A framework for non-asymptotic quantum information theory

By Marco Tomamichel. 2012

347 citations recorded by the source corpus.

This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical R\'enyi entropies. We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states. We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits. The most important property is an entropic formulation of the asymptotic equipartition property, which implies that the smooth entropies converge to the von Neumann entropy in the limit of many independent copies. The smooth entropies also satisfy duality and entropic uncertainty relations that provide limits on the power of two different observers to predict the outcome of a measurement on a quantum system. Finally, we discuss three example applications of the smooth entropy framework. We show a strong converse statement for source coding with quantum side information, characterize randomness extraction against quantum side information and prove information theoretic security of quantum key distribution using an intuitive argument based on the entropic uncertainty relation.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/92dd10ea08afe99c023b9afe9b4b7657e79abef3

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Citations in the corpus

Exponential Decay of Correlations Implies Area Law (2012, methodology): “…systems by performing a random measurement in one of them – an idea that has been used to derive the best known protocols for a variety of information theoretic tasks [43, 52, 53, 55, 56, 65, 66]) – as well as recent developments in the framework of single-shot quantum information theory [37, 60].”
Advances in quantum cryptography (2019, positive): “The 46 functional ∆AEP ( 2pǫs 3 , d ) quantifies the error committed bounding the smooth-min entropy using the asymptotic equipartition property (AEP) [62, 438].”
Quantum Information Processing with Finite Resources - Mathematical Foundations (2015, background): “ominently Nielsen and Chuang’s standard textbook [111]) deﬁnes the ﬁdelity as p F(;). Here we adopted the historical definition. The discussion on generalized ﬁdelity and puriﬁed distance is based on [134] and [138]. The puriﬁed distance was independently proposed by Gilchrist et al. [56] and Rasteg”
Quantum Conditional Mutual Information and Approximate Markov Chains (2014, methodology): “12 We note that the definition proposed in [49,50] applies more generally to any ρ with tr(ρ) ≤ 1.”
The second laws of quantum thermodynamics (2013, positive): “In the macroscopic regime, and for systems which are not highly correlated, then Fαðρ; ρβÞ≈F1ðρ; ρβÞ for all α (2, 8), which explains why the single constraint given by the usual second law is more or less adequate in this limit.”
Quantum resource theories (2018, methodology): “Since then they have been used promeninetly in quantum hypothesis testing and other studies of one-shot quantum shannon theory (Datta and Hsieh, 2013; Datta et al., 2013a,b; Konig et al., 2009; Matthews and Wehner, 2014; Radhakrishnan et al., 2016; Renes and Renner, 2011; Renner and König, 2005; Tom”
On quantum Rényi entropies: A new generalization and some properties (2013, background): “The min, max-and collision entropies occurring in various operational scenarios in quantum information theory (see [31, 37] and references therein for an overview) are not specializations of D α ( ) for any value of α [39, Sec.”
Entropic uncertainty relations and their applications (2015, background): “was first proved by Tomamichel and Renner (2011) and is fundamental to quantum key distribution (see Sec.”
Performance and structure of single-mode bosonic codes (2017, background): “[42] by noting that R‡S is a unital map.”
Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy (2013, neutral): “[67]: Dα(ρ‖σ) ≤ D(ρ‖σ) + 4(α − 1)(log ν)2, (19)”
Quantum thermodynamics (2015, background): “This can be written as (see Chapter 4 in [23])”