Strong converse theorems using Rényi entropies

Transcription

1 Strong converse theorems using Rényi entropies Felix Leditzky joint work with Mark M. Wilde and Nilanjana Datta arxiv: January 2016

2 Table of Contents 1 Weak vs. strong converse 2 Rényi entropies 3 State redistribution: definition and strong converse 4 Further results and outlook Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

3 Table of Contents 1 Weak vs. strong converse 2 Rényi entropies 3 State redistribution: definition and strong converse 4 Further results and outlook Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

4 Optimal rates of information-theoretic tasks Information-theoretic tasks: source coding, channel coding,... Operational definition of a rate of the code: compression rate, capacity,... Coding theorem establishes entropic quantity as optimal rate: (Weak) Converse every code has lim n ε n > 0 Achievability there exists a code with lim n ε n = 0 optimal rate rate Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

5 Strong converse property Weak converse: trade-off between error ε n and rate r below optimal rate r? Strong converse theorem: No! Every code at rate below the optimal rate fails with certainty! Optimal rate satisfies strong converse property. lim n ε n 1 lim n ε n 1? sharp threshold 0 r r 0 r r Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

6 Strong converse property Example: Quantum data compression Optimal compression rate [Schumacher 1995]: r = S(ρ) = Tr(ρ log ρ) Strong converse for data compression [Winter 1999]: For every code with rate r < S(ρ), we have ε n 1 exp( Kn) for some K > 0, and hence lim n ε n = 1 (since ε n [0, 1]). Exponential convergence of error strong converse in the Wolfowitz sense [Wolfowitz 1961]. How can we prove strong converse theorems? Winter: method of types Here: Rényi entropy approach [Arimoto 1973; Ogawa and Nagaoka 1999] Derive lower bound on error in terms of a Rényi entropic quantity Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

7 Table of Contents 1 Weak vs. strong converse 2 Rényi entropies 3 State redistribution: definition and strong converse 4 Further results and outlook Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

8 Rényi entropies Definition (Sandwiched Rényi divergence of order α) Let α (0, ) \ {1} and ρ, σ be quantum states with supp ρ supp σ: D α (ρ σ) := 1 α 1 log Tr ( σ (1 α)/2α ρσ (1 α)/2α) α. Additivity: D α (ρ 1 ρ 2 σ 1 σ 2 ) = D α (ρ 1 σ 1 ) + D α (ρ 2 σ 2 ) Data processing inequality: [Beigi 2013; Frank and Lieb 2013] For a quantum channel Λ and α 1/2, Limit property: D α (ρ σ) D α (Λ(ρ) Λ(σ)). D α (ρ σ) α 1 D(ρ σ) = Tr[ρ(log ρ log σ)] Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

9 Rényi entropic quantities The sandwiched Rényi divergence (α-srd) serves as a parent for the following quantities: Definition (Entropic quantities derived from α-srd) Let α > 0 and ρ AB be a bipartite quantum state with marginal ρ A. Rényi entropy S α (A) ρ := D α (ρ A 1 A ) Rényi conditional entropy (RCE) Rényi mutual information (RMI) S α (A B) ρ := min σ B Dα (ρ AB 1 A σ B ) Ĩ α (A; B) ρ := min σ B Dα (ρ AB ρ A σ B ) Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

10 Rényi entropic quantities These quantities inherit some of the properties of the α-srd: Additivity [Hayashi and Tomamichel 2014] Let ρ A1 B 1 and σ A2 B 2 be quantum states, then S α (A 1 A 2 B 1 B 2 ) ρ σ = S α (A 1 B 1 ) ρ + S α (A 2 B 2 ) σ Ĩ α (A 1 A 2 ; B 1 B 2 ) ρ σ = Ĩα(A 1 ; B 1 ) ρ + Ĩα(A 2 ; B 2 ) σ. Data processing inequality Let Λ: B C be a quantum channel, and ω AC = (id A Λ)(ρ AB ), then for α 1/2 we have S α (A B) ρ S α (A C) ω Ĩ α (A; B) ρ Ĩ α (A; C) ω. Limit property S α (A) ρ S α (A B) ρ Ĩ α (A; B) ρ α 1 S(A) ρ α 1 S(A B) ρ = S(AB) ρ S(B) ρ α 1 I (A; B) ρ = S(A) ρ S(A B) ρ Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

11 Extending the Rényi entropy calculus We prove the following new properties for these quantities: Theorem (Dimension bounds) For α 1/2 and a tripartite state ρ ABC with C quantum, S α (A BC) ρ + 2 log C S α (A B) ρ Ĩ α (A; B) ρ + 2 log C Ĩ α (A; BC) ρ whereas for ρ ABX with X classical, S α (A BX ) ρ + log X S α (A B) ρ Ĩ α (A; B) ρ + log X Ĩ α (A; BX ) ρ. Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

12 Extending the Rényi entropy calculus We prove the following new properties for these quantities: Theorem (Fidelity bounds) For α (1/2, 1), β = α/(2α 1), and bipartite states ρ AB and σ AB, S α (A) ρ S β (A) σ 2α 1 α log F (ρ A, σ A ) ( ) S α (A B) ρ S β (A B) σ 2α 1 α log F (ρ AB, σ AB ) Ĩ β (A; B) ρ Ĩ α (A; B) σ 2α 1 α log F (ρ AB, σ AB ) where F (ω, τ) := ω τ 1 is the fidelity. Eq. ( ) first appeared in [van Dam and Hayden 2002], and we generalize this result to the Rényi conditional entropy and mutual information. Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

13 Table of Contents 1 Weak vs. strong converse 2 Rényi entropies 3 State redistribution: definition and strong converse 4 Further results and outlook Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

14 State redistribution: protocol R R ψ ABCR ψ A B C R C A B T A Φ k T B C T A Q QA B Φ m T B T B target state: ψ A B C R Φ m T A T (A B C B = ABC) figure of merit: F ( ψ Φ m, (D E)(ψ Φ k ) ) # of qubits sent from Alice to Bob: log Q # of ebits consumed: log T A log T A = log k log m (if log k < log m, then ebits are gained) Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

15 State redistribution: protocol n copies of ρ ABC : Initial state: ψ n ABCR Φkn T A T B Overall map: Encoding E n, quantum communication Q, and decoding D n. Target state: ψ n A B C R Φmn T A T B Figure of merit: F n := F ( ψ n Φ mn, (D n E n )(ψ n Φ kn ) ) Quantum communication cost: q n := 1 n log Qn Entanglement cost: e n := 1 n (log k n log m n ) Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

16 State redistribution: optimal rates Definition (Achievable rates) (e, q) is achievable: For ρ ρ ABC there is a protocol {(ρ n, E n, D n )} n N with lim n F n = 1 and lim sup e n = e, n lim sup q n = q. n Theorem (Luo and Devetak 2009; Yard and Devetak 2009) The pair (e, q) is achievable if and only if q 1 2 I (A; R B) ρ q + e S(A B) ρ. Conditional mutual information (CMI): I (A; R B) ρ = S(A B) ρ S(A RB) ρ Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

17 Main result: strong converse region q + e S(A B) ρ q strong conv. reg. achievable region q 1 2 I (A; R B) ρ e strong conv. reg. Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

18 State redistribution: Strong converse theorem Main result: Theorem For every state redistribution protocol with initial state ρ ABC, we have the following bounds on F n for all n N and α (1/2, 1), setting β = α/(2α 1) and κ(α) = (1 α)/(2α) > 0: F n exp { nκ(α) [S β (AB) ρ S α (B) ρ (q n + e n )]} { F n exp nκ(α) [ Sβ (R B) ρ S ]} α (R AB) ρ 2q n As an alternative to the second bound, we also have { ]} F n exp nκ(α) [Ĩα (R; AB) ρ Ĩβ(R; B) ρ 2q n Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

19 State redistribution: Strong converse theorem Bounds on fidelity (β β(α) = α/(2α 1)) F n exp { nκ(α) [S β (AB) ρ S α (B) ρ (q n + e n )]} { F n exp nκ(α) [ Sβ (R B) ρ S ]} α (R AB) ρ 2q n Optimal rates: q + e S(A B) ρ, q 1 2 I (A; R B) ρ Rényi generalization of conditional entropy: α 1 S β(α) (AB) ρ S α (B) ρ S(AB) ρ S(B) ρ = S(A B) ρ Converse region: q n + e n < S(A B) ρ There is α 0 (1/2, 1) such that C := κ(α 0 )[S β(α0 )(AB) ρ S α0 (B) ρ (q n + e n )] > 0. Strong converse: F n exp{ nc} n 0 Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

20 State redistribution: Strong converse theorem Bounds on fidelity (β β(α) = α/(2α 1)) F n exp { nκ(α) [S β (AB) ρ S α (B) ρ (q n + e n )]} { F n exp nκ(α) [ Sβ (R B) ρ S ]} α (R AB) ρ 2q n Optimal rates: q + e S(A B) ρ, q 1 2 I (A; R B) ρ Rényi generalization of conditional mutual information: S β(α) (R B) ρ S α 1 α (R AB) ρ S(R B) ρ S(R AB) ρ = I (A; R B) ρ Converse region: 2q n < I (A; R B) ρ As before, this yields a strong converse: F n exp{ nc} n 0 for some C > 0. Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

21 Proof sketch Strong converse property for q n + e n Strategy: Prove for n = 1 and then use additivity of Rényi quantities. Define post-encoding state ) ω C QBRT A T BE := U E ( ψ ABCR Φ k T A T B where U E is a Stinespring isometry of Alice s encoding map E : ACT A QC T A with environment E. Subadditivity property for Rényi entropies: [van Dam and Hayden 2002] For α > 0 and a bipartite state ρ AB, S α (A) ρ log B S α (AB) ρ S α (A) ρ + log B For the marginal ω QBTB, this yields S α (QBT B ) ω log Q + log T A + S α (B) ρ. Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

22 Proof idea Define final state σ A B C RT A T B ED := U D ω C QBRT A T BE where U D is a Stinespring isometry of Bob s decoding map D : QBT B A B T B with environment D. Isometric invariance: S α (QBT B ) ω = S α (A B T B D) σ. Uhlmann: F = F (ψ Φ m, (D E)(ψ Φ k )) F (σ A B T B D, π m T A ρ A B χ D) Fidelity bound applied to σ A B T B D and π m T A ρ A B χ D yields S α (A B T B D) σ log T A + S β(a B ) ρ + 2α 1 α log F. Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

23 Proof idea Both bounds together: 2α 1 α log F log Q + log T A log T A S β(ab) ρ + S α (B) ρ log Q quantum communication cost (for n = 1) log T A log T A entanglement cost (for n = 1) n copies of ρ ABC : 2α 1 α log F n log Q + log T A log T A S β (A n B n ) ρ n + S α (B n ) ρ n = n (q n + e n S β (AB) ρ + S α (B) ρ ) Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

24 Table of Contents 1 Weak vs. strong converse 2 Rényi entropies 3 State redistribution: definition and strong converse 4 Further results and outlook Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

25 More strong converse theorems Using the Rényi entropy method, we derive strong converse theorems for the following protocols (details in [arxiv: ]): 1 State redistribution with feedback (allowing quantum communication from Bob to Alice) 2 Coherent state merging and quantum state splitting (special cases of state redistribution) 3 Measurement compression with quantum side information (QSI) 4 Randomness extraction against QSI 5 Data compression with QSI Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

26 Open questions Applications of the fidelity bound Crucial mathematical result in the proofs: S α (A B) ρ S β (A B) σ where α (1/2, 1) and β = α/(2α 1). 2α 1 α log F (ρ AB, σ AB ) Can we use this to derive strong converse theorems for other protocols? Example: Strong converse for degradable channels In entanglement generation, fidelity bound yields a bound on figure of merit in terms of entanglement generation capacity and a Rényi coherent information. Additivity result or other estimates for this quantity needed to infer strong converse theorem. Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27

27 References Arimoto, S. (1973). IEEE Transactions on Information Theory 19.3, pp Beigi, S. (2013). Journal of Mathematical Physics 54.12, p arxiv: [quant-ph]. Berta, M. et al. (2014). arxiv preprint arxiv: Frank, R. L. and E. H. Lieb (2013). Journal of Mathematical Physics 54.12, p arxiv: [math-ph]. Hayashi, M. and M. Tomamichel (2014). arxiv preprint. arxiv: [quant-ph]. Leditzky, F. et al. (2015). arxiv preprint arxiv: Luo, Z. and I. Devetak (2009). IEEE Transactions on Information Theory 55.3, pp Ogawa, T. and H. Nagaoka (1999). IEEE Transactions on Information Theory 45, pp Schumacher, B. (1995). Physical Review A 51.4, p Winter, A. (1999). IEEE Transactions on Information Theory 45.7, pp Wolfowitz, J. (1961). 31. Springer Berlin-Göttingen-Heidelberg. Yard, J. T. and I. Devetak (2009). IEEE Transactions on Information Theory 55.11, pp van Dam, W. and P. Hayden (2002). arxiv preprint quant-ph/ Thank you very much! Felix Leditzky (Univ. of Cambridge) Strong converse theorems 5 January / 27