Quantum Dynamic Capacity & Superadditivity
- arXiv:0811.4227 with M. Wilde
- arXiv:0901.3038 with M. Wilde
- arXiv:1708.04314 with Y. Zhu, Q. Zhung, and P. Shor
\langle\mathcal{N} \rangle \geq R[c\to c]+ Q[q\to q]+ E[qq]
⟨N⟩≥R[c→c]+Q[q→q]+E[qq]
Quantum Dynamic Protocols
[MH & M.Wilde, arXiv:0901.3038]
C(\mathcal{N})= \left\{(R,Q,E): \right.
C(N)={(R,Q,E):
Quantum Dynamic Capacity
a quantum dynamic protocol exists.}
\langle\mathcal{N}\rangle + |E|[qq] \geq R[c\to c]+ Q[q\to q]
⟨N⟩+∣E∣[qq]≥R[c→c]+Q[q→q]
EA-assisted C-Q Communication
R= I(X:B)_\sigma
R=I(X:B)σ
Q= \frac{1}{2}I(A:B|X)_\sigma
Q=21I(A:B∣X)σ
|E|= \frac{1}{2}I(A:E|X)_\sigma
∣E∣=21I(A:E∣X)σ
[MH & M.Wilde, arXiv:0811.4227]
Special Cases
\langle\mathcal{N} \rangle \geq R[c\to c]
⟨N⟩≥R[c→c]
\langle\mathcal{N} \rangle \geq Q[q\to q]
⟨N⟩≥Q[q→q]
\langle\mathcal{N} \rangle \geq R[c\to c] + Q[q\to q]
⟨N⟩≥R[c→c]+Q[q→q]
\langle\mathcal{N}\rangle +|E|[qq] \geq R[c\to c]
⟨N⟩+∣E∣[qq]≥R[c→c]
\langle\mathcal{N} \rangle+|E|[qq] \geq Q[q\to q]
⟨N⟩+∣E∣[qq]≥Q[q→q]
SD
TP
ED
EACQ Capacity Region
R+2Q\leq I(AX:B)_\sigma
R+2Q≤I(AX:B)σ
Q\leq I(A\rangle BX)_\sigma +|E|
Q≤I(A⟩BX)σ+∣E∣
R+Q\leq I(X:B)_\sigma + I(A\rangle BX)_\sigma +|E|
R+Q≤I(X:B)σ+I(A⟩BX)σ+∣E∣
\{
{
C^{(1)}_\sigma(\mathcal{N})
Cσ(1)(N)
C^{(1)}(\mathcal{N})=\bigcup_\sigma C^{(1)}_\sigma(\mathcal{N})
C(1)(N)=⋃σCσ(1)(N)
C(\mathcal{N})=\overline{\bigcup_{k=1}^{\infty} \frac{1}{k}C^{(1)}_\sigma(\mathcal{N}^{\otimes k})}
C(N)=⋃k=1∞k1Cσ(1)(N⊗k)
\text{SD:} [q\to q]+[qq]\geq 2[c\to c]
SD:[q→q]+[qq]≥2[c→c]
\text{TP:} [qq]+2[c\to c]\geq [q\to q]
TP:[qq]+2[c→c]≥[q→q]
\text{ED:} [q\to q]\geq [qq]
ED:[q→q]≥[qq]
Quantum Dynamic Capacity
R+2Q\leq I(AX:B)_\sigma
R+2Q≤I(AX:B)σ
Q+E\leq I(A\rangle BX)_\sigma
Q+E≤I(A⟩BX)σ
R+Q+E\leq I(X:B)_\sigma + I(A\rangle BX)_\sigma
R+Q+E≤I(X:B)σ+I(A⟩BX)σ
\{
{
C^{(1)}_\sigma(\mathcal{N})
Cσ(1)(N)
C^{(1)}(\mathcal{N})=\bigcup_\sigma C^{(1)}_\sigma(\mathcal{N})
C(1)(N)=⋃σCσ(1)(N)
C(\mathcal{N})=\overline{\bigcup_{k=1}^{\infty} \frac{1}{k}C^{(1)}_\sigma(\mathcal{N}^{\otimes k})}
C(N)=⋃k=1∞k1Cσ(1)(N⊗k)
Dynamic Achievable Region
Quantum Dephasing Channels
Additivity
C(\mathcal{N})=\overline{\bigcup_{k=1}^{\infty} \frac{1}{k}C^{(1)}_\sigma(\mathcal{N}^{\otimes k})}
C(N)=⋃k=1∞k1Cσ(1)(N⊗k)
\stackrel{?}{=}
=?
C^{(1)}_\sigma(\mathcal{N})
Cσ(1)(N)
Additivity
| Type | Additive? |
|---|---|
| Classical Capacity | |
| Quantum Capacity | |
| EA-assisted Classical Capacity | |
| EA-assisted C-Q Capacity |
N
N
N
Y
| Additive | CE | CQ | QE |
| C | |||
| Ｑ | |||
| Ｃ＆Ｑ |
Additivity?
| Implies the trade-off additivity? |
|---|
N
N
N
Y
Y
N
N
N
N
| Implies the trade-off additivity? | |
|---|---|
| Additive | CQE |
| QE | |
| CQ | |
| CE | |
| CE&Q | |
| CE&CQ |
Additivity?
Y
N
N
N
N
[MH & M.Wilde, arXiv:0901.3038]
Proof Ingredients
A Switch Channel such that
C^{(1)}_{CQE}(\mathcal{N}) = \text{conv}\left( C^{(1)}_{CQE}(\mathcal{N_0}),C^{(1)}_{CQE}(\mathcal{N_1}) \right)
CCQE(1)(N)=conv(CCQE(1)(N0),CCQE(1)(N1))
Proof Ingredients
\text{If}\quad C_{CQE}(\mathcal{N}_0) = C^{(1)}_{CQE}(\mathcal{N}_0)
IfCCQE(N0)=CCQE(1)(N0)
\text{then}
then
C_{CQE}(\mathcal{N}) = \text{conv}\left( C_{CQE}(\mathcal{N_0}),C_{CQE}(\mathcal{N_1}) \right)
CCQE(N)=conv(CCQE(N0),CCQE(N1))
Q additive, CQ superadditive
\text{(1):}\quad C_{Q}(\mathcal{N}_0) \geq C_{Q}(\mathcal{N}_1)
(1):CQ(N0)≥CQ(N1)
\text{(3):}\quad C_{C}(\mathcal{N}_1) > C^{(1)}_{C}(\mathcal{N}_1)
(3):CC(N1)>CC(1)(N1)
\text{(2):}\quad C_{C}(\mathcal{N}_0) < C_{C}(\mathcal{N}_1)
(2):CC(N0)<CC(N1)
Thank you for your attention!
Quantum Dynamic Capacity & Superadditivity arXiv:0811.4227 with M. Wilde arXiv:0901.3038 with M. Wilde arXiv:1708.04314 with Y. Zhu, Q. Zhung, and P. Shor