Strong Subadditivity and Entropic Inequalities

Strong subadditivity is the organizing entropy inequality for three subsystems. In QFT it applies to compatible regulated algebras and survives useful continuum combinations, but it cannot compare entropies computed with mismatched cutoffs, centers, or region assignments.

Required background. Use regulated subregion entropy. Helpful background. Positivity, Monotonicity, and Data Processing supplies the relative-entropy proof strategy.

The inequality and its algebraic content

For a state on a compatible tripartite type-I system $ABC$,

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

Equivalently, the conditional mutual information

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

is nonnegative. Lieb and Ruskai 1973, pp. 1938–1941 proved strong subadditivity. Algebraically, it is a data-processing statement: restricting a relative-entropy comparison cannot increase distinguishability.

The structure diagram shows why strong subadditivity belongs to the correlation branch yet also opens the route to recovery.

Strong subadditivity constrains a common tripartite state. Equality and small conditional mutual information lead toward Markov and recovery statements only after the channel and support hypotheses are supplied. Schematic.

QFT regions and cancellation

Consider three adjacent blocks of a harmonic chain with a single regulator. Every entropy in the four-term combination uses the same global state, lattice spacing, endpoint rule, and tensor decomposition. Local ultraviolet terms cancel in the conditional mutual information when the geometric assignment is compatible.

For overlapping continuum regions, it is safer to state the associated algebra inclusions and express the inequality through relative entropy. Merely drawing set unions and intersections does not establish that the corresponding observable algebras form the required commuting square. Gauge constraints and center choices can change the entropy decomposition.

Other familiar consequences include subadditivity,

$$S(A)+S(B)\geq S(AB),$$

and weak monotonicity. They do not imply that entropy itself is monotone under region inclusion: adding degrees of freedom may either raise or lower the entropy of a mixed state.

Equality and mismatched prescriptions

The lower diagram isolates the conditions needed for the four entropies to belong to one inequality.

All four entropy terms must be restrictions of the same state under a compatible subsystem prescription. Mixing lattice spacings, edge-mode conventions, or centers can leave uncancelled boundary terms and is not a test of strong subadditivity. Schematic.

Equality, $I(A{:}C\mid B)=0$, has structural content: in the finite faithful setting it characterizes an exact quantum Markov state and permits perfect recovery from $B$ to $BC$ by the sufficiency result of Petz 1986, pp. 123–131. The next pages separate that theorem from approximate and continuum variants.

References

• 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.
• Petz, Dénes. “Sufficient Subalgebras and the Relative Entropy of States of a von Neumann Algebra.” Communications in Mathematical Physics 105 (1986): 123–131. DOI.