Strong subadditivity of quantum entropy

Remarks

The inequality was proved by Lieb and Ruskai in 1973, using Lieb's concavity theorem. It is equivalent to the monotonicity of the quantum relative entropy under partial traces. Indeed, with $d_C = \dim H_C$, one has $S(\rho_{ABC} \| \rho_{AB} \otimes I/d_C) = \log d_C + S(\rho_{AB}) - S(\rho_{ABC})$. Likewise $S(\rho_{BC} \| \rho_B \otimes I/d_C) = \log d_C + S(\rho_B) - S(\rho_{BC})$. Tracing out $A$ cannot increase the relative entropy, so the first quantity is at least the second, which is the inequality.

Example: when $H_B = \C$ is trivial, the statement becomes subadditivity, $S(\rho_{AC}) \le S(\rho_A) + S(\rho_C)$.

The inequality can be written as $I(A : C \mid B) \ge 0$, where $I(A:C\mid B) = S(\rho_{AB}) + S(\rho_{BC}) - S(\rho_{ABC}) - S(\rho_B)$ is the conditional mutual information. Some classical entropy inequalities fail for quantum states. For example, $S(\rho_{AB}) \ge S(\rho_A)$ fails for a Bell state, where $S(\rho_{AB}) = 0$ and $S(\rho_A) = \log 2$. Strong subadditivity, in contrast, holds without change.

Going deeper

For a student

Strong subadditivity is the main inequality of quantum information theory. Most coding theorems and entropy inequalities for quantum states rest on it. It is equivalent to the data processing inequality for the quantum relative entropy and to the joint convexity of the relative entropy. The cases of equality are the quantum Markov chains, states for which $C$ can be recovered from $B$ alone by a channel.