Moderate and Large Deviation Analysis for C-Q Channels
1701.03195 | 1704.05703 |
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Communication Systems Concern
- Error Probability:
- Transmission Rate:
- Code Length:
Small Deviation
Large Deviation
Performance 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.
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.
Classical-Quantum Channels
Part I
Large Deviation
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
\(E_0(s,P)\) is concave in \(s\geq 0\)
Cheng, MH. IEEE Transactions on Information Theory, vol. 62, no. 10, pp. 5960–5965 (2016).
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:
$$\varepsilon_{\max}(\mathcal{C}_n) \geq \max_\sigma\min_{\mathbb{x}^n\in \mathcal{C}_n} \tilde{\alpha}_{\frac{1}{|\mathcal{C}_n|}}(W_{\mathbb{x}^n}||\sigma).$$
Step 2:
Two one-shot converse Hoeffding bounds for \(\tilde{\alpha}_\mu(\cdot|\cdot)\).
For Bad Codewords, Use Weak Converse Hoeffding Bound.
\(H_0:\rho^n=\rho_1\otimes\cdots\otimes\rho_n\); \(H_1:\sigma^n=\sigma_1\otimes\cdots\otimes\sigma_n\)
Blahut. IEEE TIT, 20(4):405–417, 1974.
Audenaert et. al., PRL 98:160501, 2007.
For Good Codewords, Use Sharp Converse Hoeffding Bound.
\(H_0:\rho^n=\rho_1\otimes\cdots\otimes\rho_n\), \(H_1:\sigma^n=\sigma_1\otimes\cdots\otimes\sigma_n\)
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 \((\alpha,P)\to I_\alpha\) is continuous on \([0,1]\times\mathcal{P}(\mathcal{X})\).
(b) The map \(\alpha\to I_\alpha\) is monotone increasing on \([0,1]\).
(c) The map \(\alpha\to \frac{1-\alpha}{\alpha}I_\alpha\) is strictly concave on \((0,1]\).
Property of \(E(R)\):
(a) The map \(R\to E^{(\cdot)}_{\text{sp}}\) is convex, continuous, and non-increasing.
(b) \(E^{(\cdot)}_{\text{sp}}\) is differentiable w.r.t. \(R\).
(c) \({E'}^{(\cdot)}_{\text{sp}}\) is continuous.
Part II
Moderate Deviation
Moderate Deviation:
Cheng and Hsieh, arXiv: 1701.03195.
[Achievability] \(\limsup_{n\to\infty}\frac{1}{na_n^2}\log\varepsilon_n(R_n)\leq -\frac{1}{2V_W}\)
[Converse] \(\liminf_{n\to\infty}\frac{1}{na_n^2}\log\varepsilon_n(R_n)\geq -\frac{1}{2V_W}\)
Chubb, Tomamichel and Tan, arXiv: 1701.03114.
Achievability:
Step 1:
$$\varepsilon_n(R_n) \leq 4\exp\left(-n\left[\max_{0\leq s\leq 1} \tilde{E}_0(s,P)-sR_n\right]\right)$$
Hayashi, PRA 76(6): 06230,12007.
Achievability:
Step 2:
Apply Taylor Expansion to \(\tilde{E}_0(s,P)\) at \(s=0\).
Property of \(\tilde{E}_0(s,P)\):
(a) Partial derivatives of \(\tilde{E}_0\) are continuous.
(b) \(\tilde{E}_0\) is concave in \(s\geq 0 \).
Converse
Similar to Quantum SP Bounds
1. A New Sharp Hoeffiding Bound.
2. Weak Hoeffiding Bound needs special attention.
New Sharp Converse Hoeffding Bound
\(H_0:\rho^n=\rho_1\otimes\cdots\otimes\rho_n\), \(H_1:\sigma^n=\sigma_1\otimes\cdots\otimes\sigma_n\)
Chaganty-Sethuraman , The Annals of Probability, 21(3):1671–1690, 1993.
Summary
Open Questions
1. Beyond C-Q Channel?
2. EA Channel?
Thank you!
Quantum Moderate and Large Deviation Analysis
By Lawrence Min-Hsiu Hsieh
Quantum Moderate and Large Deviation Analysis
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