Moderate and Large Deviation Analysis 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.
Classical-Quantum Channels
W: \mathcal{X} \to \mathbb{C}^{d\times d}
Part I
Large Deviation
\lim_{n\to\infty} \frac{-1}{n} \log\varepsilon_n(R) = E(R)
Error Exponent Analysis
E_{\text{rc}}(R) \leq E(R) \leq E_{\text{sp}}(R)
Shannon, Bell System Technical Journal, 38(3):611–656, 1959.
Claissical Sphere-Packing Bounds
E_{\text{sp}}(R):=\sup_{s\geq 0} \left\{ \max_P E_0(s,P) -sR\right\}
\tilde{E}_{\text{sp}}(R):=\max_P\min_V\{D(V\|W|P):I(P,V)\leq R\}
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.
{E}_{\text{sp}}(R)= \tilde{E}_{\text{sp}}(R)
Quantum Sphere-Packing Bounds
E_{\text{sp}}(R):=\sup_{s\geq 0} \left\{ \max_P E_0(s,P) -sR\right\}
\tilde{E}_{\text{sp}}(R):=\max_P\min_V\{D(V\|W|P):I(P,V)\leq R\}
Dalai. IEEE TIT, 59(12):8027–8056, 2013.
Winter. PhD Thesis, Universitate Bielefeld, 1999.
Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703
{E}_{\text{sp}}(R)\leq \tilde{E}_{\text{sp}}(R)
E_0(s,P) := -\log {\rm Tr} \left[
\left( \sum_{x\in\mathcal{X}} P(x) W_x^{\frac1{1+s}}\right)^{1+s}
\right]
\(E_0(s,P)\) is concave in \(s\geq 0\)
Cheng, MH. IEEE Transactions on Information Theory, vol. 62, no. 10, pp. 5960–5965 (2016).
\tilde{E}_{\text{sp}}(R,P)=\sup_{0<\alpha\leq 1}\min_\sigma\{\frac{1-\alpha}{\alpha}\left(D^\flat_\alpha(W\|\sigma|P)-R\right)\}
{E}_{\text{sp}}(R,P)\leq\sup_{0<\alpha\leq 1}\min_\sigma\{\frac{1-\alpha}{\alpha}\left(D_\alpha(W\|\sigma|P)-R\right)\}
Theorem:
{E}_{\text{sp}}(R)\leq \tilde{E}_{\text{sp}}(R)
Proof:
D_\alpha(\rho\|\sigma):= \frac{1}{\alpha-1}\log \text{Tr}\rho^\alpha \sigma^{(1-\alpha)} \leq D^\flat_\alpha(\rho\|\sigma):=\frac{1}{\alpha-1}\text{Tr}[e^{\alpha\log\rho+(1-\alpha)\log\sigma}]
Cheng, Hsieh, and Tomamichel. arXiv: 1704.05703
Dalai's Sphere-Packing Bound
\log \frac{1}{\varepsilon_n(R)} \leq n E_{\text{sp}}(R) + O(\sqrt{n})
Dalai. IEEE TIT, 59(12):8027–8056, 2013.
Shannon, Gallager, and Berlekamp. Information and Control, 10(1):65–103, 1967.
Theorem:
\log \frac{1}{\varepsilon_n(R)} \leq n E_{\text{sp}}(R) + O(\log{n})
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 \(\tilde{\alpha}_\mu(\cdot|\cdot)\).
\tilde{\alpha}_\mu(\rho\|\sigma) = \min_\Pi\{\alpha(\Pi,\rho): \beta(\Pi,\sigma)\leq \mu\}
For Bad Codewords, Use Weak Converse Hoeffding Bound.
\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \kappa_1 e^{-\kappa_2\sqrt{n}-n\phi_n(R'|\rho^n\|\sigma^n)}
\phi_n(r|\rho^n\|\sigma^n):= \sup_{\alpha\in(0,1]} \left\{\frac{1-\alpha}{\alpha}(\frac{1}{n}D_\alpha(\rho^n\|\sigma^n)-r)\right\}
\(H_0:\rho^n=\rho_1\otimes\cdots\otimes\rho_n\); \(H_1:\sigma^n=\sigma_1\otimes\cdots\otimes\sigma_n\)
Blahut. IEEE TIT, 20(4):405–417, 1974.
Audenaert et. al., PRL 98:160501, 2007.
For Good Codewords, Use Sharp Converse Hoeffding Bound.
\tilde{\alpha}_{e^{-nR}}(\rho^n\|\sigma^n) \geq \frac{A}{n^{-t}} e^{-n\phi_n(R'|\rho^n\|\sigma^n)}
\(H_0:\rho^n=\rho_1\otimes\cdots\otimes\rho_n\), \(H_1:\sigma^n=\sigma_1\otimes\cdots\otimes\sigma_n\)
Bahadur and Rao, The Annals of Mathematical Statistics, 31(4):1015–1027, 1960.
Altug and Wagner, IEEE TIT, 60(3): 1592–1614, 2014.
t> 1/2
If Channel is Symmetric
W_x = V^{x-1}W_1 (V^\dagger)^{x-1},
\log \frac{1}{\varepsilon_n(R)} \leq n E_{\text{sp}}(R) + \frac{1}{2}(1+|E'_{\text{sp}}(R)|)\log{n}+o(1)
then Sphere-packing bound is exact.
Property of \(E(R)\):
I_{\alpha}^{(1)}(P,W):= \inf_\sigma D_\alpha(P\circ W\|P\otimes\sigma)
I_{\alpha}^{(2)}(P,W):= \inf_\sigma D_\alpha(W\|\sigma|P)
(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 \(E(R)\):
E^{(1)}_{\text{sp}}(R,P):= \sup_{0<\alpha\leq 1} \frac{1-\alpha}{\alpha} \left(I_{\alpha}^{(1)}(P,W)-R\right)
(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.
E^{(2)}_{\text{sp}}(R,P):= \sup_{0<\alpha\leq 1} \frac{1-\alpha}{\alpha} \left(I_{\alpha}^{(2)}(P,W)-R\right)
E_{\text{sp}}(R,P):= \sup_{s>0} \left(E_{0}(s,P)-sR\right)
Part II
Moderate Deviation
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
Open Questions
1. Beyond C-Q Channel?
2. EA Channel?
Thank you!
Quantum Moderate and Large Deviation Analysis
By Lawrence Min-Hsiu Hsieh
