Error Exponent Analysis in Quantum Source and Channel Coding

1. [arXiv:1803.07505]: Cheng (NTU&UTS), Hanson (Cambridge), Datta (Cambridge), MH
3. [arXiv:1701.03195]: Cheng (NTU&UTS), MH
2. [arXiv:1704.05703]: Cheng (NTU&UTS), MH, Tomamichel (UTS)
In Source/Channel Coding,
- Error Probability:
- Transmission Rate:
- Code Length:
Small Deviation
Large Deviation
Channel Coding Trade-offs
Moderate Deviation
Small Deviation
Large Deviation
Source Coding Trade-offs
Moderate Deviation
Three Regimes:
Small Deviation
Large Deviation
Moderate Deviation
Small Deviation:
a.k.a. Second-Order Analysis
Strassen, Transactions of the Third Prague Conference on Information Theory, pp. 689–723, 1962.
Tomamichel and Tan, CMP 338(1):103–137, 2015.
Tomamichel and Hayashi, IEEE IT 59(11):7693-7710, 2013.
Nomura and Han. IEEE IT 60(9):5553–5572, 2014.
Source
Channel
Large Deviation:
a.k.a. Error Exponent Analysis
Shannon, Bell System Technical Journal, 38(3):611–656, 1959.
Burnashev and Holevo, Problems of information transmission, 34(2):97–107, 1998.
Moderate Deviation:
Altug and Wagner. IEEE TIT 60(8):4417–4426, 2014.
Chubb, Tomamichel and Tan, arXiv: 1701.03114.
Cheng and Hsieh, arXiv: 1701.03195.
Outlines
Part I
Channel Coding
Classical-Quantum Channels
Error Exponent Analysis
Shannon, Bell System Technical Journal, 38(3):611–656, 1959.
Claissical Sphere-Packing Bounds
Shannon, Gallager, and Berlekamp. Information and Control, 10(1):65–103, 1967.
Haroutunian. Problemy Peredachi Informatsii, 4(4):37–48, 1968, (in Russian).
Blahut. IEEE TIT, 20(4):405–417, 1974.
Quantum Sphere-Packing Bounds
Dalai. IEEE TIT, 59(12):8027–8056, 2013.
Winter. PhD Thesis, Universitate Bielefeld, 1999.
Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703
Theorem:
Proof:
Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703
Dalai's Sphere-Packing Bound
Dalai. IEEE TIT, 59(12):8027–8056, 2013.
Shannon, Gallager, and Berlekamp. Information and Control, 10(1):65–103, 1967.
Theorem:
Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703, 2017.
Altug and Wagner, IEEE TIT, 60(3): 1592–1614, 2014.
Proof:
Step 1:
εmax(Cn)≥σmaxxn∈Cnminα~∣Cn∣1(Wxn∣∣σ).
Step 2:
Two one-shot converse Hoeffding bounds for α~μ(⋅∣⋅).
For Bad Codewords, Use Weak Converse Hoeffding Bound.
H0:ρn=ρ1⊗⋯⊗ρn; H1:σn=σ1⊗⋯⊗σn
Blahut. IEEE TIT, 20(4):405–417, 1974.
Audenaert et. al., PRL 98:160501, 2007.
For Good Codewords, Use Sharp Converse Hoeffding Bound.
H0:ρn=ρ1⊗⋯⊗ρn, H1:σn=σ1⊗⋯⊗σn
Bahadur and Rao, The Annals of Mathematical Statistics, 31(4):1015–1027, 1960.
Altug and Wagner, IEEE TIT, 60(3): 1592–1614, 2014.
If Channel is Symmetric
then Sphere-packing bound is exact.
Property of E(R):
(a) The map (α,P)→Iα is continuous on [0,1]×P(X).
(b) The map α→Iα is monotone increasing on [0,1].
(c) The map α→α1−αIα is strictly concave on (0,1].
Property of E(R):
(a) The map R→Esp(⋅) is convex, continuous, and non-increasing.
(b) Esp(⋅) is differentiable w.r.t. R.
(c) E′sp(⋅) is continuous.
Part I.B
Moderate Deviation
Moderate Deviation:
Cheng and Hsieh, arXiv: 1701.03195.
[Achievability] limsupn→∞nan21logεn(Rn)≤−2VW1
[Converse] liminfn→∞nan21logεn(Rn)≥−2VW1
Chubb, Tomamichel and Tan, arXiv: 1701.03114.
Achievability:
Step 1:
εn(Rn)≤4exp(−n[0≤s≤1maxE~0(s,P)−sRn])
Hayashi, PRA 76(6): 06230,12007.
Achievability:
Step 2:
Apply Taylor Expansion to E~0(s,P) at s=0.
Property of E~0(s,P):
(a) Partial derivatives of E~0 are continuous.
(b) E~0 is concave in s≥0.
Converse
Similar to Quantum SP Bounds
1. A New Sharp Hoeffiding Bound.
2. Weak Hoeffiding Bound needs special attention.
New Sharp Converse Hoeffding Bound
H0:ρn=ρ1⊗⋯⊗ρn, H1:σn=σ1⊗⋯⊗σn
Chaganty-Sethuraman , The Annals of Probability, 21(3):1671–1690, 1993.
Summary



Part II
Source Coding (with quantum side information)
Error Exponent Analysis
Shannon, Bell System Technical Journal, 38(3):611–656, 1959.
Theorem
Theorem
Theorem
Moderate Deviation:
Duality
Open Questions
1. Beyond C-Q Channel?
2. EA Channel?
3. Duality?
Thank you!
Error Exponent Analysis in Quantum Source and Channel Coding
By Lawrence Min-Hsiu Hsieh
Error Exponent Analysis in Quantum Source and Channel Coding
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