Doubly minimized Petz Renyi mutual information: Properties and operational interpretation from direct exponent

TL;DR

The paper advances the quantum information theory of Rényi mutual information by studying the doubly minimized Petz Rényi MI $I_\alpha^{\downarrow\downarrow}(A:B)_\rho$ for bipartite states. It establishes fundamental properties, including joint convexity of the defining optimization, additivity over product states for $\alpha$ in $[\tfrac{1}{2},2]$, minimizer uniqueness on a broad range, fixed-point characterizations, and differentiability with respect to $\alpha$, along with asymptotic optimality of universal permutation-invariant states. The central operational contribution is a direct-exponent interpretation: for $\alpha\in(\tfrac{1}{2},1)$, the direct exponent of certain binary quantum state discrimination tasks is given by a supremum over $s$ of $(1-s)/s$ times the difference between $I_s^{\downarrow\downarrow}(A:B)_\rho$ and the rate $R$. This generalizes classical results to the quantum setting and links a structural information measure to concrete error exponents, with implications for discrimination tasks and related quantum information processing.

Abstract

The doubly minimized Petz Renyi mutual information of order $α$ is defined as the minimization of the Petz divergence of order $α$ of a fixed bipartite quantum state relative to any product state. In this work, we establish several properties of this type of Renyi mutual information, including its additivity for $α\in [1/2,2]$. As an application, we show that the direct exponent of certain binary quantum state discrimination problems is determined by the doubly minimized Petz Renyi mutual information of order $α\in (1/2,1)$. This provides an operational interpretation of this type of Renyi mutual information, and generalizes a previous result for classical probability distributions to the quantum setting.

Figures (1)

• Figure 1: Comparison of PRMIs for a pure state. Suppose $d_A=2,d_B=2$, and let $\{|i\rangle_A\}_{i=0}^1,\{|i\rangle_B\}_{i=0}^1$ be orthonormal vectors in $A,B$. Let $\rho_{AB}\coloneqq|\rho\rangle\!\langle\rho|_{AB}$, where $|\rho\rangle_{AB}\coloneqq \sqrt{p}|0,0\rangle_{AB}+\sqrt{1-p}|1,1\rangle_{AB}$ and $p\coloneqq 0.2$. The solid lines depict the behavior of three PRMIs for $\rho_{AB}$, computed according to Proposition \ref{['prop:prmi0']} (o), Proposition \ref{['prop:prmi1']} (q), and Theorem \ref{['thm:prmi2']} (t), respectively. For comparison, the values of certain Rényi entropies of $\rho_A=p|0\rangle\!\langle0|_A+(1-p)|1\rangle\!\langle1|_A$ are indicated by dashed lines. The plot shows that the three PRMIs differ from each other for all $\alpha\in [0,1)\cup (1,\infty)$.

Theorems & Definitions (88)

• Proposition 1: Universal permutation invariant state
• Proposition 2: Petz divergence
• Proposition 3: Non-minimized PRMI
• Remark 1: Order in the list
• Proposition 4: Singly minimized PRMI
• Lemma 5: Operator inequality from subadditivity of geometric operator mean
• Theorem 6: Joint concavity/convexity
• Theorem 7: Doubly minimized PRMI
• Remark 2: Inequivalence of PRMIs
• Remark 3: Classical case