The strong converse exponent of composable randomness extraction against quantum side information

TL;DR

This paper determines the exact strong converse exponent for randomness extraction against quantum side information under a composable fidelity criterion, expressing the exponent through a club-sandwiched conditional entropy. The authors develop a dual framework involving Log–Euclidean and sandwiched entropies, and introduce a double-pinning technique to show that the LE bound converges to the club-sandwiched bound, yielding a tight operational interpretation of this entropy family. The main result is the closed-form exponent $E_{pa}(\rho_{XE},R) = \max_{\alpha\in[1/2,1]} \frac{1-\alpha}{\alpha}\left( R - \widetilde{H}_{\alpha}^{\frac{1-2\alpha}{1-\alpha}}(X|E)_{\rho} \right)$, valid for rates above a critical threshold where the exponent becomes linear in $R$. This work provides a robust toolset for strong-converse analyses in quantum information, including a principled interpretation of club-sandwiched entropies and methods potentially applicable to broader exponent problems.

Abstract

We find a tight characterization of the strong converse exponent for randomness extraction against quantum side information. In contrast to previous tight bounds, we employ a composable error criterion given by the fidelity (or purified distance) to a uniform distribution in product with the marginal state. The characterization is in terms of a club-sandwiched conditional entropy recently introduced by Rubboli, Goodarzi and Tomamichel and used by Li, Li and Yu to establish the strong converse exponent for the case of classical side information. This provides the first precise operational interpretation of this family of conditional entropies in the quantum setting.

Figures (1)

• Figure 1: The figure shows the strong converse exponent $E_{\mathrm{pa}}(\rho_{XE},R)$ derived in Theorem \ref{['thm: main error exponent']} for the state $\rho_{XE} = \sum_{xy}0.1\ketbra{00}{00}_{XE}+0.1\ketbra{01}{01}_{XE}+0.7\ketbra{10}{10}_{XE}+0.1\ketbra{11}{11}_{XE}$ as a function of the rate $R$. The strong converse exponent is zero when the rate is below the von Neumann conditional entropy $H(X|E)$. Indeed, in this case, the fidelity goes exponentially to one. When the rate is above the von Neumann conditional entropy $H(X|E)$, the strong converse exponent is non-zero since the fidelity decreases exponentially to zero. In addition, the strong converse exponent is linear in $R$ for the rate above a critical rate. Indeed, in this case, the optimization in Theorem \ref{['thm: main error exponent']} is achieved for $\alpha=1/2$, in which case the expression becomes linear.

Theorems & Definitions (46)

• Theorem 1
• Lemma 2
• proof
• Lemma 3: Coarse-graining reduces the sandwiched conditional entropy
• proof
• Remark 4
• Lemma 5
• proof
• Lemma 6
• proof