Multipartite Information and Entropy Cones

An entropy vector records the entropy of every nonempty union of a fixed set of parties. Entropy cones organize the homogeneous linear inequalities obeyed by such vectors—but the generic quantum, stabilizer, and leading-order holographic cones are different sets. A special-state inequality is a diagnostic of that state class, not a universal QFT law.

Required background. Strong Subadditivity and Entropic Inequalities supplies conditional mutual information, purification, and the equality conditions used to test entropy vectors. Helpful background. Mutual Information for Disjoint Regions supplies the matched-regulator and boundary-cancellation logic used for QFT combinations.

The shared structure map identifies the geometry, state, and measure inputs. The canonical comparison table should be consulted before combining entropies from different subsystem prescriptions. Research-sensitive cone statements below include primary literature available through 10 August 2026.

Entropy vectors and their closure

For an $N$-party density operator, define

$$\mathbf s(\rho) =\bigl(S(X)_\rho\bigr)_{\varnothing\neq X\subseteq[N]} \in\mathbb R^{2^N-1}.$$

Let $\Sigma_N^*$ be the set of all such vectors over finite-dimensional states and all local dimensions. The quantum entropy cone is its topological closure $\overline{\Sigma_N^*}$. The distinction matters: $\overline{\Sigma_N^*}$ is a convex cone, whereas the raw set $\Sigma_N^*$ is not a cone for $N\geq3$ Christandl, Durhuus, and Wolff 2023, pp. 240201-1–240201-3, Theorem 1.

Every finite-dimensional quantum entropy vector obeys, among their permutations and purified forms,

$$\begin{aligned} S(A)&\geq0,\\ S(A)+S(B)&\geq S(AB),\\ S(AB)&\geq|S(A)-S(B)|,\\ S(AB)+S(BC)&\geq S(B)+S(ABC). \end{aligned}$$

These are positivity, subadditivity, Araki–Lieb, and strong subadditivity Lieb and Ruskai 1973, Theorem 1, pp. 1938–1941. Purification turns strong subadditivity into weak monotonicity. Their permutations characterize the closed generic quantum entropy cone through three parties; for four or more parties, no complete set of generic quantum entropy inequalities is known Bao et al. 2015, §1, pp. 2–4.

For three parties, 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 permit either sign. If $ABC$ itself is pure, complementary entropies make $I_3=0$ identically; nonzero values require a mixed reduction or an additional purifier.

Which cone is being tested?

State class	Guaranteed linear constraints beyond positivity	Established completeness	Scope ceiling
Generic finite-dimensional quantum states	Subadditivity, Araki–Lieb, strong subadditivity, and weak monotonicity	Complete through $N=3$; unknown for $N\geq4$	Applies to every density operator with the declared parties
Stabilizer states	Generic constraints plus a proved family of group-theoretic linear-rank inequalities; at four parties this adds Ingleton	Four-party stabilizer cone is characterized by strong subadditivity, weak monotonicity, and Ingleton	Not every classical linear-rank inequality is known to hold, and none of these extra constraints applies to arbitrary nonstabilizer states
Leading Ryu–Takayanagi entropies	Generic constraints plus monogamy $I_3\leq0$ and further holographic inequalities	Complete for $N\leq5$; the full six-party cone remains unknown	Static smooth classical bulk geometries at leading order, not unrestricted QFT entropy

The stabilizer result proves the linear-rank inequalities generated by its group-theoretic common-information construction; it does not prove every classical linear-rank inequality for stabilizer entropies Linden, Matúš, Ruskai, and Winter 2013, Theorems 6 and 13 and the discussion after Theorem 11. At four parties, strong subadditivity, weak monotonicity, and Ingleton are complete. Bao et al. 2015, §§2.2–3, especially eqs. (3)–(6) construct the cone through four parties and the five-party inequalities; Hernández-Cuenca 2019, §§1 and 3 proves those inequalities complete for five parties by realizing every extreme-ray orbit. Monogamy follows for entropies computed by the Ryu–Takayanagi prescription,

$$I_3(A{:}B{:}C)\leq0,$$

not for arbitrary states of a holographic CFT and not automatically after unrestricted quantum corrections Hayden, Headrick, and Maloney 2013, abstract and §3, eq. (3.1).

Through the stated literature cutoff, machine-assisted work has extended exploration rather than the generic theorem set. In particular, He, Lee, and Ooguri 2026, §§3–4 give exact graph realizations for three candidate six-party rays and numerical evidence concerning three others. The exact realizations are constructive results; failure of a numerical search is evidence, not by itself a proof of a new facet.

Three- and four-region Gaussian application

A covariance calculation makes a complete entropy vector reproducible. Use one bosonic mode per party with quadrature order $(q_1,p_1,\ldots,q_N,p_N)$ and $[q_j,p_k]=i\delta_{jk}$. Consider the Gaussian state

$$V_N(t)=\frac12\mathbb I_{2N}+t\,w_Nw_N^T, \qquad w_N=(1,0,1,0,\ldots,1,0)^T, \qquad t\geq0.$$

It is the vacuum with a common classical Gaussian displacement added to every $q_j$, so it is a valid separable Gaussian state. A $k$-party reduction has the same form with $w_k$. An orthogonal mode transformation isolates the collective position quadrature, giving one nonvacuum symplectic eigenvalue

$$\nu_k=\frac12\sqrt{1+2kt}$$

and $k-1$ eigenvalues equal to $1/2$. Consequently every union $X$ with $|X|=k$ has

$$S(X)=H_k=h(\nu_k),$$

where

$$h(\nu)= \left(\nu+\frac12\right)\log\left(\nu+\frac12\right) -\left(\nu-\frac12\right)\log\left(\nu-\frac12\right).$$

At $t=1$,

$$H_1=0.793945, \quad H_2=1.076022, \quad H_3=1.254902, \quad H_4=1.386294.$$

The complete three-party vector is therefore

$$\mathbf s_3 =(H_1,H_1,H_1; H_2,H_2,H_2; H_3),$$

and the complete four-party vector is specified by assigning $H_k$ to each of the $\binom4k$ unions of size $k$. Direct tests give

$$\begin{aligned} S(A)+S(B)-S(AB) &=2H_1-H_2=0.511868,\\ I(A{:}C\mid B) &=2H_2-H_1-H_3=0.103197,\\ I(A{:}C\mid BD) &=2H_3-H_2-H_4=0.047488. \end{aligned}$$

The standard quantum inequalities pass with positive margins. But

$$I_3=3H_1-3H_2+H_3=0.408671>0.$$

This single calculation fulfills the adversarial test: the state is a valid generic—indeed separable—Gaussian state and must not be rejected for violating holographic monogamy. What fails is the attempted classification as a leading-RT entropy vector. Numerical reproduction should report the convention $V_{\rm vac}=\mathbb I/2$, the value of $t$, symplectic-eigenvalue tolerance, and the inequality residuals above.

QFT regions and ultraviolet cancellation

For QFT regions, a cutoff or split-factor entropy vector is meaningful only when every component uses the same lattice spacing, algebra, center, state, and boundary convention. At any fixed cutoff, the ordinary finite-dimensional inequalities must hold; a violation larger than numerical error signals an inconsistent reduction or implementation.

The raw vector generally diverges in the continuum. A balanced linear combination can be finite when each local boundary contribution has zero total coefficient. For three mutually separated regions, a boundary belonging to $A$ enters $I_3$ with coefficient

$$+1\ \text{from }S(A) -1\ \text{from }S(AB) -1\ \text{from }S(AC) +1\ \text{from }S(ABC)=0.$$

The same check holds for the boundaries of $B$ and $C$. It can fail for touching or overlapping regions, unmatched centers, corners with different local data, or entropy components computed at different cutoffs. The shared validity map separates those regulator failures from the state-class failure demonstrated above.

Report the ordered parties, every union, purifier convention, state class, regulator, center, and numerical uncertainty. Satisfying one holographic half-space is only a necessary condition within that special class; it neither proves a gravity dual nor determines the underlying field theory.

Exercises

1. Raw set versus cone closure

Explain why tensor products show that integer multiples of an entropy vector are attainable. Why does that fact not prove that the raw set $\Sigma_N^*$ is a cone?

Solution

If $\rho$ has entropy vector $\mathbf s$, then $\rho^{\otimes m}$ has entropy vector $m\mathbf s$ because von Neumann entropy is additive under tensor products. Thus every positive integer multiple is attainable. A cone must also contain $\lambda\mathbf s$ for every real $\lambda\geq0$, including arbitrarily small noninteger multiples. Tensor products do not supply those vectors. For $N\geq3$, nonhomogeneous constraints near the origin show that such down-scaling can fail, even though the closure is conic.

2. Reproduce the Gaussian vector

For $V_N(t)$, diagonalize a $k$-party reduction by isolating the normalized collective quadrature $q_\Sigma=k^{-1/2}\sum_{j=1}^kq_j$. Derive $\nu_k$ and reproduce the three-party subadditivity, strong-subadditivity, and $I_3$ residuals at $t=1$.

Solution

The common-noise term adds variance $kt$ only to $q_\Sigma$. Its conjugate $p_\Sigma$ retains variance $1/2$, while the remaining $k-1$ normal modes are vacuum modes. Hence

$$\nu_k=\sqrt{(1/2+kt)(1/2)} =\frac12\sqrt{1+2kt}.$$

Inserting $t=1$ into $h(\nu_k)$ gives the displayed $H_1,H_2,H_3$. Therefore

$$2H_1-H_2=0.511868>0,$$ $$2H_2-H_1-H_3=0.103197>0,$$

and

$$3H_1-3H_2+H_3=0.408671>0.$$

The state passes generic quantum inequalities and violates only the additional holographic monogamy condition.

3. Audit ultraviolet cancellation

Assume each smooth boundary component contributes the same local divergent term $d(\epsilon)$ to every entropy containing that boundary. Show boundary-by-boundary cancellation in $I_3$ for three separated regions. Name two changes that invalidate the argument.

Solution

A boundary of $A$ contributes to $S(A)$, $S(AB)$, $S(AC)$, and $S(ABC)$ with coefficients $+1,-1,-1,+1$, whose sum is zero. The same reasoning applies independently to $B$ and $C$. Cancellation fails if regions touch and create new common-boundary terms, or if different union entropies use different regulators or center choices. Corners or overlapping algebras can also introduce local structures not covered by the assumed common $d(\epsilon)$.

4. Classify, do not reject

Let $A$, $B$, and $C$ be distinct classical registers, each holding the same fair random bit $X$, represented by a diagonal quantum state. Compute $I_3$. Which cone claim fails, and which conclusions remain valid?

Solution

Every nonempty union determines the same fair bit, so

$$S(A)=S(B)=S(C)=S(AB)=S(AC)=S(BC)=S(ABC)=\log2.$$

Therefore

$$I_3=(3-3+1)\log2=\log2>0.$$

The state is a valid separable quantum state and obeys the generic entropy inequalities. It lies outside the leading-RT holographic cone because it violates monogamy. The correct conclusion is a state-class classification, not rejection of the state or of quantum mechanics.

References

• Bao, Ning, Sepehr Nezami, Hirosi Ooguri, Bogdan Stoica, James Sully, and Michael Walter. “The Holographic Entropy Cone.” Journal of High Energy Physics 09 (2015): 130. DOI. Open preprint.
• Christandl, Matthias, Bergfinnur Durhuus, and Lasse Harboe Wolff. “Tip of the Quantum Entropy Cone.” Physical Review Letters 131 (2023): 240201. DOI. Open preprint.
• Hayden, Patrick, Matthew Headrick, and Alexander Maloney. “Holographic Mutual Information Is Monogamous.” Physical Review D 87 (2013): 046003. DOI. Open preprint.
• He, Temple, Jaeha Lee, and Hirosi Ooguri. “Exploring the Holographic Entropy Cone via Reinforcement Learning.” Journal of High Energy Physics 06 (2026): 267. DOI. Open preprint.
• Hernández Cuenca, Sergio. “The Holographic Entropy Cone for Five Regions.” Physical Review D 100 (2019): 026004. DOI. Open preprint.
• 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.
• Linden, Noah, František Matúš, Mary Beth Ruskai, and Andreas Winter. “The Quantum Entropy Cone of Stabiliser States.” In Theory of Quantum Computation, Communication, and Cryptography, LIPIcs 22 (2013): 270–284. DOI. Open preprint.