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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 2 / 23 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. 4 / 23 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 5 / 23 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. 6 / 23 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. 6 / 23 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. 7 / 23 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. 7 / 23 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 . 8 / 23 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→∞ 9 / 23 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. 10 / 23 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)). 11 / 23 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 12 / 23 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) = 13 / 23 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 14 / 23 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. 15 / 23 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 16 / 23 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. 17 / 23 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 )). 18 / 23 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 )). 18 / 23 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. 19 / 23 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 20 / 23 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) 21 / 23 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? 22 / 23 Thank you very much! Arimoto, S. (1973). 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