Subadditivity

The von Neumann entropy obeys a family of inequalities that constrain how information can be distributed among the parts of a quantum system. The most basic is subadditivity; its powerful strengthening, strong subadditivity, is one of the deepest results in quantum information theory.

Subadditivity

For any bipartite state $\rho_{AB}$ with reduced states $\rho_A$ and $\rho_B$ ,

\boxed{\,S(AB) \le S(A) + S(B),\,}

with equality if and only if $\rho_{AB} = \rho_A \otimes \rho_B$ is a product state. The whole is no more uncertain than the sum of its parts; correlations reduce the joint entropy below the naive sum. This inequality is exactly what makes the quantum mutual information

I(A\!:\!B) = S(A) + S(B) - S(AB) \ge 0

non-negative. Equality (zero mutual information) marks the complete absence of correlation.

The Araki–Lieb triangle inequality

Subadditivity has a lower-bound companion, the Araki–Lieb inequality:

S(AB) \ge \big|\,S(A) - S(B)\,\big|.

Together they sandwich the joint entropy:

\big|S(A) - S(B)\big| \le S(AB) \le S(A) + S(B).

The lower bound has a purely quantum flavor. Classically $H(X,Y) \ge \max\{H(X), H(Y)\}$ , so the joint entropy is at least as large as either marginal. Quantumly that can fail: for a pure entangled state $S(AB) = 0$ while $S(A) = S(B) > 0$ , and indeed $0 = |S(A) - S(B)|$ saturates the Araki–Lieb bound. The joint state can be less uncertain than its parts — there is no quantum analogue of the classical fact that conditioning leaves non-negative entropy.

Strong subadditivity

The crown jewel concerns three systems $A$ , $B$ , $C$ :

\boxed{\,S(ABC) + S(B) \le S(AB) + S(BC).\,}

Equivalently, in terms of conditional entropy, $S(A\,|\,BC) \le S(A\,|\,B)$ : discarding system $C$ cannot decrease your uncertainty about $A$ given the rest. Strong subadditivity (SSA), proved by Lieb and Ruskai in 1973, is equivalent to a striking operational statement: the conditional mutual information is non-negative,

I(A\!:\!C\,|\,B) = S(AB) + S(BC) - S(ABC) - S(B) \ge 0.

Why strong subadditivity matters

SSA is the workhorse behind nearly every converse theorem in quantum information. It implies:

Monotonicity of mutual information under discarding subsystems: $I(A\!:\!BC) \ge I(A\!:\!B)$ .
Data-processing inequality: quantum operations cannot increase distinguishability or mutual information — you cannot create correlation by acting locally.
Non-negativity of conditional mutual information, the cornerstone of the theory of quantum Markov chains and approximate recoverability.

These consequences place hard limits on communication, compression, and error correction. The inequalities are not merely formal: they encode the physical impossibility of generating information from nothing.

These inequalities will reappear throughout the module — in the Holevo bound, in data compression, and whenever we ask how much information a quantum channel can carry. For now, the take-away is the chain of bounds $\,|S(A) - S(B)| \le S(AB) \le S(A) + S(B)\,$ and its three-party reinforcement.