strict quantum flow.

This setup encodes and geometrizes all monotonicity, subadditivity, and higher-order information constraints arising from quantum entropy, providing a direct method for translating boundary entropic data to geometric (flow) data in holographic theories. In addition, it reflects monogamy and other information constraints, and its structure adapts in the presence of quantum phenomena such as entanglement islands and baby universes (Headrick et al., 26 Oct 2025).

Notion	Mathematical Description	Physical Meaning
Entropohedron (holographic)	$$F_S = \{ f \in \mathbb{R}^N : \forall A, |\sum_{x \in A} f(x)| \leq S(A) \}$$	Allowed "entanglement flux" patterns compatible with entropy data

3. Arrangements, Polyhedra, and Entropic Cones

In multipartite holographic systems, the entropohedron connects to combinatorial and geometric objects constructed from information quantities (Hubeny et al., 2018). The holographic entropy arrangement is an arrangement of hyperplanes in entropy space, with each hyperplane corresponding to a primitive information quantity (e.g., mutual or multipartite information) vanishing. The intersection of the half-spaces defined by sign-definite inequalities on these quantities carves out a convex polyhedral cone: the holographic entropy polyhedron, or "entropohedron" in this context.

Formally, if $Q^{(k)}$ are the list of primitive, sign-definite information quantities, the entropohedron is

$$P = \bigcap_k \{ \vec{S} \in \mathbb{R}^D : Q^{(k)}(\vec{S}) \geq 0 \}$$

This construction systematizes the derivation and classification of fundamental entropic constraints. In the holographic context, it has been shown that the entropohedron constructed in this manner coincides, for up to four parties, with the holographic entropy cone: the convex hull of entropy vectors realisable by geometric (bulk) configurations (Hubeny et al., 2018). The combinatorial properties and symmetries of the arrangement reflect the algebraic and topological features of allowed multipartite correlations.

Structure	Definition	Role
Holographic entropy arrangement	Collection of hyperplanes defined by vanishing information quantities	Encodes entanglement diagnostics
Holographic entropohedron (polyhedron)	Region allowed by all (universal) holographic entropy inequalities	Spectrum of allowed entropy vectors

4. Dynamical Entropohedron in Polyhedral Phase Space

The term "Entropohedron" is used in the dynamical context by designating minimal systems with built-in entropy generation and strong chaotic dynamics (Coleman-Smith et al., 2012). In the paper of the isochoric pentahedron (the simplest polyhedron beyond the tetrahedron), the Kapovich-Millson phase space is used, where the volume of the pentahedron serves as Hamiltonian. The system displays fast mixing, strong chaos (as measured by Lyapunov exponents and Kolmogorov-Sinai entropy), and the dominance of locally unstable trajectories. These features make the pentahedron a prototype for chaotic, high-entropy micro-dynamics in quantum gravity.

This usage of "entropohedron" highlights the geometric origin of entropy and chaos at the most elementary level, with implications for black hole information and fast scrambling phenomena in quantum gravity. The entropohedron, in this guise, represents a minimal unit exhibiting entropy generation at the classical-quantum boundary.

5. Mathematical Properties, Transformations, and Operations

The entropohedron in various incarnations enjoys several universal mathematical properties:

• Convexity: In all settings, the entropohedron is convex (either a cone or a polytope), with extremal points corresponding to saturations of subsets of constraints.
• Decomposition: In the context of polymatroids, the space decomposes into modular and tight parts, reducing paper to a lower-dimensional "essential" entropohedron (Matúš et al., 2013).
• Convolution Closure: The closure of the entropy region (the entropohedron) is closed under convolution with modular polymatroids.
• Projection and Lifting: Tracing out or purifying parties in quantum information projects or lifts the entropohedron appropriately, reshaping the region of admissible entropy vectors or fluxes.
• Saturation and Facets: Faces correspond to simultaneous saturation of entropy inequalities, mapping to informational concepts such as monogamy or conditional/mutual information.

Operation	Property	Impact on Entropohedron
Convolution	Closed under convolution with modulars	Generates new entropic polymatroids
Projection (marginalization)	Projection in entropy/flux space	Corresponds to tracing out systems
Purification	Hyperplane embedding	Adds conservation constraints

6. Significance, Open Questions, and Comparisons

The entropohedron is significant because it encodes the allowed structure of entropy assignments and distributions across mathematical physics, from classical information theory to quantum gravity. It captures fundamental constraints (polymatroidal, quantum, or geometric) and reveals both geometric and combinatorial structure.

• In classical information theory, its boundaries are shaped by yet-to-be-fully-classified non-Shannon inequalities.
• In quantum/holographic settings, it reflects the interplay between geometry, information, and quantum correlations.
• The gap between known entropic inner approximations and outer inequality-based bounds in the classical case suggests profound underlying complexity, even for four variables (Matúš et al., 2013).
• In dynamical systems, it illustrates that even minimal classical models of quantum geometry display fast-mixing and chaos (Coleman-Smith et al., 2012).

Comparison between entropohedra arising from different settings—the polymatroidal (classical), bit-thread (quantum/holographic), and dynamical (chaotic/phase space)—reveals both universalities (convexity, encoding entropy constraints) and sharp contrasts in how the structure encodes information, correlations, or dynamics.

7. Mathematical Formulas and Summary Table

Key defining formulae for the entropohedron in the featured contexts:

Setting	Mathematical Definition
Polymatroidal	$\overline{\Gamma_N^*}$ (set of monotonic submodular functions; cone)
Bit thread/holography	$$F_S = \{ f \in \mathbb{R}^N : \forall A, |\sum_{x \in A} f(x)| \leq S(A) \}$$
Arrangement/cone	$$P = \bigcap_k \{ \vec{S} \in \mathbb{R}^D : Q^{(k)}(\vec{S}) \geq 0 \}$$
Dynamical	Phase space region with high Kolmogorov-Sinai entropy, strong Lyapunov instability

8. Concluding Perspective

The entropohedron stands as a geometric, algebraic, and information-theoretic crucible for the paper of entropy and correlations. Its boundaries are the "facets" of allowed entropy assignments, dictated by fundamental mathematical or physical laws, whether those be polymatroidal submodularity, quantum information inequalities, geometric flow constraints, or the chaos of minimal quantum-gravitational building blocks. Characterizing the entropohedron, both in low and high dimensions, remains a deep challenge with continuing impact on quantum information, combinatorics, network theory, and the foundations of quantum gravity (Coleman-Smith et al., 2012, Matúš et al., 2013, Headrick et al., 26 Oct 2025, Hubeny et al., 2018).