A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks

We consider two fundamental tasks in quantum information theory, data compression with quantum side information, as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. These characterizations-in contrast to earlier results-enable us to derive tight second-order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree (up to logarithmic terms) with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our technique also naturally yields bounds on operational quantities for finite block lengths.

[1]  I. Devetak,et al.  Classical data compression with quantum side information , 2003 .

[2]  Nilanjana Datta,et al.  Smooth Entropies and the Quantum Information Spectrum , 2009, IEEE Transactions on Information Theory.

[3]  Richard G. Baraniuk,et al.  Redundancy Rates of Slepian-Wolf Coding ∗ , 2004 .

[4]  Ioannis Kontoyiannis.  Second-order noiseless source coding theorems , 1997, IEEE Trans. Inf. Theory.

[5]  林 正人.  Quantum information : an introduction , 2006 .

[6]  M. Nussbaum,et al.  THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING , 2006, quant-ph/0607216.

[7]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[8]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[9]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[10]  Sergio Verdú,et al.  Approximation theory of output statistics , 1993, IEEE Trans. Inf. Theory.

[11]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[12]  Ke Li,et al.  Second Order Asymptotics for Quantum Hypothesis Testing , 2012, ArXiv.

[13]  Masahito Hayashi,et al.  Non-asymptotic analysis of privacy amplification via Rényi entropy and inf-spectral entropy , 2012, 2013 IEEE International Symposium on Information Theory.

[14]  Vincent Y. F. Tan,et al.  The dispersion of Slepian-Wolf coding , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[15]  Hiroki Koga,et al.  Information-Spectrum Methods in Information Theory , 2002 .

[16]  Robert König,et al.  The Operational Meaning of Min- and Max-Entropy , 2008, IEEE Transactions on Information Theory.

[17]  H. Yuen.  Quantum detection and estimation theory , 1978, Proceedings of the IEEE.

[18]  M. Berta.  Single-shot Quantum State Merging , 2009, 0912.4495.

[19]  M. Tomamichel.  A framework for non-asymptotic quantum information theory , 2012, 1203.2142.

[20]  Robert König,et al.  Universally Composable Privacy Amplification Against Quantum Adversaries , 2004, TCC.

[21]  Marco Tomamichel,et al.  Duality Between Smooth Min- and Max-Entropies , 2009, IEEE Transactions on Information Theory.

[22]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[23]  Marco Tomamichel,et al.  A Fully Quantum Asymptotic Equipartition Property , 2008, IEEE Transactions on Information Theory.

[24]  Masahito Hayashi,et al.  Second-Order Asymptotics in Fixed-Length Source Coding and Intrinsic Randomness , 2005, IEEE Transactions on Information Theory.

[25]  H. Vincent Poor,et al.  Channel Coding Rate in the Finite Blocklength Regime , 2010, IEEE Transactions on Information Theory.

[26]  H. Vincent Poor,et al.  Channel coding: non-asymptotic fundamental limits , 2010 .

[27]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[28]  I. Tyurin.  Refinement of the upper bounds of the constants in Lyapunov's theorem , 2010 .

[29]  R. Renner,et al.  One-shot classical-quantum capacity and hypothesis testing. , 2010, Physical review letters.

[30]  Nilanjana Datta,et al.  Min- and Max-Relative Entropies and a New Entanglement Monotone , 2008, IEEE Transactions on Information Theory.

[31]  Joseph M. Renes,et al.  One-Shot Classical Data Compression With Quantum Side Information and the Distillation of Common Randomness or Secret Keys , 2010, IEEE Transactions on Information Theory.

[32]  Masahito Hayashi,et al.  An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses , 2007, IEEE Transactions on Information Theory.

[33]  Masahito Hayashi,et al.  Information Spectrum Approach to Second-Order Coding Rate in Channel Coding , 2008, IEEE Transactions on Information Theory.

[34]  M. Hayashi.  Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing , 2002, quant-ph/0208020.

[35]  K. Audenaert,et al.  Quantum state discrimination bounds for finite sample size , 2012, 1204.0711.

[36]  Masahito Hayashi,et al.  General formulas for capacity of classical-quantum channels , 2003, IEEE Transactions on Information Theory.

[37]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[38]  Tomohiro Ogawa,et al.  Strong converse and Stein's lemma in quantum hypothesis testing , 2000, IEEE Trans. Inf. Theory.

[39]  Nilanjana Datta,et al.  The Quantum Capacity of Channels With Arbitrarily Correlated Noise , 2009, IEEE Transactions on Information Theory.