A Fully Quantum Asymptotic Equipartition Property

By Marco Tomamichel, Roger Colbeck, Renato Renner. IEEE Transactions on Information Theory, 2008

337 citations recorded by the source corpus.

The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, a fully quantum generalization of this property is shown, where both the output of the experiment and side information are quantum. An explicit bound on the convergence is given, which is independent of the dimensionality of the side information. This naturally leads to a family of REacutenyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.

Subfields: Artificial Intelligence, Electrical and Electronic Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dc958f5f6a22db9659013bea82137ef9bfe1fa70

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Citations in the corpus

Entropic uncertainty relations—a survey (2009, methodology): “Using the SDP formalism of (Ballester et al. , 2008) it is easily seen (Wehner et al. , 2008) that this gives us In the limit of large n , it is known that for independent states, the min-entropy behaves approximately like the Shannon entropy (Renner, 2005; Tomamichel et al. , 2008).”
Advances in quantum cryptography (2019, neutral): “) assumption where the total state of the system after the quantum phase has product form, the smooth min-entropy can be bounded by the von Neumann entropy of a single measurement, using the socalled quantum asymptotic equipartition property [492].”
The Operational Meaning of Min- and Max-Entropy (2008, background): “It can be shown, however, that the smooth variants of these definitions only deviate by an additive term which is logarithmic in the smoothness parameter (see [5]).”
Quantum Conditional Mutual Information and Approximate Markov Chains (2014, methodology): “) While the proof in [50] applies to the case where σ is a density operator, the slightly extended claim provided here follows directly from Remark A.”
On quantum Rényi entropies: A new generalization and some properties (2013, methodology): “It was studied for example in [13, 16, 22, 27, 39] and the term “generalized R´enyi relative entropy” occurs for example in [22].”
The Quantum Reverse Shannon Theorem Based on One-Shot Information Theory (2009, background): “For a more detailed discussion of the smooth entropy formalism we refer to [12, 17, 18, 38, 39].”
Tight finite-key analysis for quantum cryptography (2011, methodology): “The min-entropy of the eavesdropper is in turn bounded using the quantum asymptotic equipartition property [7, 49], introducing a penalty scaling with 1 / √ n on the key rate.”
Quantum Information Processing with Finite Resources - Mathematical Foundations (2015, background): “the smooth relative max-divergence, which we deﬁne for any r 2S (A) and s 2S(A) as De max(rks):=min T r˜ 2Be(r)D ¥(r˜ks): (6.76) The following gives an upper bound on the smooth relative max-entropy [38, 137]. Lemma 6.9. Let r 2S (A);s 2S(A) and l 2( ¥;D max(rks)]. Then, De max(rks)l; where e = q 2T”
Exponential Decay of Correlations Implies Area Law (2012, methodology): “The second idea, necessary to handle the remaining gaps that might exist between von Neumann and max entropies, is to use the hypothesis of exponential decay of correlations, together with the quantum equipartition property [59, 60], to upper bound the max-entropy by a sum of von Neumann entropies.”
An intuitive proof of the data processing inequality (2011, methodology): “Then we can specialize the smooth min-entropy to the von Neumann entropy by the quantum asymptotic equipartition property (QAEP) (Theorem 2) [19].”