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IEEE International Symposium on Information Theory (ISIT)

Second-order asymptotics of quantum data compression from partially-smoothed conditional entropy

Dina Abdelhadi, Joseph M. Renes

Published Jun 1, 2020• 0 citations
Publisher

Abstract

Anshu et al. recently introduced "partially" smoothed information measures and used them to derive tighter bounds for several information-processing tasks, including quantum state merging and privacy amplification against quantum adversaries [arXiv:1807.05630 [quant-ph]]. Yet, a tight second- order asymptotic expansion of the partially smoothed conditional min-entropy in the i.i.d. setting remains an open question. Here we establish the second-order term in the expansion for pure states, and find that it differs from that of the original "globally" smoothed conditional min-entropy. Remarkably, this reveals that the second-order term is not uniform across states, since for other classes of states the second-order term for partially and globally smoothed quantities coincides. By relating the task of quantum compression to that of quantum state merging, our derived expansion allows us to determine the second-order asymptotic expansion of the optimal rate of quantum data compression. This closes a gap in the bounds determined by Datta and Leditzky [IEEE Trans. Inf. Theory 61, 582 (2015)], and shows that the straightforward compression protocol of cutting off the eigenspace of least weight is indeed asymptotically optimal at second order.

DOI
10.1109/isit44484.2020.9173948

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Anshu et al. recently introduced "partially" smoothed information measures and used them to derive tighter bounds for several information-processing tasks, including quantum state merging and privacy amplification against quantum adversaries [arXiv:1807.05630 [quant-ph]]. Yet, a tight second- order asymptotic expansion of the partially smoothed conditional min-entropy in the i.i.d. setting remains an open question. Here we establish the second-order term in the expansion for pure states, and find that it differs from that of the original "globally" smoothed conditional min-entropy. Remarkably, this reveals that the second-order term is not uniform across states, since for other classes of states the second-order term for partially and globally smoothed quantities coincides. By relating the task of quantum compression to that of quantum state merging, our derived expansion allows us to determine the second-order asymptotic expansion of the optimal rate of quantum data compression. This closes a gap in the bounds determined by Datta and Leditzky [IEEE Trans. Inf. Theory 61, 582 (2015)], and shows that the straightforward compression protocol of cutting off the eigenspace of least weight is indeed asymptotically optimal at second order.
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.
Recently, Anshu et al. introduced “partially” smoothed information measures and used them to derive tighter bounds for several information-processing tasks, including quantum state merging and privacy amplification against quantum adversaries [IEEE Trans. Inf. Theory 66, 5022 (2020)]. Yet, a tight second-order asymptotic expansion of the partially smoothed conditional min-entropy in the i.i.d. setting remains an open question. Here we establish the second-order term in the expansion for pure states, and find that it differs from that of the original “globally” smoothed conditional min-entropy. Remarkably, this reveals that the second-order term is not uniform across states, since for other classes of states the second-order term for partially and globally smoothed quantities coincides. In view of the tight bounds on the entanglement cost of state merging in terms of the partially-smoothed conditional min-entropy by Anshu et al., this indicates that the second-order asymptotic rate for that task will not separate so neatly into a term depending on the state (the variance) and a term depending on the error (the quantile). Finally, by relating the task of quantum compression to that of quantum state merging, our derived expansion allows us to determine the second-order asymptotic expansion of the optimal rate of quantum data compression. This closes a gap in the bounds determined by Datta and Leditzky [IEEE Trans. Inf. Theory 61, 582 (2015)], and shows that the straightforward compression protocol of cutting off the eigenspace of least weight is indeed asymptotically optimal at second order.
Smooth entropies are a tool for quantifying resource trade-offs in information theory and cryptography. However, in typical multi-partite problems some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states. We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed. This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting. As a consequence, we can derive asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting. For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well.
We determine the strong converse exponent for the entanglement cost of quantum state merging, showing that it is characterized by the optimized ααα-zzz conditional Rényi entropies with z=α/2∈[1/2,1]z=α/2\in[1/2,1]z=α/2∈[1/2,1]. This contrasts with the sandwiched conditional Rényi entropies that typically govern strong converse exponents in quantum information theory. As a consequence, we derive the strong converse exponent of the partially smoothed conditional min-entropy in purified distance. This exponent is governed by club-sandwiched conditional entropies, whereas global smoothing leads to a sandwiched expression.
Smooth entropies are a tool for quantifying resource trade-offs in (quantum) \ninformation theory and cryptography. In typical bi- and multi-partite problems, \nhowever, some of the sub-systems are often left unchanged and this is not \nreflected by the standard smoothing of information measures over a ball of \nclose states. We propose to smooth instead only over a ball of close states \nwhich also have some of the reduced states on the relevant sub-systems fixed. \nThis partial smoothing of information measures naturally allows to give more \nrefined characterizations of various information-theoretic problems in the \none-shot setting. In particular, we immediately get asymptotic second-order \ncharacterizations for tasks such as privacy amplification against classical \nside information or classical state splitting. For quantum problems like state \nmerging the general resource trade-off is tightly characterized by partially \nsmoothed information measures as well. However, for quantum systems we can so \nfar only give the asymptotic first-order expansion of these quantities.
Smooth entropies are a tool for quantifying resource trade-offs in (quantum) information theory and cryptography. In typical bi- and multi-partite problems, however, some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states. We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed. This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting. In particular, we immediately get asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting. For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well.