Multipartite Information and Entropy Cones

Multipartite entropy vectors organize all subsystem entropies of a state, while entropy cones encode the homogeneous linear inequalities they obey. The generic quantum cone, stabilizer cone, and holographic cone are different objects; inequalities valid in a special state class must not be presented as universal QFT constraints.

Required background. Use Strong Subadditivity and Entropic Inequalities. Helpful background. Mutual Information for Disjoint Regions supplies regulator-safe combinations.

Evidence cutoff. Research-sensitive comparisons on this page include primary literature available through 10 August 2026.

Entropy vectors and multipartite information

For $N$ regulated parties, collect $S(X)$ for every nonempty union $X$. The vector lies in the quantum entropy cone after closure over all states and dimensions. Positivity, subadditivity, and strong subadditivity impose linear inequalities. For three parties, the tripartite information is

$$I_3(A{:}B{:}C) =S(A)+S(B)+S(C) -S(AB)-S(AC)-S(BC)+S(ABC).$$

Generic quantum states allow either sign. The holographic monogamy inequality $I_3\leq0$ is therefore a constraint on a special class, not on QFT states in general.

The structure map places entropy cones on the multipartite branch.

Multipartite information is a linear combination of entropies; a cone is the set of vectors attainable in a declared state class. Generic quantum, stabilizer, and holographic cones obey different additional inequalities. Schematic.

QFT regions and ultraviolet cancellation

For three or four separated Gaussian regions, compute every union entropy with the same lattice, algebra, and center choice. Linear combinations may cancel local ultraviolet terms, but only when each boundary contribution enters with zero net coefficient. Verify this boundary-by-boundary rather than assuming that every balanced-looking expression is finite.

Entropy vectors from QFT at one cutoff can be tested against universal quantum inequalities. A violation usually signals mismatched subsystem prescriptions or numerical error. Satisfaction of a holographic inequality, however, does not establish a holographic dual; many nonholographic states occupy the same half-space.

Bao et al. 2015, §§2–4 construct the holographic entropy cone and its graph-model inequalities. Research through the cutoff includes machine-assisted exploration, such as He, Lee, and Ooguri 2026, §§2–4; these results concern the holographic cone and do not enlarge the generic QFT theorem set.

State-class boundaries

The validity figure highlights the danger of silently replacing one cone by another.

Universal quantum inequalities apply broadly; additional stabilizer or holographic facets require their special state classes. Unmatched regulators, overlapping algebra choices, or an altered purifier can move an apparent entropy vector and spoil the comparison. Schematic.

Report the parties, all unions, purifier convention, state class, regulator, and numerical uncertainty. A cone membership test is a necessary consistency check within its class, not a one-number classification of the underlying field theory.

References

• Bao, Ning, ChunJun Cao, Sergio M. Carroll, and Aidan Chatwin-Davies. “The Holographic Entropy Cone.” Journal of High Energy Physics 09 (2015): 130. DOI. Open preprint.
• He, Wan Zhen, Yen-Chi Lee, and Hirosi Ooguri. “Exploring the Holographic Entropy Cone via Reinforcement Learning.” Journal of High Energy Physics 06 (2026): 267. DOI.

Further reading

• Lieb, Elliott H., and Mary Beth Ruskai. “Proof of the Strong Subadditivity of Quantum-Mechanical Entropy.” Journal of Mathematical Physics 14 (1973): 1938–1941. DOI.