Moderate Deviations for C-Q Channels
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Communication Systems Concern
- Error Probability:
- Transmission Rate:
- Code Length:
\varepsilon
ε
R
R
n
n
\varepsilon
ε
R
R
C
C
1
1
n=1
n=1
n=\infty
n=∞
Small Deviation
R\to C
R→C
\varepsilon>0
ε>0
Large Deviation
\varepsilon\to 0
ε→0
R \neq C
R=C
Performance Trade-offs
Moderate Deviation
R\to C
R→C
\varepsilon\to 0
ε→0
Three Regimes:
Small Deviation
Large Deviation
Moderate Deviation
R\to C
R→C
\varepsilon \neq0
ε=0
\varepsilon\to 0
ε→0
R \neq C
R=C
\varepsilon\to 0
ε→0
R\to C
R→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})
Rn(ε)=C+nVΦ−1(ε)+O(nlogn)
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)}
εn(R)=e−Θ(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)}
εn(Rn)=e−Θ(nan2)
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.
{an}:an→0, ann→∞.
R_n= C- a_n
Rn=C−an
Cheng and Hsieh, arXiv: 1701.03195.
Moderate Deviation:
\varepsilon_n(R_n) = e^{-\Theta(na_n^2)}
εn(Rn)=e−Θ(nan2)
\{a_n\}: a_n\to 0,\ a_n\sqrt{n} \to \infty.
{an}:an→0, ann→∞.
R_n= C- a_n,
Rn=C−an,
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.
\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\}
nan21logεn(Rn)≤nan24−an21maxs{E~0(s,P)−sR}
Achievability:
Step 2:
Apply Taylor Expansion to 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}}
E~0(s,P)=sCW−2s2VW+6s3∂s3∂3E~0(s,P)∣s=sˉ
Property of E~0(s,P):
\tilde{E}_0(s,P):= -\log \sum_x P_x\text{Tr} W_x^{1-s}(PW)^s
E~0(s,P):=−log∑xPxTrWx1−s(PW)s
(a) Partial derivatives of E~0 are continuous.
(b) E~0 is concave in s≥0.
\text{(c)}\left.\frac{\partial}{\partial s}\tilde{E}_0(s,P)\right|_{s=0} = I(P,W).
(c)∂s∂E~0(s,P)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).
(d)∂s2∂2E~0(s,P)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)}
α~e−nRn(ρn∥σn)≥sn⋆n−1/2Ae−nϕn(Rn′∣ρn∥σn)
H0:ρn=ρ1⊗⋯⊗ρn, H1:σn=σ1⊗⋯⊗σn
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
α~e−nR(ρn∥σn)≥n−tAe−nϕn(R′∣ρn∥σn),t>1/2
Summary
Thank you!
Moderate Deviations for C-Q Channels 1701.03195
AQIS 2017: Moderate Deviations for Classical-Quantum Channels
By Lawrence Min-Hsiu Hsieh
AQIS 2017: Moderate Deviations for Classical-Quantum Channels
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