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IEEE International Symposium on Information Theory (ISIT)

Smooth Rényi Entropy of Ergodic Quantum Information Sources

Berry Schoenmakers, Jilles Tjoelker, Pim Tuyls, Evgeny A. Verbitskiy

Published Jun 1, 2007• 6 citations
Publisher

Abstract

We investigate the recently introduced notion of smooth Renyi entropy for the case of ergodic information sources, thereby generalizing previous work which concentrated mainly on i.i.d. information sources. We will actually consider ergodic quantum information sources, of which ergodic classical information sources are a special case. We prove that the average smooth Renyi entropy rate will approach the entropy rate of a stationary, ergodic source, which is equal to the Shannon entropy rate for a classical source and the von Neumann entropy rate for a quantum source.

DOI
10.1109/isit.2007.4557235

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We investigate the recently introduced notion of smooth Renyi entropy for the case of ergodic information sources, thereby generalizing previous work which concentrated mainly on i.i.d. information sources. We will actually consider ergodic quantum information sources, of which ergodic classical information sources are a special case. We prove that the average smooth Renyi entropy rate will approach the entropy rate of a stationary, ergodic source, which is equal to the Shannon entropy rate for a classical source and the von Neumann entropy rate for a quantum source.
We investigate therecently introduced notion of smoothRenyi entropy forthecaseofergodic information sources, thereby generalizing previous workwhichconcentrated mainly oni.i.d. information sources. We willactually consider ergodic quantuminformation sources, ofwhichergodic classical infor- mationsources area special case. We provethattheaverage smoothRenyientropy ratewill approach theentropy rateofa stationary, ergodic source, whichisequal totheShannon entropy rateforaclassical source andthevonNeumannentropy rate foraquantumsource. I.INTRODUCTION Theelegant notion ofsmooth Renyi entropy wasintroduced recently byRennerandWolfin(6)forclassical information sources, andthenatural extension toquantuminformation sources wasdefined byRenner andKonigin(5). Inthese two papers andfurther workbyRennerandWolf(7), (4), many properties ofsmooth Renyi entropyandsmooth min-entropy andsmooth max-entropy inparticular havebeenstudied in detail. A central property ofsmooth Renyi entropy proved inthese worksisthat formemoryless (i.i.d.) information sources, the average smooth Renyi entropy ratewill approach theentropy rate ofthesource, whichisequal totheShannon entropy fora classical source andthevonNeumannentropy foraquantum source. Whereas, ingeneral, theaverage (conventional) Renyi entropy rateofamemoryless source doesnotconverge tothe source's entropy rate. Inthis paper weextend thestudy ofsmooth Renyi entropy tothemoregeneral class ofstationary, ergodic sources rather thanmemoryless sources. We willprovethatforboththe classical andthequantum casethat theaverage smooth Renyi entropy ratewill approach theShannon andthevonNeumann entropy rate, respectively. We will dosobyfirst treating the classical caseandthenreducing thequantumcasetothe classical onewithout losing generality. Ingeneral, smooth Renyi entropy oforder a > 1,anda oc(min-entropy) inparticular, isofcryptographic relevance (e.g., forrandomness-extraction), andsmoothRenyientropy oforder a < 1,anda = 0(max-entropy) inparticular, are relevant todatacompression (minimum encoding length). In these contexts, theimportance ofsmooth Renyi entropy isthat its rate isbasically equal totheShannon/von Neumannentropy rate forani.i.d. source (andforergodic sources aswell, aswe showinthis paper). Thisisnotthecaseforconventional Renyi entropy. Moregenerally, asshowninthepapers byRenner et al.mentioned above, smooth Renyientropy behaves muchas Shannon/von Neumannentropy does. Inthis paperwefocus ontheunconditional case, whereas muchoftheabovementioned workbyRenner etal.treats the moregeneral conditional case. We leave theextension tothe conditional caseforfuture work.However, we doconsider twonotions ofe-closeness, onebasedontrace distance (also knownasvariational orstatistical distance) andonebased on non-normalized density matrices (orprobability distributions), wherethelatter ismoresuitable tohandle theconditional case.' Thus, webelieve that ourresults canbeextended tothe conditional caseaswell.
Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two different techniques, the information spectrum method and the smooth entropy framework, have been developed independently. They are based on new entropy measures, called spectral entropy rates andsmoothentropies, respectively, that generalize Shannon entropy (in the classical case) and von Neumann entropy (in the more general quantum case). Here, we show that the two techniques are closely related. More precisely, the spectral entropy rate can be seen as the asymptotic limit of the smooth entropy. Our results apply to the quantum setting and thus include the classical setting as a special case.
Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are in- dependent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two different techniques, the informationspectrummethodandthesmoothentropyframework, have been developed independently. They are based on new en- tropymeasures, calledspectralentropy ratesandsmooth entropies, respectively,that generalizeShannon entropy (in the classicalcase) and von Neumann entropy (in the more general quantum case). Here, we show that the two techniques are closely related. More precisely, the spectral entropy rate can be seen as the asymptotic limit of the smooth entropy. Our results apply to the quantum set- ting and thus include the classical setting as a special case.
We define a large class of quantum sources and prove a quantum analog of the asymptotic equipartition property. Our proof relies on using local measurements on the quantum source to obtain an associated classical source. The classical source provides an upper bound for the dimension of the relevant subspace of the quantum source, via the Shannon–McMillan noiseless coding theorem. Along the way we derive a bound for the von Neumann entropy of the quantum source in terms of the Shannon entropy of the classical source, and we provide a definition of ergodicity of the quantum source. Several explicit models of quantum sources are also presented.