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arXiv:0811.1221 · IEEE Transactions on Information Theory

A Fully Quantum Asymptotic Equipartition Property

Marco Tomamichel, Roger Colbeck, Renato Renner

Published Nov 24, 2009• 356 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tit.2009.2032797

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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.
This chapter begins our first technical foray into the asymptotic theory of information. We start with the classical setting in an effort to build up our intuition of asymptotic behavior before delving into the asymptotic theory of quantum information. The central concept of this chapter is the asymptotic equipartition property. The name of this property may sound somewhat technical at first, but it is merely an application of the law of large numbers to a sequence drawn independently and identically from a distribution p X ( x ) for some random variable X . The asymptotic equipartition property reveals that we can divide sequences into two classes when their length becomes large: those that are overwhelmingly likely to occur and those that are overwhelmingly likely not to occur. The sequences that are likely to occur are the typical sequences, and the ones that are not likely to occur are the atypical sequences. Additionally, the size of the set of typical sequences is exponentially smaller than the size of the set of all sequences whenever the random variable generating the sequences is not uniform. These properties are an example of a more general mathematical phenomenon known as “measure concentration,” in which a smooth function over a high-dimensional space or over a large number of random variables tends to concentrate around a constant value with high probability. The asymptotic equipartition property immediately leads to the intuition behind Shannon's scheme for compressing classical information. The scheme first generates a realization of a random sequence and asks the question: Is the produced sequence typical or atypical? If it is typical, compress it. Otherwise, throw it away. The error probability of this compression scheme is non-zero for any fixed length of a sequence, but it vanishes in the asymptotic limit because the probability of the sequence being in the typical set converges to one, while the probability that it is in the atypical set converges to zero. This compression scheme has a straightforward generalization to the quantum setting, in which we wish to compress qubits instead of classical bits. The bulk of this chapter is here to present the many technical details needed to make rigorous statements in the asymptotic theory of information. We begin with an example, follow with the formal definition of a typical sequence and a typical set, and prove the three important properties of a typical set.
This chapter begins our first technical foray into the asymptotic theory of information. We start with the classical setting in an effort to build up our intuition of asymptotic behavior before delving into the asymptotic theory of quantum information. The central concept of this chapter is the asymptotic equipartition property. The name of this property may sound somewhat technical at first, but it is merely an application of the law of large numbers to a sequence drawn independently and identically from a distribution pX(x) for some random variable X . The asymptotic equipartition property reveals that we can divide sequences into two classes when their length becomes large: those that are overwhelmingly likely to occur and those that are overwhelmingly likely not to occur. The sequences that are likely to occur are the typical sequences, and the ones that are not likely to occur are the atypical sequences. Additionally, the size of the set of typical sequences is exponentially smaller than the size of the set of all sequences whenever the random variable generating the sequences is not uniform. These properties are an example of a more general mathematical phenomenon known as “measure concentration,” in which a smooth function over a high-dimensional space or over a large number of random variables tends to concentrate around a constant value with high probability.
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 Rényi 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.
We establish two results on the asymptotic equipartition property (AEP) for quantum channels. First, the AEP formulated using smoothing over completely positive trace-preserving maps fails in general. Second, the AEP with smoothing over subchannels (completely positive trace-nonincreasing maps) is equivalent to the strong converse for parallel quantum channel Stein's lemma, to a sharp threshold for the channel hockey-stick divergence, and to an AEP for the channel Lorenz divergence. The counterexample reveals a surprising separation between channel and subchannel smoothing. Although states and classical channels admit a sharp, dimension-independent completion bound, which we establish, the corresponding single-shot gap for quantum channels is unbounded already for qubits. This separation persists asymptotically: we construct qutrit channel pairs with finite max-relative entropy whose CPTP-smoothed rate strictly exceeds the regularized channel relative entropy. The asymptotic gap between channel and subchannel smoothing is unbounded across the family, even when the subchannel approximation error decays exponentially. To establish the equivalences, we develop a uniform filter that converts Lorenz smoothing into a single nearby subchannel. The four conjectured formulations thus share a common asymptotic threshold, whose validity for general channel pairs remains open. Together, these results separate this threshold question from the obstruction imposed by exact trace preservation.
We investigate the asymptotic equipartition property (AEP) in the context of multipartite entanglement measures on pure states. In particular, we formulate AEP for subadditive entanglement measures that admit certain weak conditions. This is motivated by the uniqueness of the entanglement entropy in the asymptotic limit in the bipartite case. From an operational perspective, the result is relevant in the LOCCqscenario (asymptotic local operations and classical communication with a sublinear amount of quantum communication). Analogously to the classical AEP, we show that the regularization of smooth, weakly additive entanglement measures - under mild additional assumptions -yields entanglement measures, that are weakly additive and asymptotically continuous. We then evaluate the mentioned regularization and smoothing procedure for a known family of Rényi type multipartite entanglement measures, showing that the resulting entanglement measures reduce to convex combinations of bipartite entanglement entropies.
We consider an extension of the conditional min- and max-entropies to infinite-dimensional separable Hilbert spaces. We show that these satisfy characterizing properties known from the finite-dimensional case, and retain information-theoretic operational interpretations, e.g., the min-entropy as maximum achievable quantum correlation, and the max-entropy as decoupling accuracy. We furthermore generalize the smoothed versions of these entropies and prove an infinite-dimensional quantum asymptotic equipartition property. To facilitate these generalizations we show that the min- and max-entropy can be expressed in terms of convergent sequences of finite-dimensional min- and max-entropies, which provides a convenient technique to extend proofs from the finite to the infinite-dimensional setting.