[Paper Review] Optimal Second-Order Rates for Quantum Soft Covering and Privacy Amplification

Yuchen Shen, Li Gao+1 more|arXiv (Cornell University)|Feb 23, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper establishes one-shot characterizations for quantum soft covering and privacy amplification using trace distance as the security metric. It shows that the optimal second-order rates are governed by the quantum mutual information variance and conditional information variance, respectively, with precise operational quantities given by hypothesis testing entropy and information, avoiding smoothing techniques.

ABSTRACT

We study quantum soft covering and privacy amplification against quantum side information. The former task aims to approximate a quantum state by sampling from a prior distribution and querying a quantum channel. The latter task aims to extract uniform and independent randomness against quantum adversaries. For both tasks, we use trace distance to measure the closeness between the processed state and the ideal target state. We show that the minimal amount of samples for achieving an $\varepsilon$-covering is given by the $(1-\varepsilon)$-hypothesis testing information (with additional logarithmic additive terms), while the maximal extractable randomness for an $\varepsilon$-secret extractor is characterized by the conditional $(1-\varepsilon)$-hypothesis testing entropy. When performing independent and identical repetitions of the tasks, our one-shot characterizations lead to tight asymptotic expansions of the above-mentioned operational quantities. We establish their second-order rates given by the quantum mutual information variance and the quantum conditional information variance, respectively. Moreover, our results extend to the moderate deviation regime, which are the optimal asymptotic rates when the trace distances vanish at sub-exponential speed. Our proof technique is direct analysis of trace distance without smoothing.

Motivation & Objective

• To close a gap in quantum information theory by providing one-shot characterizations for quantum soft covering and privacy amplification against quantum side information.
• To derive tight second-order asymptotic expansions for these tasks using trace distance as the figure of merit.
• To establish optimal rates in the moderate deviation regime where trace distances vanish sub-exponentially.
• To avoid the use of smoothing techniques in trace distance analysis, enabling direct and tighter bounds.
• To demonstrate that hypothesis testing entropy and information are the natural one-shot operational quantities when trace distance is used without smooth entropies.

Proposed method

• The authors use direct trace distance analysis without smoothing, relying on hypothesis testing information and entropy to characterize operational quantities.
• They define the $(1-\varepsilon)$-hypothesis testing information $I_{\text{h}}^{1-\varepsilon}(X:B)_\rho$ as the key quantity for soft covering, and the $(1-\varepsilon)$-conditional hypothesis testing entropy $H_{\text{h}}^{1-\varepsilon}(X|E)_\rho$ for privacy amplification.
• The proof technique involves bounding the trace distance using the sandwiched Rényi divergence and applying Lemma 3 to derive lower bounds on codebook size and extractable randomness.
• They derive tight second-order expansions by relating the one-shot quantities to the quantum mutual information variance and conditional information variance.
• The analysis is extended to the moderate deviation regime by characterizing rates when the trace distance decays no faster than $O(1/\sqrt{n})$.
• The framework is validated through inequalities involving the sandwiched Rényi divergence $D_{\text{s}}^{1-\varepsilon}(\rho_{XB} \| \rho_X \otimes \rho_B)$ and hypothesis testing divergence $D_{\text{h}}^{1-\varepsilon}(\rho_{XB} \| \rho_X \otimes \rho_B)$.

Experimental results

Research questions

• RQ1 What is the optimal one-shot characterization of the minimal codebook size for quantum soft covering under trace distance constraint?
• RQ2 How can the maximal extractable randomness in privacy amplification be characterized using trace distance without smoothing?
• RQ3 What are the second-order asymptotic rates for quantum soft covering and privacy amplification when the error is bounded by a constant $\varepsilon \in (0,1)$?
• RQ4 How do the rates behave in the moderate deviation regime, where the trace distance vanishes sub-exponentially?
• RQ5 Can hypothesis testing information and entropy serve as natural one-shot operational quantities when using trace distance instead of smooth entropies?

Key findings

• The minimal codebook size for $\varepsilon$-covering is characterized by the $(1-\varepsilon)$-hypothesis testing information $I_{\text{h}}^{1-\varepsilon}(X:B)_\rho$, up to logarithmic additive terms.
• The maximal extractable randomness in $\varepsilon$-secret privacy amplification is given by the $(1-\varepsilon)$-conditional hypothesis testing entropy $H_{\text{h}}^{1-\varepsilon}(X|E)_\rho$.
• The second-order rate for quantum soft covering is $V(X:B|\rho)$, the quantum mutual information variance, under i.i.d. repetitions.
• The second-order rate for privacy amplification is $V(X:E|\rho)$, the quantum conditional information variance, under i.i.d. repetitions.
• In the moderate deviation regime, the optimal rates are characterized by the same variances, with trace distance decaying as $O(1/\sqrt{n})$.
• The proof avoids smoothing techniques and directly analyzes trace distance, yielding tighter and more direct bounds than prior approaches.