Quantum Sphere-Packing Bounds and Moderate Deviation Analysis for Classical-Quantum Channels

Transcription

1 Quantum Sphere-Packing Bounds and Moderate Deviation Analysis for Classical-Quantum Channels (, ) Joint work with Min-Hsiu Hsieh and Marco Tomamichel Hao-Chung Cheng University of Technology Sydney National Taiwan University Beyond IID 2017 in Information Theory 1

2 Motivations Communications over a classical-quantum channel: product states Encoder decoder 2

3 Three Parameters trade-off Blocklength the total number of channel uses Rate the amount of information (bits) transmitted per channel use Optimal error probability of the best coding scheme Investigate the interplay between the optimum probability of error ε, transmission rate R, and the blocklength n What is the performance with finite n? 3

4 Interplay of Two Parameters [Small deviation] Interplay between R and n given a fixed ε [Large deviation] Interplay between ε and n given a fixed R 4

5 Large Deviation Regime Shannon s noisy coding theorem Reliability Function Error exponent analysis Difficult to derive even in the classical case 5

6 Characterizations Converse: Sphere-packing exponent Achievability: Random coding exponent Auxiliary function 6 C. Shannon, R. Gallager, E. Berlekamp, Inform. Cont., 1967 R. Gallager, IEEE IT, 1965

7 Classical Exponent Functions 7

8 Questions in C-Q Are they the same? What is the correct form for c-q channels? Can we obtain a finite blocklength result? Is a tighter prefactor possible? 8

9 Exponent Functions for C-Q Quantum Sibson s identity Sandwiched Rényi divergence: Log-Euclidean Rényi divergence: Petz Rényi divergence: 9 Wilde, Winter, Yang, CMP, 2014 Mosonyi, Ogawa, arxiv:

10 Previous Results for C-Q Channels Winter 1999 Dalai 2013 prefactor 10 A. Winter, (PhD Thesis) arxiv:quant-ph/ M. Dalai, IEEE IT, 2013

11 Why important? Tight in finite blocklength regime Rate close to capacity is concave for is convex and decreasing Moderate deviation regime: see also by Christopher Chubb, Vincent Tan, Marco Tomamichel 11 Cheng and Hsieh, IEEE IT, 2016

12 Main Result: Sphere-Packing Bound Dailai 2013: 12

13 Symmetric C-Q Channels 13 Altug, Wagner, Refinement of Random Coding Bound, IEEE, 60(10): , 2014

14 Constant Composition Codes E.g. Codebook 14 Scarlett, PhD Thesis (University of Cambridge), 2014

15 α-mutual Information Rényi information Augustin information 15 A. Renyi, Proc. Symp. on Math., Stat. and Prob., 1961 U. Augustin, 1969 S. Verdú, ITA, 2015

16 Crucial Properties Continuity: (a) The map is continuous on (b) For every, is increasing on (c) For every, is strictly concave on (d) The map is continuous and increasing on Saddle-point: (a) For every, is a unique saddle-point (b) For every, is differentiable with (c) For every, is continuous 16

17 Proof Sketch Channel coding Hypothesis testing Tight concentration inequality Relate to sphere-packing exponent Properties of the exponents 17

18 Step 1: Channel Coding Hypothesis Testing Hypothesis testing reduction (meta-converse) Minimum type-i error 18 R. Blahut, Hypothesis testing and information theory, IEEE IT, 20(4): , 1974

19 Step 2: Converse Bound Nussbaum-Szkoła mapping: Chebyshev-type bound 19

20 A Tight Concentration Inequality Bahadur-Ranga Rao's inequality Let 20 Bahadur, Ranga Rao, On deviations of the sample mean, Ann. Math. Stat., 31: , 1960

21 Step 3: Relate to SP Exponent Using saddle-point property Choose Remove the rate back-off 21

22 Proof Sketch Sharp bound good codes prefactor: poly-prefactor general codes constant composition codes Chebyshev bound bad codes prefactor: Criterion: 22

23 Interplay of Three Parameters [Small deviation] Interplay between R and n given a fixed ε [Large deviation] Interplay between ε and n given a fixed R [Moderate deviation] Interplay between three parameters Q: Is the reliable communication possible as rate approaches capacity? 23

24 Illustrations Error Medium errors Large errors Small errors Rate 24 Chubb, Tan, Tomamichel, Moderate deviation analysis for classical communication over quantum channels, arxiv:

25 Main Results Assumption: From small deviation From large deviation Let Let 25

26 Achievability Upper bound for error probability Hayashi 07 Properties of (a) The map is smooth (b) For every, is concave on (c) First-order derivative: (d) Second-order derivative: Asymptotic expansion 26 M. Hayashi, PRA, 2007

27 Converse Dalai 13 Properties of (a) The map is smooth (b) For every, is concave on (c) First-order derivative: (d) Second-order derivative: Asymptotic expansion 27

28 Previous on Converse (1/2) Weak sphere-packing exponent Weak sphere-packing bound For any, 28 E. Haroutunian, Problem. Peredachi Inform., 1968, R. Blahut, IEEE IT, 1974, J. Omura, Inform. Control, 1975 N. Sharma, N. A. Warsi, PRL, 2013 Y. Altug, A. Wagner, IEEE IT, 2014

29 Previous on Converse (2/2) Properties of (a) The map is smooth (b) For every, is concave on (c) First-order derivative: (d) Second-order derivative: Relative entropy variance: Asymptotic expansion: 29

30 Proof Sketch: Converse (1/2) Dalai 13: 30 Bahadur, Ranga Rao, On deviations of the sample mean, Ann. Math. Stat., 31: , 1960 Chaganty, Sethuraman, Strong large deviation and local limit theorems, Ann. Prob., 21: , 1993

31 Proof Sketch: Converse (2/2) Refined SP bound Asymptotic expansion around C good codes general codes constant composition codes Weak SP bound bad codes Criterion: 31

32 Different Regimes Second-order Analysis Interplay between R and n given a fixed ε [Moderate] Error Exponent Analysis Interplay between ε and n given a fixed R 32 [Small deviation] [Large deviation]

33 Summary 33

34 Discussions and Future Work Exact asymptotics: Unknown in classical cases Beyond C-Q channels Infinite input alphabet Cannot use constant composition codes U. Augustin and Barış Nakiboğlu arxiv: , Entanglement-assisted codes 34