Doubly minimized sandwiched Renyi mutual information: Properties and operational interpretation from strong converse exponent

TL;DR

The paper develops a comprehensive theory for the doubly minimized sandwiched Renyi mutual information of quantum states, proving a new duality for α ≥ 2/3 and establishing additivity in the same range. It shows that this SRMI variant has a direct operational meaning in binary quantum state discrimination through strong-converse exponents, including single-letter characterizations and second-order refinements. Central to the approach are permutation-invariant universal states and pinching techniques, which yield asymptotic optimality results and facilitate exact asymptotics near α=1. The results extend prior work by covering α ∈ [2/3,∞], clarifying the relationship to Ptèz-type quantities, and providing a complete second-order analysis with the quantum information variance V(A:B)_ρ. Together, these contributions advance the operational and mathematical understanding of Renyi-type mutual information in quantum information theory.

Abstract

In this paper, we deepen the study of properties of the doubly minimized sandwiched Renyi mutual information, which is defined as the minimization of the sandwiched divergence of order $α$ of a fixed bipartite state relative to any product state. In particular, we prove a novel duality relation for $α\in [\frac{2}{3},\infty]$ by employing Sion's minimax theorem, and we prove additivity for $α\in [\frac{2}{3},\infty]$. Previously, additivity was only known for $α\in [1,\infty]$, but has been conjectured for $α\in [\frac{1}{2},\infty]$. Furthermore, we show that the doubly minimized sandwiched Renyi mutual information of order $α\in [1,\infty]$ attains operational meaning in the context of binary quantum state discrimination as it is linked to certain strong converse exponents.

Figures (1)

• Figure 1: Comparison of SRMIs 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 SRMIs for $\rho_{AB}$, computed according to the expressions in Proposition \ref{['prop:srmi0']} (p), Proposition \ref{['prop:srmi1']} (r), and Theorem \ref{['thm:srmi2']} (r), 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 SRMIs differ from each other for all $\alpha\in (0,1)\cup (1,\infty]$.

Theorems & Definitions (74)

• Proposition 1: Universal permutation invariant state
• Proposition 2: Sandwiched divergence
• Proposition 3: Non-minimized SRMI
• Proposition 4: Singly minimized SRMI
• Theorem 5: Doubly minimized SRMI
• Remark 1: Previous results on properties of the doubly minimized SRMI
• Remark 2: Inequivalence of SRMIs
• Theorem 6: Strong converse exponent
• Remark 3: Necessity of permutation invariance of alternative hypothesis
• Corollary 7: Asymptotic minimum type-I error