A decoupling approach to classical data transmission over quantum channels

Most coding theorems in quantum Shannon theory can be proven using the decoupling technique: to send data through a channel, one guarantees that the environment gets no information about it; Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method: sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a "dequantizing" theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher-Westmoreland Theorem to the one-shot scenario, where a quantum channel can be applied only once

Variations on Classical and Quantum Extractors

Many constructions of randomness extractors are known to work in the presence of quantum side information, but there also exist extractors which do not [Gavinsky {\it et al.}, STOC'07]. Here we find that spectral extractors $\psi$ with a bound on the second largest eigenvalue $\lambda_{2}(\psi^{\dagger}\circ\psi)$ are quantum-proof. We then discuss fully quantum extractors and call constructions that also work in the presence of quantum correlations decoupling. As in the classical case we show that spectral extractors are decoupling. The drawback of classical and quantum spectral extractors is that they always have a long seed, whereas there exist classical extractors with exponentially smaller seed size. For the quantum case, we show that there exists an extractor with extremely short seed size $d=O(\log(1/\epsilon))$, where $\epsilon>0$ denotes the quality of the randomness. In contrast to the classical case this is independent of the input size and min-entropy and matches the simple lower bound $d\geq\log(1/\epsilon)$.Comment: 7 pages, slightly enhanced IEEE ISIT submission including all the proof

Quantum-proof randomness extractors via operator space theory

Quantum-proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion. Furthermore they are also a useful tool in quantum Shannon theory. It is known that some extractor constructions are quantum-proof whereas others are provably not [Gavinsky et al., STOC'07]. We argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries: we first phrase the definition of extractors as a bounded norm condition between normed spaces, and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces. From this we show that very high min-entropy extractors as well as extractors with small output are always (approximately) quantum-proof. We also study a generalization of extractors called randomness condensers. We phrase the definition of condensers as a bounded norm condition and the definition of quantum-proof condensers as a completely bounded norm condition. Seeing condensers as bipartite graphs, we then find that the bounded norm condition corresponds to an instance of a well studied combinatorial problem, called bipartite densest subgraph. Furthermore, using the characterization in terms of operator spaces, we can associate to any condenser a Bell inequality (two-player game) such that classical and quantum strategies are in one-to-one correspondence with classical and quantum attacks on the condenser. Hence, we get for every quantum-proof condenser (which includes in particular quantum-proof extractors) a Bell inequality that can not be violated by quantum mechanics.Comment: v3: 34 pages, published versio

Randomized Partial Decoupling Unifies One-Shot Quantum Channel Capacities

We analyze a task in which classical and quantum messages are simultaneously communicated via a noisy quantum channel, assisted with a limited amount of shared entanglement. We derive the direct and converse bounds for the one-shot capacity region. The bounds are represented in terms of the smooth conditional entropies and the error tolerance, and coincide in the asymptotic limit of infinitely many uses of the channel. The direct and converse bounds for various communication tasks are obtained as corollaries, both for one-shot and asymptotic scenarios. The proof is based on the randomized partial decoupling theorem, which is a generalization of the decoupling theorem. Thereby we provide a unified decoupling approach to the one-shot quantum channel coding, by fully incorporating classical communication, quantum communication and shared entanglement

Quantum Side Information: Uncertainty Relations, Extractors, Channel Simulations

In the first part of this thesis, we discuss the algebraic approach to classical and quantum physics and develop information theoretic concepts within this setup. In the second part, we discuss the uncertainty principle in quantum mechanics. The principle states that even if we have full classical information about the state of a quantum system, it is impossible to deterministically predict the outcomes of all possible measurements. In comparison, the perspective of a quantum observer allows to have quantum information about the state of a quantum system. This then leads to an interplay between uncertainty and quantum correlations. We provide an information theoretic analysis by discussing entropic uncertainty relations with quantum side information. In the third part, we discuss the concept of randomness extractors. Classical and quantum randomness are an essential resource in information theory, cryptography, and computation. However, most sources of randomness exhibit only weak forms of unpredictability, and the goal of randomness extraction is to convert such weak randomness into (almost) perfect randomness. We discuss various constructions for classical and quantum randomness extractors, and we examine especially the performance of these constructions relative to an observer with quantum side information. In the fourth part, we discuss channel simulations. Shannon's noisy channel theorem can be understood as the use of a noisy channel to simulate a noiseless one. Channel simulations as we want to consider them here are about the reverse problem: simulating noisy channels from noiseless ones. Starting from the purely classical case (the classical reverse Shannon theorem), we develop various kinds of quantum channel simulation results. We achieve this by using classical and quantum randomness extractors that also work with respect to quantum side information.Comment: PhD thesis, ETH Zurich. 214 pages, 13 figures, 1 table. Chapter 2 is based on arXiv:1107.5460 and arXiv:1308.4527 . Section 3.1 is based on arXiv:1302.5902 and Section 3.2 is a preliminary version of arXiv:1308.4527 (you better read arXiv:1308.4527). Chapter 4 is (partly) based on arXiv:1012.6044 and arXiv:1111.2026 . Chapter 5 is based on arXiv:0912.3805, arXiv:1108.5357 and arXiv:1301.159