We consider privacy amplification against quantum side information by using regular random binning as an effective extractor. For constant-type sources, we obtain error exponent in terms of the so-called quantum Augustin information. Via type decomposition, we then recover the error exponent for independent and identically distributed sources proved by Dupuis [arXiv:2105.05342]. As an application, we obtain an achievable secrecy exponent for classical-quantum wiretap channel coding in terms of the Augustin information, which solves an open problem in [IEEE Trans. Inf. Theory, 65(12):7985, 2019]. Our approach is to establish an operational equivalence between privacy amplification and quantum soft covering; this may be of independent interest. The full version of this paper can be found in [arXiv:2309.11073]
We consider privacy amplification against quantum side information by using regular random binning as an effective extractor. For constant-type sources, we obtain error exponent in terms of the so-called quantum Augustin information. Via type decomposition, we then recover the error exponent for independent and identically distributed sources proved by Dupuis [arXiv:2105.05342]. As an application, we obtain an achievable secrecy exponent for classical-quantum wiretap channel coding in terms of the Augustin information, which solves an open problem in [IEEE Trans. Inf. Theory, 65(12):7985, 2019]. Our approach is to establish an operational equivalence between privacy amplification and quantum soft covering; this may be of independent interest. The full version of this paper can be found in [arXiv:2309.11073]
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
A fundamental quantity of interest in Shannon theory, classical or quantum, is the optimal error exponent of a given channel W and rate R: the constant E(W,R) which governs the exponential decay of decoding error when using ever larger codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Here I show that a bound by Hayashi [CMP 333, 335 (2015)] for an analogous quantity in privacy amplification implies a lower bound on the error exponent of communication over symmetric classical-quantum channels. The resulting bound matches Dalai’s [IEEE TIT 59, 8027 (2013)] sphere-packing upper bound for rates above a critical value, and reproduces the well-known classical result for symmetric channels. The argument proceeds by first relating the error exponent of privacy amplification to that of compression of classical information with quantum side information, which gives a lower bound that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [arXiv:2111.01075 [quant-ph]], at least for linear randomness extractors.
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 ε-covering is given by the (1 − ε)-hypothesis testing information (with additional logarithmic additive terms), while the maximal extractable randomness for an ε-secret extractor is characterized by the conditional (1 − ε)-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.
We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite- blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the Eγdivergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for lossy compression. Our construction leads to protocols of optimal rate in the asymptotic i.i.d. limit for a variety of compression scenarios. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Inf. Theory 64,577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a “curious duality” in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.
We establish a one-shot strong converse bound for privacy amplification against quantum side information with trace distance as a security criterion. Firstly, our result shows that the trace distance converges to 1 exponentially fast for any finite blocklength when the rate of the extractable randomness exceeds the conditional von Neumann entropy of Alice given Eavesdropper. Secondly, we obtain a second-order converse bound for the maximal extractable randomness. Thirdly, our result extends to the moderate deviation regime. Namely, when the rate of the extractable randomness approaches the conditional von Neumann entropy from above with speed slower than O(1/n, the trace distance still converges to 1 asymptotically. Lastly, our result yields strong converse to entropy accumulation, which complements the recent result by Dupuis [arXiv:2105.05342]. The full manuscript can be assessed in [1], [2].
We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders α∈[21,∞). For α∈[21,1), the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for α∈[1,∞), it is characterized by the order-α sandwiched R{é}nyi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched R{é}nyi divergence for α∈(2,∞), expressed in terms of the corresponding order-α sandwiched R{é}nyi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the K-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for α∈[21,1).