Quantum Sphere-Packing Bounds with Polynomial Prefactors

1704.05703



 

Communication Systems Concern

  • Error Probability:
  • Transmission Rate:
  • Code Length:
\varepsilon
ε\varepsilon
R
RR
n
nn
\varepsilon
ε\varepsilon
R
RR
C
CC
1
11
n=1
n=1n=1
n=\infty
n=∞n=\infty

Small Deviation

R\to C
R→CR\to C
\varepsilon>0
ε>0\varepsilon>0

Large Deviation

\varepsilon\to 0
ε→0\varepsilon\to 0
R \neq C
R≠CR \neq C

Performance Trade-offs

Moderate Deviation

R\to C
R→CR\to C
\varepsilon\to 0
ε→0\varepsilon\to 0

Large Deviation:

a.k.a. Error Exponent Analysis

\varepsilon_n(R) = e^{-\Theta(n)}
εn(R)=e−Θ(n)\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.

\lim_{n\to\infty} \frac{-1}{n} \log\varepsilon_n(R) = E(R)
limn→∞−1nlogεn(R)=E(R)\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)
Erc(R)≤E(R)≤Esp(R)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\}
Esp(R):=sups≥0{maxPE0(s,P)−sR} 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\}
E~sp(R):=maxPminV{D(V∥W∣P):I(P,V)≤R}\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)
Esp(R)=E~sp(R){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\}
Esp(R):=sups≥0{maxPE0(s,P)−sR} 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\}
E~sp(R):=maxPminV{D(V∥W∣P):I(P,V)≤R}\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)
Esp(R)≤E~sp(R){E}_{\text{sp}}(R)\leq \tilde{E}_{\text{sp}}(R)
\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~sp(R,P)=sup0<α≤1minσ{1−αα(Dα♭(W∥σ∣P)−R)}\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)\}
Esp(R,P)≤sup0<α≤1minσ{1−αα(Dα(W∥σ∣P)−R)}{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)
Esp(R)≤E~sp(R){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}]
Dα(ρ∥σ):=1α−1logTrρασ(1−α)≤Dα♭(ρ∥σ):=1α−1Tr[eαlogρ+(1−α)logσ]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})
log1εn(R)≤nEsp(R)+O(n)\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})
log1εn(R)≤nEsp(R)+O(logn)\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\}
α~μ(ρ∥σ)=minΠ{α(Π,ρ):β(Π,σ)≤μ}\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)}
α~e−nR(ρn∥σn)≥κ1e−κ2n−nϕn(R′∣ρn∥σn)\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\}
ϕn(r∣ρn∥σn):=supα∈(0,1]{1−αα(1nDα(ρn∥σn)−r)}\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)}
α~e−nR(ρn∥σn)≥An−te−nϕn(R′∣ρn∥σn)\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
t>1/2t> 1/2

If Channel is Symmetric 

W_x = V^{x-1}W_1 (V^\dagger)^{x-1},
Wx=Vx−1W1(V†)x−1,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)
log1εn(R)≤nEsp(R)+12(1+∣Esp′(R)∣)logn+o(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)
Iα(1)(P,W):=infσDα(P∘W∥P⊗σ)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)
Iα(2)(P,W):=infσDα(W∥σ∣P)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)
Esp(1)(R,P):=sup0<α≤11−αα(Iα(1)(P,W)−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)
Esp(2)(R,P):=sup0<α≤11−αα(Iα(2)(P,W)−R)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)
Esp(R,P):=sups>0(E0(s,P)−sR)E_{\text{sp}}(R,P):= \sup_{s>0} \left(E_{0}(s,P)-sR\right)

Open Questions

1. Beyond C-Q Channel?

2. EA Channel?

Thank you!

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