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arXiv:2009.08823 · IEEE Transactions on Information Theory

Equivalence of Three Classical Algorithms With Quantum Side Information: Privacy Amplification, Error Correction, and Data Compression

Toyohiro Tsurumaru

Published Nov 8, 2021• 8 citations
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Abstract

Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study these three algorithms (PA, EC, and DC) in the presence of quantum side information, and show that they all become equivalent in the one-shot scenario. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.

DOI
10.1109/tit.2021.3126160

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Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study these three algorithms (PA, EC, and DC) in the presence of quantum side information, and show that they all become equivalent in the one-shot scenario. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.
Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study quantum algorithms of these three types (PA, EC, and DC) in the one-shot scenario, and show that they all become equivalent if modified properly. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.
Information processing protocols are typically built out of simpler parts, called primitives, and two of the most important such primitives are privacy amplification (PA) and data compression. The former extracts the truly secret part of some classical data, while the latter squeezes it into the smallest possible form. We show these tasks are dual in the setting of quantum information processing. Specifically, the tasks of PA of classical information against quantum adversaries and classical data compression with quantum side information are dual in the sense that the ability to perform one implies the ability to perform the other. The duality arises because the two protocols are connected by complementarity and the uncertainty principle in the quantum setting. Applications include a new uncertainty principle formulated in terms of smooth min- and max-entropies, which are useful in the study of one-shot protocols, as well as new conditions for approximate quantum error correction.
We show that the Mayers-Shor-Preskill approach and Renner's approach to proving the security of quantum key distribution (QKD) are essentially the same. We begin our analysis by considering a special case of QKD called privacy amplification (PA). PA itself is an important building block of cryptography, both classical and quantum. The standard theoretical tool used for its security proof is called the leftover hashing lemma (LHL). We present a direct connection between the LHL and the coding theorem of a certain quantum error correction code. Then we apply this result to proving the equivalence between the two approaches to proving the security of QKD.
The task of compressing classical information in the one-shot scenario is studied in the setting where the decompressor additionally has access to some given quantum side information. In this hybrid classical-quantum version of the famous Slepian-Wolf problem, the smooth max-entropy is found to govern the number of bits into which classical information can be compressed so that it can be reliably recovered from the compressed version and quantum side information. Combining this result with known results on privacy amplification then yields bounds on the amount of common randomness and secret key that can be recovered in one-shot from hybrid classical-quantum systems using one-way classical communication.
The task of compressing classical information in the one-shot scenario is studied in the setting where the decompressor additionally has access to some given quantum side information. In this hybrid classical-quantum version of the famous Slepian-Wolf problem, the smooth max entropy is found to govern the number of bits into which classical information can be compressed so that it can be reliably recovered from the compressed version and quantum side information. Combining this result with known results on privacy amplification then yields tight bounds on the amount of common randomness and secret key that can be recovered in one shot from hybrid classical-quantum systems using one-way classical communication.
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.