Second-Order Coding Rates for
Entanglement-Assisted Communication
Nilanjana Datta1 , Marco Tomamichel2 , and Mark M. Wilde3
1 Statistical
Laboratory, Centre for Mathematical Sciences,
University of Cambridge
2 School of Physics, The University of Sydney
3 Center for Computation and Technology, Dept. of Physics & Astronomy,
Louisiana State University
ISIT 2015, Hong Kong
Introduction
Main Result
Proof Techniques
C(W ) := max I(X : Y )T ,
PX
where
Conclusion
TXY = PX WY |X
• Shannon’s capacity formula is surprisingly robust if we give
sender and receiver additional free resources.
•
•
•
•
shared randomness
entanglement
back-communication
non-signaling correlations
• For quantum channels some of these capacities are different.
• Unassisted capacity: no single-letter formula known.
• Entanglement-assisted (EA) capacity: natural
generalization of Shannon’s formula (Bennett et al.’02).
Cea (N ) = max I(A : B)τ ,
ρA
where
τAB = NA0 →B |ψ ρ ihψ ρ |AA0
• We study non-asymptotic fundamental limits for EA
communication over quantum channels.
2 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Ingredients: Channel
• Quantum channel: completely positive trace-preserving map
N ≡ NA→B from (linear operators on) A to B.
N
A
B
We assume A and B are finite-dimensional Hilbert spaces.
• The channel is memoryless:
A1
N
B1
A2
N
B2
≡
An
N
A
N ⊗n
B
Bn
3 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Ingredients: Code
• Entanglement-Assisted Code: quadruple
n
o
Cn = M, |ϕiA0 B 0 , {EAm0 →A }m∈M , {Λm
}
BB 0 m∈M .
1
Set of messages: M.
2
Resource state: |ϕiA0 B 0 .
3
encoder E: a quantum channel
EAm0 →A for each message m.
4
decoder D: a positive operator
valued measure where Λm
BB 0
indicates that we decode to m.
4 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Ingredients: Code
• Entanglement-Assisted Code: quadruple
n
o
Cn = M, |ϕiA0 B 0 , {EAm0 →A }m∈M , {Λm
}
BB 0 m∈M .
1
Set of messages: M.
2
Resource state: |ϕiA0 B 0 .
3
encoder E: a quantum channel
EAm0 →A for each message m.
4
decoder D: a positive operator
valued measure where Λm
BB 0
indicates that we decode to m.
B0
|ϕi
A0
5 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Ingredients: Code
• Entanglement-Assisted Code: quadruple
n
o
Cn = M, |ϕiA0 B 0 , {EAm0 →A }m∈M , {Λm
}
0
BB m∈M .
1
Set of messages: M.
2
Resource state: |ϕiA0 B 0 .
3
encoder E: a quantum channel
EAm0 →A for each message m.
A0
decoder D: a positive operator
valued measure where Λm
BB 0
indicates that we decode to m.
M
4
A1
A2
E
An
6 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Ingredients: Code
• Entanglement-Assisted Code: quadruple
n
o
Cn = M, |ϕiA0 B 0 , {EAm0 →A }m∈M , {Λm
BB 0 }m∈M .
1
Set of messages: M.
2
Resource state: |ϕiA0 B 0 .
3
encoder E: a quantum channel
EAm0 →A for each message m.
4
decoder D: a positive operator
valued measure where Λm
BB 0
indicates that we decode to m.
B0
B1
B2
D
c
M
Bn
7 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Ingredients: Error Probability
B0
A1
|ϕi
A0
A2
N
N
B1
B2
D
E
M
c
M
An
N
Bn
• Average Probability of Error: M uniformly random in M,
perr (Cn , N ⊗n ) := Pr M 6= M̂ .
8 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Ingredients: Non-Asymptotic Achievable Region
• Non-Asymptotic Achievable Region: A triple {R, n, ε} is
achievable if there exists a code Cn for N ⊗n with
1
log |M| ≥ R,
n
and perr (Cn , N ⊗n ) ≤ ε
• Tolerated Error: Fixed ε ∈ (0, 1).
∗ (n; ε) is the maximum
• Boundary of Achievable Region: RN
rate R such that {R, n, ε} is achievable.
∗ (n; ε) for large n and fixed ε.
• We investigate n 7→ RN
• The O(·) and o(·) describe the asymptotics for n → ∞. The
implied constants may depend on N and ε.
• Φ is the cumulative (normal) Gaussian distribution function.
9 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Known Results: Classical
• Shannon’48 and Wolfowitz’60 established that
∗
RW
(n; ε) = C(W ) + o(1)
(as n → ∞) .
• Refined by Strassen’62, Hayashi’09, and Polyanskiy et al.’10.
• Under some regularity conditions (Polyanskiy’10, T & Tan’13):
r
∗
RW
(n; ε)
= C(W ) +
1
V (W ) −1
log n
Φ (ε) +
+O
.
n
2n
n
0.6
rate, R
0.5
third-order approximation
capacity
0.4
0.3
converse rate
achievable rate
0.2
0.1
20
40
60
80
100
number of channel uses, n
10 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Known Results: Quantum
• Bennett et al.’99–02 established
∗
lim lim RN
(n; ε) = Cea (N ) .
ε→0 n→∞
• The strong converse follows from the quantum reverse
Shannon theorem (Bennett et al.’14, Berta et al.’11):
∗
RN
(n; ε) = Cea (N ) + o(1)
• Gupta and Wilde’15 proved
1 ∗
RN
(n; ε) = Cea (N ) + O √
n
• Our main result reveals the second-order term scaling as
√
1/ n in the above for covariant channels.
11 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Main Result: Covariant Channels
• Covariant channels: NA→B ◦ UA (g) = UB (g) ◦ NA→B for all
g ∈ G for some group G, with UA (g) irreducible.
• They include qubit Pauli and depolarizing channels.
Result
For any covariant quantum channel N and ε ∈ (0, 1), we have
r
log n Vea (N ) −1
∗
RN
(n, ε) = Cea (N ) +
Φ (ε) + O
.
n
n
12 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Main Result: Covariant Channels
Result
For any covariant quantum channel N and ε ∈ (0, 1), we have
r
log n Vea (N ) −1
∗
RN
(n, ε) = Cea (N ) +
Φ (ε) + O
.
n
n
• Let ψAA0 be the maximally entangled state. Then,
entanglement-assisted channel capacity and dispersion are
Cea (N ) = D NA→B (ψAA0 )ψA0 ⊗ NA→B (ψA ) ,
Vea (N ) = V NA→B (ψAA0 )ψA0 ⊗ NA→B (ψA ) .
• relative entropy: D(ρkτ ) := tr ρ(log ρ − log τ ) .
• its variance: V (ρkτ ) := tr ρ(log ρ − log τ − D(ρkτ ))2 .
13 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Main Result: Achievable Rate
Result
For any quantum channel N and ε ∈ (0, 1), we have
r
log n ε (N )
Vea
∗
RN (n, ε) ≥ Cea (N ) +
Φ−1 (ε) + O
.
n
n
• Entanglement-Assisted Capacity:
Cea (N ) := max D NA→B (ρAA0 )ρA0 ⊗ NA→B (ρA ) .
ρA
• Capacity achieving input states:
ΠA := arg max D NA→B (ρAA0 )ρA0 ⊗ NA→B (ρA ) .
ρA
• Entanglement-Assisted Channel Dispersion:
(
Vea (N ) :=
min V (NA→B (ρAA0 )kρA0 ⊗ NA→B (ρA ))
if ε <
1
2
max V (NA→B (ρAA0 )kρA0 ⊗ NA→B (ρA ))
if ε ≥
1
2
ρA ∈ΠA
ρA ∈ΠA
.
14 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Binary Hypothesis Testing
• The ε-hypothesis testing relative entropy:
Dhε (ρkσ) := − log βε (ρkσ) ,
βε (ρkσ) := min tr(Qσ) : 0 ≤ Q ≤ I, tr(Qρ) ≥ 1 − ε .
• Its asymptotics for i.i.d. states ρ⊗n and σ ⊗n for large n are
the main ingredient.
• Second-order expansion (T & Hayashi’13 and Li’14):
p
Dhε ρ⊗n σ ⊗n = nD(ρkσ) + nV (ρkσ) Φ−1 (ε) + O(log n) .
15 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Achievability: Code (1)
• Key Idea: random superdense coding on type subspaces.
• Resource state: Use HA0 ≡ HB 0 ≡ HA and decompose
⊗n
⊗n
HA
⊗ HB
=
M
λ
λ
HA
n ⊗ HB n
λ
where Hλ is the subspace spanned by vectors of type λ.
• We use the state
λ
Xp
ϕAn B n =
p(λ) ψA
nBn ,
λ
λ
λ
λ
where ψA
n B n is maximally entangled on HAn ⊗ HB n .
16 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Achievability: Code (2)
• Random code: For each m ∈ M, randomly chose
sm = {bt , xt , zt }t where bt ∈ {0, 1}, and
xt , zt ∈ {0, 1, . . . , dt − 1}.
†
• Encoder: Define E m ( · ) = UA (sm ) · UA
(sm ) where
UA (s) :=
M
(−1)bt X(xt )Z(zt ),
t
where X and Z are Heisenberg-Weyl operators.
• Decoder: “Pretty-good” measurement.
17 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Achievability: One-Shot to Asymptotics
λ
Xp
Recall: ϕAn B n =
p(λ) ψA
nBn ,
λ
Lemma
For any δ ∈ (0, 2ε ), we have
∗
RN
(ε) ≥
1 ε−2δ
⊗n
DH
NA→B
0 (ϕAn B n )
n
X
λ
⊗n
λ
n
n
p(λ)
π
⊗
N
(π
)
B
A
A→B 0
λ
1−ε
1
− log 2 ,
n
δ
λ is the maximally mixed state on the space Hλ .
where πA
n
An
• Every pure state ρ⊗n
AA0 can be written in the above form.
• Moreover, we show that
X
⊗n
λ
λ
p(λ) πB
(π
)
. poly(n) · (φB ⊗ NA→B 0 (φA ))⊗n
n ⊗ N
n
A→B 0 A
λ
18 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Converse
• We use the one-shot converse by Matthews & Wehner’14:
1
⊗n
⊗n
∗
RN
(n; ε) ≤ max Dhε NA→B
(ρAn A0 n )ρA0 n ⊗ NA→B
(ρAn )
ρAn n
• The functional is quasi-concave in ρAn , allowing to reduce the
optimization to states invariant under the channel symmetry.
• Generally these are the permutation invariant states.
• For covariant channels the only invariant state is the fully
mixed state ψA . Then,
∗
RN
(n; ε) ≤
1 ε
Dh (NA→B (ψAA0 ))⊗n (ψA0 ⊗ NA→B (ψA )⊗n ) .
n
• Second order expansion for hypothesis testing yields result.
19 / 20
Introduction
Main Result
Proof Techniques
Conclusion
Conclusion
• We have established the second-order approximation for the
EA capacity of covariant channels.
• We also show a second-order achievability for general
channels.
• We conjecture that this is tight also for general channels.
• Establishing a general second-order converse bound requires
new techniques, and constitutes an interesting open problem.
• Reduction from permutation invariant to product states using
de Finetti argument is not tight enough.
20 / 20