Holographic Entropy Inequalities and Entropy Cones

The classical holographic entropy cone is the set of leading RT/HRT entropy vectors allowed by bulk min-cut geometry. It is a proper subset of the quantum entropy cone: holographic states satisfy universal inequalities such as strong subadditivity plus additional geometric inequalities such as monogamy of mutual information. Graph models provide exact finite combinatorial representatives of the static cone and make candidate inequalities auditable. A valid quantum entropy vector outside this cone is not inconsistent—it shows that the extra inequality is holographic rather than universal.

Required background. Wedge nesting and information inequalities supplies the geometric cut logic, and multipartite entropy cones supplies the abstract entropy-vector definitions.

Helpful background. Convex duality organizes facets; bit-thread multiflows provide flow proofs; mutual information fixes the basic combination; topological and long-range entanglement supplies non-holographic structures; and field Bell nonlocality prevents confusing entropy constraints with Bell inequalities.

Entropy vectors and graph cuts

For $N$ named boundary regions, collect the nonempty-subset entropies into

$$\mathbf S=(S_I)_{\varnothing\neq I\subseteq\{1,\ldots,N\}}.$$

At leading static holographic order, $S_I$ is a minimum cut capacity. A graph model has boundary vertices, optional bulk vertices, and nonnegative edge weights. For a boundary subset $I$,

$$S_I=\min_{\text{cuts separating }I\text{ from }I^c} \sum_{e\in\text{cut}}w_e.$$

Graph entropies form a convex cone under disjoint union and positive rescaling. Static holographic geometries can be represented by suitable weighted graphs for entropy-cone purposes, and graph models can be thickened into geometries, establishing their role beyond a suggestive analogy (Bao et al. 2015, §§2–3).

Monogamy on a four-leaf star

Take a central bulk vertex joined by unit-weight edges to boundary vertices $A,B,C,O$, where $O$ purifies the named system. A minimum cut for a subset $I$ places the central vertex on whichever side cuts fewer spokes, so

$$S_I=\min(|I|,4-|I|).$$

Therefore

$$S_A=S_B=S_C=S_{ABC}=1, \qquad S_{AB}=S_{AC}=S_{BC}=2.$$

Monogamy of mutual information is

$$S_{AB}+S_{AC}+S_{BC} \geq S_A+S_B+S_C+S_{ABC}.$$

The star gives $6\geq4$. Equivalently, the tripartite information

$$I_3(A:B:C)=S_A+S_B+S_C-S_{AB}-S_{AC}-S_{BC}+S_{ABC}$$

equals $-2\leq0$. Minimal-surface cut-and-paste proves this inequality for leading classical holographic entropies (Hayden, Headrick, and Maloney 2013, §§2–3). It is not a universal property of quantum states.

A five-party cyclic inequality

Now use a six-leaf unit star with named vertices $A_1,\ldots,A_5$ and purifier $O$. Again

$$S_I=\min(|I|,6-|I|).$$

The five-party cyclic inequality, with indices understood modulo five, is

$$\sum_{i=1}^{5}S(A_iA_{i+1}A_{i+2}) \geq \sum_{i=1}^{5}S(A_iA_{i+1}) +S(A_1A_2A_3A_4A_5).$$

Every triple has entropy $3$, every adjacent pair has entropy $2$, and the five-party set has entropy $S_O=1$. Hence

$$\text{left side}=5\times3=15, \qquad \text{right side}=5\times2+1=11.$$

The inequality is satisfied with margin four. This explicit graph verifies both the cut rule and a higher-party facet-type constraint. A check on one graph does not prove the inequality for the whole cone; contraction maps, cut rearrangements, or multiflows provide general proofs for appropriate families.

A valid quantum vector outside the holographic cone

Consider the perfectly correlated classical mixture, viewed as a quantum state diagonal in a product basis,

$$\rho_{ABC}=\frac12|000\rangle\langle000| +\frac12|111\rangle\langle111|.$$

Every nonempty marginal has entropy $\log2$:

$$S_A=S_B=S_C=S_{AB}=S_{AC}=S_{BC}=S_{ABC}=\log2.$$

It satisfies the ordinary quantum entropy inequalities, but

$$I_3(A:B:C)=\log2>0.$$

Equivalently, monogamy would demand $3\log2\geq4\log2$, which is false. This is the adversarial control requested by the page: a valid quantum entropy vector lies outside the classical holographic cone. The correct conclusion is that MMI restricts leading classical holographic states; it is not a universal axiom of quantum information.

Quantum and finite-$N$ corrections can also move a holographic entropy vector outside the classical cone by an amount of order $G_N^0$. One must not apply an $O(G_N^{-1})$ facet inequality to the full entropy without tracking the order and corrected prescription.

Cone boundaries and physical qualifications

The cone depends on the class of allowed bulk geometries, whether the state is static or covariant, and the number of parties. Purification identifies complementary entropies only when a purifier is included. Gauge centers and regulator choices affect continuum matter contributions but not the leading classical graph calculation. Phase transitions make the entropy function piecewise linear in graph weights or piecewise smooth in geometry; they are part of the cone boundary, not numerical noise.

Reflected entropy and negativity proposals concern different measures and do not automatically inherit von Neumann entropy-cone facets.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Bao, N., Nezami, S., Ooguri, H., Stoica, B., Sully, J., and Walter, M. (2015). “The holographic entropy cone.” Journal of High Energy Physics 2015(9), 130. DOI.
• Hayden, P., Headrick, M., and Maloney, A. (2013). “Holographic mutual information is monogamous.” Physical Review D 87, 046003. DOI.