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Register a new userA Quantum Multiparty Packing Lemma and the Relay Channel
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Optimally encoding classical information in a quantum system is one of the oldest and most fundamental challenges of quantum information theory. Holevos bound places a hard upper limit on such encodings, while the Holevo-Schumacher-Westmoreland (HSW) theorem addresses the question of how many classical messages can be packed into a given quantum system. In this article, we use Sens recent quantum joint typicality results to prove a one-shot multiparty quantum packing lemma generalizing the HSW theorem. The lemma is designed to be easily applicable in many network communication scenarios. As an illustration, we use it to straightforwardly obtain quantum generalizations of well-known classical coding schemes for the relay channel: multihop, coherent multihop, decode-forward, and partial decode-forward. We provide both finite blocklength and asymptotic results, the latter matching existing classical formulas. Given the key role of the classical packing lemma in network information theory, our packing lemma should help open the field to direct quantum generalization.
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Quantum channel estimation and discrimination are fundamentally related information processing tasks of interest in quantum information science. In this paper, we analyze these tasks by employing the right logarithmic derivative Fisher information and the geometric Renyi relative entropy, respectively, and we also identify connections between these distinguishability measures. A key result of our paper is that a chain-rule property holds for the right logarithmic derivative Fisher information and the geometric Renyi relative entropy for the interval $alphain(0,1) $ of the Renyi parameter $alpha$. In channel estimation, these results imply a condition for the unattainability of Heisenberg scaling, while in channel discrimination, they lead to improved bounds on error rates in the Chernoff and Hoeffding error exponent settings. More generally, we introduce the amortized quantum Fisher information as a conceptual framework for analyzing general sequential protocols that estimate a parameter encoded in a quantum channel, and we use this framework, beyond the aforementioned application, to show that Heisenberg scaling is not possible when a parameter is encoded in a classical-quantum channel. We then identify a number of other conceptual and technical connections between the tasks of estimation and discrimination and the distinguishability measures involved in analyzing each. As part of this work, we present a detailed overview of the geometric Renyi relative entropy of quantum states and channels, as well as its properties, which may be of independent interest.
There are different inequivalent ways to define the Renyi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csiszar has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Renyi capacity, defined in terms of the sandwiched quantum Renyi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel $W$ with composition $P$ and rate $R$ as $sc(W,R,P)$, our main result is that [ sc(W,R,P)=sup_{alpha>1}frac{alpha-1}{alpha}left[R-chi_{alpha}^*(W,P)right], ] where $chi_{alpha}^*(W,P)$ is the $P$-weighted sandwiched Renyi divergence radius of the image of the channel.
We show that the new quantum extension of Renyis alpha-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter alpha: for alpha<1, the right choice seems to be the traditional definition, whereas for alpha>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi alpha-relative entropies are asymptotically attainable by measurements for alpha>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.
This paper introduces coherent quantum channel discrimination as a coherent version of conventional quantum channel discrimination. Coherent channel discrimination is phrased here as a quantum interactive proof system between a verifier and a prover, wherein the goal of the prover is to distinguish two channels called in superposition in order to distill a Bell state at the end. The key measure considered here is the success probability of distilling a Bell state, and I prove that this success probability does not increase under the action of a quantum superchannel, thus establishing this measure as a fundamental measure of channel distinguishability. Also, I establish some bounds on this success probability in terms of the success probability of conventional channel discrimination. Finally, I provide an explicit semi-definite program that can compute the success probability.
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