Download Strong converse theorem for state redistribution using Rényi

Strong converse theorem for
state redistribu on using Rényi entropies
arXiv:1506.02635
Felix Leditzkya , Mark M. Wildeb , Nilanjana Da aa
a: University of Cambridge, b: Louisiana State University
ISIT Barcelona, 15 July 2016
Table of Contents
1 Introduc on: Weak vs. strong converse
2 State redistribu on
3 Rényi entropies
4 Main result: Strong converse for state redistribu on
5 Summary and open ques ons
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Table of Contents
1 Introduc on: Weak vs. strong converse
2 State redistribu on
3 Rényi entropies
4 Main result: Strong converse for state redistribu on
5 Summary and open ques ons
3 / 23
Introduc on: Weak vs. strong converse
▶ Consider: informa on-theore c task with op mal rate r∗ .
▶ Code with rate r, blocklength n, and error ϵ n .
lim ϵ n
lim ϵ n
n→∞
n→∞
1
1
?
0
r∗
Weak converse
r
0
r∗
r
Strong converse
▶ This talk: Strong converse for state redistribu on.
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Table of Contents
1 Introduc on: Weak vs. strong converse
2 State redistribu on
3 Rényi entropies
4 Main result: Strong converse for state redistribu on
5 Summary and open ques ons
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State redistribu on
Target state
Ini al state
R
R
ψABCR
ψA′ B′ C′ R
C
A
B
C′
TA
Φk
TB
T′A
A′
Φm
B′
T′B
▶ Consider a tripar te state ρABC shared between Alice (AC) and
Bob (B), with purifica on ψABCR where R is a reference system.
▶ Furthermore, Alice and Bob share a maximally entangled state
ΦkTA TB of Schmidt rank k.
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State redistribu on
Target state
Ini al state
R
R
ψABCR
ψA′ B′ C′ R
C
A
B
C′
TA
Φk
TB
T′A
A′
Φm
B′
T′B
▶ State redistribu on: Transfer A-part of ρ from Alice to Bob.
▶ Use shared entanglement, local encoding E (Alice) and decoding
D (Bob), and quantum communica on Q from Alice to Bob.
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State redistribu on
R
C’
C
A
TA
B
Ω
E
T’A
Q
T’B
D
TB
ω
A’
B’
σ
▶ Use shared entanglement, local encoding E (Alice) and decoding
D (Bob), and quantum communica on Q from Alice to Bob.
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State redistribu on
R
C’
C
A
TA
B
Ω
E
T’A
Q
T’B
D
TB
ω
A’
B’
σ
▶ State redistribu on generalizes many quantum
informa on-processing tasks like Schumacher compression,
coherent state merging, and quantum state spli ng.
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State redistribu on
▶ Consider now n i.i.d. copies of ini al state ρ, and a shared
maximally entangled state Φkn of Schmidt rank kn .
▶ Target state: n i.i.d. copies of ρ, and some maximally entangled
state Φmn of Schmidt rank mn .
▶ State redistribu on protocol outputs state σ n .
▶ Figure of merit: Fidelity
(
)
Fn := F σ n , Φmn ⊗ ψ⊗n ,
where F(ω, τ) := ∥ω1/2 τ 1/2 ∥1 .
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State redistribu on
▶ Opera onal quan
es:
▷ Quantum communica on cost qn :=
1
n
log |Qn |
▷ Entanglement cost
en := n1 (log kn − log mn ) = n1 (log |TnA | − log |T′n
A |)
▶ Note that en can be nega ve, in which case entanglement is
gained in the protocol rather than consumed.
Defini on (Achievable rates)
A rate pair (e, q) is achievable if there exists a sequence of state
redistribu on protocols such that
lim sup en = e,
n→∞
lim sup qn = q,
n→∞
lim Fn = 1.
n→∞
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State redistribu on: Op mal rates
Theorem (Op mal rates)
Luo and Devetak (2009) and Yard and Devetak (2009)
The rate pair (e, q) is achievable if and only if
q + e ≥ S(A|B) ρ
and
q≥
1
I(A; C|B) ρ ,
2
where S(A|B) ρ = S(AB) ρ − S(B) ρ is the condi onal entropy and
I(A; C|B) ρ = S(A|B) ρ − S(A|BC) ρ is the condi onal mutual
informa on.
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State redistribu on: Op mal rates
▶ Preceding theorem proves a weak converse:
lim sup(qn + en ) < S(A|B) ρ
∨
lim sup qn <
1
I(A; C|B) ρ
2
=⇒ lim Fn < 1
▶ Our goal is to prove strong converse, that is,
Fn ≤ exp(−Kn)
for some suitable constant K > 0, yielding lim Fn = 0.
▶ Note: A strong converse theorem has also been obtained by
Berta et al. (2016) using smooth entropies.
▶ We use the so-called Rényi entropy method (cf. Arimoto (1973)
and Ogawa and Nagaoka (1999)).
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Table of Contents
1 Introduc on: Weak vs. strong converse
2 State redistribu on
3 Rényi entropies
4 Main result: Strong converse for state redistribu on
5 Summary and open ques ons
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Rényi entropies
Defini on (Sandwiched Rényi divergence)
Müller-Lennert et al. (2013) and Wilde et al. (2014)
Let ρ be a quantum state and σ be posi ve semidefinite with
supp ρ ⊆ supp σ. For α ∈ (0, 1) ∪ (1, ∞), we set γ = (1 − α)/2α
and define the α-sandwiched Rényi divergence as
e α (ρ∥σ) :=
D
1
log Tr (σ γ ρσ γ )α .
α−1
Quantum generaliza on of the classical Rényi divergence
∑
1
P(x)α Q(x)1−α .
log
x
α−1
⊕
⊕
e α (ρX ∥σ X ) = Dα (P∥Q) for ρX = x P(x), σ X = x Q(x).
That is, D
Dα (P∥Q) =
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Rényi entropies
Proper es of the sandwiched Rényi divergence:
▶ Recovers the quantum rela ve entropy:
e α (ρ∥σ) = D(ρ∥σ),
lim D
α→1
where D(ρ∥σ) = Tr(ρ(log ρ − log σ)).
▶ Addi vity:
e α (ρ1 ⊗ ρ2 ∥σ 1 ⊗ σ 2 ) = D
e α (ρ1 ∥σ 1 ) + D
e α (ρ2 ∥σ 2 )
D
▶ Data processing inequality: Frank and Lieb (2013)
For every quantum opera on Λ and α ≥ 1/2,
e α (ρ∥σ) ≥ D
e α (Λ(ρ)∥Λ(σ)).
D
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Rényi entropies
Defini on (Derived entropies)
Let ρAB be a bipar te quantum state with marginal ρA . Then we
define the following quan
es:
e α (ρA ∥1A );
▶ Rényi entropy: Sα (A) ρ := −D
▶ Rényi condi onal entropy:
e
e α (ρAB ∥1A ⊗ σ B );
Sα (A|B) ρ := − minσB D
▶ Reyni mutual informa on:
eIα (A; B) ρ := minσB D
e α (ρAB ∥ρA ⊗ σ B ).
These recover the von Neumann quan
es in the limit α → 1, and
sa sfy addi vity and data processing inequality as well.
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Table of Contents
1 Introduc on: Weak vs. strong converse
2 State redistribu on
3 Rényi entropies
4 Main result: Strong converse for state redistribu on
5 Summary and open ques ons
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Main result: Strong converse theorem
Theorem
Let α ∈ (1/2, 1) and ϐ be such that 1/α + 1/ϐ = 2. Then for all
(
)
n ∈ N we have the following bounds on Fn = F σ n , Φmn ⊗ ψ⊗n :
Fn ≤ exp {−nκ(α) [Sϐ (AB) ρ − Sα (B) ρ − (qn + en )]}
{
[
]}
Fn ≤ exp −nκ(α) e
Sϐ (R|B) ρ − e
Sα (R|AB) ρ − 2qn
{
[
]}
Fn ≤ exp −nκ(α) eIα (R; AB) ρ − eIϐ (R; B) ρ − 2qn
where κ(α) = (1 − α)/2α, and qn and en are the quantum
communica on cost and entanglement cost, respec vely.
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Main result: Strong converse theorem
Fn ≤ exp {−nκ(α) [Sϐ (AB) ρ − Sα (B) ρ − (qn + en )]}
{
[
]}
Fn ≤ exp −nκ(α) e
Sϐ (R|B) ρ − e
Sα (R|AB) ρ − 2qn
{
[
]}
Fn ≤ exp −nκ(α) eIα (R; AB) ρ − eIϐ (R; B) ρ − 2qn
▶ Recall that state redistribu on is possible iff q + e ≥ S(A|B) ρ .
α→1
▶ We have Sϐ (AB) ρ − Sα (B) ρ −−−→ S(A|B) ρ .
▶ Hence, if qn + en < S(A|B) ρ , there is α0 < 1 such that
K(α0 ) := κ(α0 )[Sϐ(α0 ) (AB) ρ − Sα0 (B) ρ − (qn + en )] > 0,
and Fn ≤ exp(−nK(α0 )).
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Main result: Strong converse theorem
Fn ≤ exp {−nκ(α) [Sϐ (AB) ρ − Sα (B) ρ − (qn + en )]}
{
[
]}
Fn ≤ exp −nκ(α) e
Sϐ (R|B) ρ − e
Sα (R|AB) ρ − 2qn
]}
{
[
Fn ≤ exp −nκ(α) eIα (R; AB) ρ − eIϐ (R; B) ρ − 2qn
▶ Recall that state redistribu on is possible iff 2q ≥ I(A; C|B) ρ .
α→1
▶ e
Sϐ (R|B) ρ − e
Sα (R|AB) ρ −−−→ I(A; C|B) ρ and
α→1
eIα (R; AB) ρ − eIϐ (R; B) ρ −
−−→ I(A; C|B) ρ .
▶ Hence, if 2q < I(A; C|B) ρ , there is α0 < 1 and K(α0 ) > 0 such
that Fn ≤ exp(−nK(α0 )).
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Key ingredient in the proof of the main result
Theorem (Fidelity bound)
Let ρAB and σ AB be bipar te quantum states, and for α ∈ (1/2, 1) let
ϐ be such that 1/α + 1/ϐ = 2, then
(i) Sα (A) ρ − Sϐ (A)σ ≥ κ(α)−1 log F(ρA , σ A )
(ii) e
Sα (A|B) ρ − e
Sϐ (A|B)σ ≥ κ(α)−1 log F(ρAB , σ AB ).
If ρA = σ A , then also:
(iii) eIϐ (A; B) ρ − eIα (A; B)σ ≥ κ(α)−1 log F(ρAB , σ AB ).
Here, κ(α) = (1 − α)/2α as before.
▶ Generalizes van Dam and Hayden (2002), who proved (i).
▶ Our proof method: Hölder inequality for Scha en p-norms.
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Table of Contents
1 Introduc on: Weak vs. strong converse
2 State redistribu on
3 Rényi entropies
4 Main result: Strong converse for state redistribu on
5 Summary and open ques ons
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Summary
▶ We established a full strong converse theorem for state
redistribu on.
▶ Employed “Rényi entropy method” based on entropic quan
es
derived from sandwiched Rényi divergence, together with
fidelity bounds.
▶ Same method yields strong converse for:
▷ measurement compression with quantum side informa on
▷ randomness extrac on
▷ data compression with quantum side informa on
(see full paper at arXiv:1506.02635)
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Open ques ons
▶ Are these bounds op mal?
−→ Rényi quan
es as strong converse exponents?
▶ Can we use the fidelity bounds for proving strong converse
theorems for other tasks?
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Thank you very much!
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