Strong Subadditivity and Monogamy

Strong subadditivity is the nonnegativity of conditional mutual information for a tripartite quantum state. It is an entropy theorem, not a slogan that “entanglement is monogamous.” The AMPS application uses an equivalent weak-monotonicity inequality and a declared $EBA$ tensor factorization. If gravity does not supply that common state or factorization to the needed accuracy, the premise—not the theorem—must be revised.

Required background. Quantum Information supplies density operators, purification, relative entropy, and partial trace used in the theorem and proof.

Helpful background. Semiclassical Gravity supplies the old-hole regime; Quantum Fields in Curved Spacetime supplies the near-horizon pair state; Black-Hole Information supplies the Page and AMPS applications.

Strong-subadditivity theorem

Theorem. Let $\rho_{ABC}$ be a normalized density operator on a finite-dimensional tensor product $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$. Then

$$S(AB)+S(BC)\geq S(B)+S(ABC).$$

Equivalently, the conditional mutual information is nonnegative:

$$I(A{:}C\vert B) =S(AB)+S(BC)-S(B)-S(ABC)\geq0.$$

The finite-dimensional hypotheses ensure all entropies are finite. The result extends to appropriate von Neumann-algebraic settings, but a continuum or gravitational use must state that extension or regulate the algebras. Lieb and Ruskai give the original theorem and proof Lieb and Ruskai 1973, theorem and proof, pp. 1938–1941.

Proof through data processing

Quantum relative entropy is

$$D(\rho\Vert\sigma) =\operatorname{Tr}\rho(\ln\rho-\ln\sigma).$$

Its monotonicity under the partial trace $\operatorname{Tr}_C$ gives

$$D(\rho_{ABC}\Vert\rho_A\otimes\rho_{BC}) \geq D(\rho_{AB}\Vert\rho_A\otimes\rho_B).$$

Expanding each side yields

$$-S(ABC)+S(A)+S(BC) \geq -S(AB)+S(A)+S(B).$$

Canceling $S(A)$ and rearranging proves strong subadditivity. The deep input is monotonicity of relative entropy, whose matrix-analytic proof is equivalent in strength to the theorem; this derivation exposes why discarding $C$ cannot increase distinguishability. Harlow collects this formulation and its entropy corollaries in Harlow 2016, Appendix C.

The support condition in relative entropy causes no loophole. If the support of the first state is not contained in that of the second, the relative entropy is defined as infinity and data processing remains valid. For the tensor-product comparison states above, the marginals of a legitimate finite-dimensional density operator provide the required support on the terms that contribute. Zero eigenvalues are handled by continuity.

Equality also has content. In finite dimensions, $I(A{:}C\vert B)=0$ precisely characterizes a short quantum Markov chain: there is a CPTP map $\mathcal R_{B\to BC}$ for which

$$\rho_{ABC} = (\mathrm{id}_A\otimes\mathcal R_{B\to BC})(\rho_{AB}).$$

The equality-to-recovery condition and the direct-sum tensor structure of $B$ are given in Hayden, Jozsa, Petz, and Winter 2004, Eq. (11) and Theorem 6. Thus conditional mutual information measures information shared between $A$ and $C$ that is not mediated by $B$. This explains why the classically correlated three-bit state below saturates the inequality—once $B$ is known, the endpoints have no residual correlation—whereas the coherent GHZ state does not.

For approximate applications, a small positive conditional mutual information supports an approximate recovery statement, but the recovery error, norm, and dimensions must be carried explicitly. The AMPS use is the opposite regime: its two asserted entropy decreases would require a negative conditional-entropy combination forbidden by the theorem. Small gravitational corrections help only if their controlled error is comparable to the finite entropy gap.

Weak monotonicity and AMPS

Purify $ABE$ by a system $D$. Apply strong subadditivity to the triplet $B,A,D$:

$$S(BA)+S(AD)\geq S(A)+S(BAD).$$

Purity gives $S(AD)=S(BE)$ and $S(BAD)=S(E)$, so

$$S(AB)+S(BE)\geq S(A)+S(E).$$

This is weak monotonicity. In the AMPS notation, late-radiation purification says $S(BE)\lt S(E)$, while a near-pure entangled horizon pair says $S(AB)\lt S(A)$. Adding the two strict inequalities contradicts weak monotonicity. The black-hole application and its assumptions are stated in Almheiri et al. 2013, § 2.

The derivation requires $A$, $B$, and $E$ to be subsystems of the same global state. Borrowing $AB$ from one effective description and $BE$ from an incompatible one invalidates the inference. Approximate commuting algebras require an error bound smaller than the entropy gap.

Pure pairs and entanglement monogamy

If $AB$ is exactly pure, any extension factorizes:

$$\rho_{ABE}=\lvert\psi\rangle\langle\psi\rvert_{AB}\otimes\rho_E.$$

Therefore $B$ has no correlation with independent $E$. The reason is purity of the pair, not maximal entanglement: the same factorization holds for a nonmaximally entangled pure pair, or even a pure product pair. For a pure entangled $AB$, all of $B$‘s mixedness is already purified by $A$. This is the relevant monogamy statement for a thermally weighted Hawking pair, which is generally not maximally entangled. Strong subadditivity is more general: it constrains arbitrary mixed quantum and classical correlations through conditional mutual information.

It should not be confused with inequalities for concurrence or squared entanglement measures in multiqubit systems. Those use different quantities and hypotheses. AMPS can be expressed either through weak monotonicity or through closeness to a pure entangled pair, but replacing one theorem by a vague monogamy slogan hides the factorization and approximation errors.

Product, Bell, GHZ, and classical checks

For a product state $\rho_A\otimes\rho_B\otimes\rho_C$, all cross-correlations vanish and $I(A{:}C\vert B)=0$. For a Bell pair on $AB$ with a pure spectator $C$, the same conditional mutual information is zero: conditioning on $B$ leaves no additional $A$–$C$ correlation.

For the pure GHZ state $(\lvert000\rangle+\lvert111\rangle)/\sqrt2$,

$$S(AB)=S(BC)=S(B)=\ln2, \qquad S(ABC)=0,$$

so $I(A{:}C\vert B)=\ln2$ and the inequality is strict. Yet each two-qubit reduced state is separable. This demonstrates that conditional mutual information is not simply pairwise entanglement.

For the classical correlated mixture

$$\rho_{ABC}=\tfrac12\lvert000\rangle\langle000\rvert +\tfrac12\lvert111\rangle\langle111\rvert,$$

all four entropies in the conditional mutual information equal $\ln2$, so the result is zero. Once $B$ is known, $A$ and $C$ are conditionally independent. The contrast with GHZ locates the coherent three-party information.

Gravity qualification

Gauge constraints and gravitational dressing can obstruct an exact inside–outside tensor factorization. That does not permit a violation of strong subadditivity for a legitimate common density operator. It means the argument must identify commuting algebras, centers, edge modes, or code-subspace embeddings and use the correct algebraic entropy theorem. A proposed nonfactorization resolution must also reproduce the semiclassical correlators that motivated $A$ and $B$.

Generic entropy theory belongs to Quantum Information, local horizon states to Quantum Fields in Curved Spacetime, and explicit gravitational algebra constructions elsewhere. The theorem alone does not select a firewall.

Exercise

Compute $I(A{:}C\vert B)$ for the classical distribution with equal probabilities on $000$, $010$, $101$, and $111$. Is $A$ conditionally independent of $C$ given $B$?

Solution

$A$ and $C$ are always equal, while $B$ is an independent fair bit. Thus $S(AB)=2\ln2$, $S(BC)=2\ln2$, $S(B)=\ln2$, and $S(ABC)=2\ln2$. Therefore $I(A{:}C\vert B)=\ln2$. Knowing $B$ does not remove the perfect $A$–$C$ correlation, so conditional independence fails and the inequality is strict.

References

• Almheiri, Ahmed, Donald Marolf, Joseph Polchinski, and James Sully. 2013. “Black Holes: Complementarity or Firewalls?” Journal of High Energy Physics 2013 (02): 062. Open text.
• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Hayden, Patrick, Richard Jozsa, Dénes Petz, and Andreas Winter. 2004. “Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality.” Communications in Mathematical Physics 246: 359–374. DOI. Open text.
• Lieb, Elliott H., and Mary Beth Ruskai. 1973. “Proof of the Strong Subadditivity of Quantum-Mechanical Entropy.” Journal of Mathematical Physics 14: 1938–1941. DOI.