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IEEE International Symposium on Information Theory (ISIT)

Exponents in smoothing the max-relative entropy and of randomness extraction against quantum side information

Ke Li, Yongsheng Yao, Masahito Hayashi

Published Jun 26, 2022• 0 citations
Publisher

Abstract

This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award.The smooth max-relative entropy is a basic tool in quantum information theory and cryptography. In this paper, we derive the exact exponent for the decay of the small modification of the quantum state in smoothing the max-relative entropy. We then apply this result to the problem of privacy amplification against quantum side information and obtain an upper bound for the exponent of the decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general.Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of order between 1 and 2, which has not been addressed previously in the quantum setting.

DOI
10.1109/isit50566.2022.9834595

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This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award.The smooth max-relative entropy is a basic tool in quantum information theory and cryptography. In this paper, we derive the exact exponent for the decay of the small modification of the quantum state in smoothing the max-relative entropy. We then apply this result to the problem of privacy amplification against quantum side information and obtain an upper bound for the exponent of the decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general.Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of order between 1 and 2, which has not been addressed previously in the quantum setting.
The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the decay of the small modification of the quantum state in smoothing the max-relative entropy. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding.
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.
The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of orders∈(1,2]s\in (1,2]s∈(1,2], which has not been addressed previously in the quantum setting.
We establish the exact exponent for smoothing the max-relative entropy on arbitrary von Neumann algebras. The proof first develops the required large-deviation and smoothing estimates in the semifinite setting and then passes to general von Neumann algebras through Haagerup-Junge-Xu reduction. We next study catalytic quantum information decoupling when the reference system is described by an arbitrary von Neumann algebra. A key ingredient is an operator layer-cake formula valid on arbitrary von Neumann algebras. This formula leads to a convex-split estimate for normal states with a general von Neumann algebraic reference system. Combining this estimate with the smoothing exponent result, we obtain lower and upper bounds on the catalytic-decoupling reliability function. These bounds coincide below the corresponding critical rate, yielding the exact decoupling reliability function in that regime.
The fidelity-based smooth min-relative entropy is a distinguishability measure that has appeared in a variety of contexts in prior work on quantum information, including resource theories like thermodynamics and coherence. Here we provide a comprehensive study of this quantity. First we prove that it satisfies several basic properties, including the dataprocessing inequality. We also establish connections between the fidelity-based smooth min-relative entropy and other widely used information-theoretic quantities, including smooth min-relative entropy and smooth sandwiched Rényi relative entropy, of which the sandwiched Rényi relative entropy and smooth max-relative entropy are special cases. After that, we use these connections to establish the second-order asymptotics of the fidelity-based smooth min-relative entropy and all smooth sandwiched Rényi relative entropies, finding that the first-order term is the quantum relative entropy and the second-order term involves the quantum relative entropy variance. Utilizing the properties derived, we also show how the fidelity-based smooth min-relative entropy provides one-shot bounds for operational tasks in general resource theories in which the target state is mixed, with a particular example being randomness distillation. The above observations then lead to second-order expansions of the upper bounds on distillable randomness, as well as the precise second-order asymptotics of the distillable randomness of particular classical–quantum states. Finally, we establish semi-definite programs for smooth maxrelative entropy and smooth conditional min-entropy, as well as a bilinear program for the fidelity-based smooth min-relative entropy, which we subsequently use to explore the tightness of a bound relating the last to the first.
We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders α∈[12,∞)α\in[\frac{1}{2},\infty)α∈[21​,∞). For α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1), the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for α∈[1,∞)α\in[1,\infty)α∈[1,∞), it is characterized by the order-ααα 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,∞)α\in(2,\infty)α∈(2,∞), expressed in terms of the corresponding order-ααα 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 KKK-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1).