Strong Converse for Privacy Amplification against Quantum Side Information

Abstract

We establish a one-shot strong converse bound for privacy amplification against quantum side information using trace distance as a security criterion. This strong converse bound implies that in the independent and identical scenario, the trace distance exponentially converges to one in every finite blocklength when the rate of the extracted randomness exceeds the quantum conditional entropy. The established one-shot bound has an application to bounding the information leakage of classical-quantum wiretap channel coding and private communication over quantum channels. That is, the trace distance between Alice and Eavesdropper's joint state and its decoupled state vanishes as the rate of randomness used in hashing exceeds the quantum mutual information. On the other hand, the trace distance converges to one when the rate is below the quantum mutual information, resulting in an exponential strong converse. Our result also leads to an exponential strong converse for entropy accumulation, which complements a recent result by Dupuis [arXiv:2105.05342]. Lastly, our result and its applications apply to the moderate deviation regime. Namely, we characterize the asymptotic behaviors of the trace distances when the associated rates approach the fundamental thresholds with speeds slower than O( 1 / √ n).

Error Exponent and Strong Converse for Quantum Soft Covering

“…As a consequence, our results in achievability and strong converse combined implies that the optimal rate of quantum soft covering is the quantum mutual information I(X : B) ρ . We remark that our results hold for every blocklength n ∈ N, providing a large deviation analysis when the operating rate is fixed [28,31,[39][40][41][42][43][44][45]. Our results also extend to the moderate deivation regime when the rates approaches I(X : B) ρ moderately quickly [33,34].…”

“…Our results also extend to the moderate deivation regime when the rates approaches I(X : B) ρ moderately quickly [33,34]. Lastly, it is interesting to note that the sandwiched Rényi information I * α(X : B) ρ used in the exponent appears in classical-quantum channel coding as well [23,31,[45][46][47], while the sandwiched Augustin information Ȋ * α(X : B) ρ has appeared in other contexts using constant composition codes [28,40,48,49].…”

“…Combining (4.1), (4.2), and (4.3) proves our claim with substitution α = 1 1+s . Lemma 6 (A trace inequality [31,Lemma 3]). For any positive semi-definite K and L, the following holds,…”

Optimal Second-Order Rates for Quantum Soft Covering and Privacy Amplification

“…We would like to point out that although there are essentially no differences between the three deviation regimes in the one-shot setting, there are at least two different types of operational quantities of interest; their characterizations might be different. As observed in[26] and our previous works[27,38], indeed, the Rényi-type quantities are more favorable in characterizing the optimal error given a fixed size or cardinality such as |Z| and |C| considered in this paper. On the other hand, if one concerns the size or cardinality given a fixed error, the hypothesis-testing-type quantities or the information-spectrum-type quantities might be more direct for characterizations.…”

“…Indeed, much progress has been made in deriving the exact second-order rates for numerous quantum information-theoretic protocols. Nevertheless, this problem remains open for certain tasks such as privacy amplification against quantum side information (or called randomness extraction) [15,21], where the operational quantities usually being used as a security criterion is the trace distance [19,[21][22][23][24][25][26][27]. This manifests the fact that existing analysis on operational quantities in terms of the trace distance as in various quantum information-theoretic tasks [19,[28][29][30][31][32] still has room for improvement.…”