Moderate Deviations for C-Q Channels
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Communication Systems Concern
- Error Probability:
- Transmission Rate:
- Code Length:
\varepsilon
R
n
\varepsilon
R
C
1
n=1
n=\infty
Small Deviation
R\to C
\varepsilon>0
Large Deviation
\varepsilon\to 0
R \neq C
Performance Trade-offs
Moderate Deviation
R\to C
\varepsilon\to 0
Three Regimes:
Small Deviation
Large Deviation
Moderate Deviation
R\to C
\varepsilon \neq0
\varepsilon\to 0
R \neq C
\varepsilon\to 0
R\to C
Small Deviation:
a.k.a. Second-Order Analysis
R_n(\varepsilon) = C + \sqrt{\frac{V}{n}}\Phi^{-1}(\varepsilon) + O(\frac{\log n}{n})
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
\varepsilon_n(R) = e^{-\Theta(n)}
Shannon, Bell System Technical Journal, 38(3):611–656, 1959.
Burnashev and Holevo, Problems of information transmission, 34(2):97–107, 1998.
Moderate Deviation:
\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}
Altug and Wagner. IEEE TIT 60(8):4417–4426, 2014.
Chubb, Tomamichel and Tan, arXiv: 1701.03114.
\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
R_n= C- a_n
Cheng and Hsieh, arXiv: 1701.03195.
Moderate Deviation:
\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}
\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
R_n= C- a_n,
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.
\frac{1}{n a_n^2}\log\varepsilon_n(R_n)\leq \frac{4}{n a_n^2} - \frac{1}{a_n^2}\max_{s} \left\{ \tilde{E}_0(s,P)-sR\right\}
Achievability:
Step 2:
Apply Taylor Expansion to \(\tilde{E}_0(s,P)\) at \(s=0\).
\tilde{E}_0(s,P) = s C_W - \frac{s^2}{2} V_W + \frac{s^3}{6} \frac{\partial^3 \tilde{E}_0(s,P)}{\partial s^3}|_{s=\bar{s}}
Property of \(\tilde{E}_0(s,P)\):
\tilde{E}_0(s,P):= -\log \sum_x P_x\text{Tr} W_x^{1-s}(PW)^s
(a) Partial derivatives of \(\tilde{E}_0\) are continuous.
(b) \(\tilde{E}_0\) is concave in \(s\geq 0 \).
\text{(c)}\left.\frac{\partial}{\partial s}\tilde{E}_0(s,P)\right|_{s=0} = I(P,W).
\text{(d)}\left.\frac{\partial^2}{\partial s^2}\tilde{E}_0(s,P)\right|_{s=0} = V(P,W).
Converse
Similar to Quantum SP Bounds
1. A New Sharp Hoeffiding Bound.
2. Weak Hoeffiding Bound needs special attention.
New Sharp Converse Hoeffding Bound
\tilde{\alpha}_{e^{-nR_n}}(\rho^n\|\sigma^n) \geq \frac{A}{{s_n^\star} n^{-1/2}} e^{-n\phi_n(R_n'|\rho^n\|\sigma^n)}
\(H_0:\rho^n=\rho_1\otimes\cdots\otimes\rho_n\), \(H_1:\sigma^n=\sigma_1\otimes\cdots\otimes\sigma_n\)
Chaganty-Sethuraman , The Annals of Probability, 21(3):1671–1690, 1993.
\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \frac{A}{ n^{-t}} e^{-n\phi_n(R'|\rho^n\|\sigma^n)}, \quad t>1/2
Summary
Thank you!
AQIS 2017: Moderate Deviations for Classical-Quantum Channels
By Lawrence Min-Hsiu Hsieh
