A new operator extension of strong subadditivity of quantum entropy

TL;DR

This work introduces an operator extension of weak monotonicity, which is equivalent to strong subadditivity (SSA) of quantum entropy. The authors construct an operator inequality $\log \rho_{AB} - \log \rho_A + \log \rho_{BC} - \log \rho_C \le 0$ using Stinespring dilations of the Accardi-Cecchini coarse graining and rely on the operator monotonicity of $t \mapsto \log t$ to deduce the scalar weak monotonicity upon taking expectations; this extends to two independent density matrices and to Rényi-type generalizations via $t^{\alpha}$ for $\alpha\in[0,1]$. The operator inequality implies the operator extension of SSA and offers a simpler route to SSA, with a related discussion connecting to algebraic quantum field theory and highlighting distinctions between finite-dimensional and QFT settings. The paper also points to potential further matrix-inequality consequences arising from dilations of quantum channels and frames open questions about broader applicability of these dilational methods.

Abstract

Let $S(ρ)$ be the von Neumann entropy of a density matrix $ρ$. Weak monotonicity asserts that $S(ρ_{AB}) - S(ρ_A) + S(ρ_{BC}) - S(ρ_C)\geq 0$ for any tripartite density matrix $ρ_{ABC}$, a fact that is equivalent to the strong subadditivity of entropy. We prove an operator inequality, which, upon taking an expectation value with respect to the state $ρ_{ABC}$, reduces to the weak monotonicity inequality. Generalizations of this inequality to the one involving two independent density matrices, as well as their Rényi-generalizations, are also presented.

Theorems & Definitions (7)

• Lemma 1
• proof
• Remark 1
• Theorem 2
• Corollary 1
• proof
• Theorem 3