Semidefinite programming strong converse bounds for quantum channel capacities

Transcription

1 Semidefinite programming strong converse bounds for quantum channel capacities Xin Wang UTS: Centre for Quantum Software and Information Joint work with Runyao Duan and Wei Xie arxiv: & Beyond I.I.D. 2017, IMS, Singapore

2 Communication over channels (Shannon, 1948) The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point. Optimal success probability p s (N, m) = sup E,D 1 m m p(k = ˆk) k=1 Classical capacity is the maximum rate for asymptotically error-free classical data transmission using N many times, i.e., C(N ) = sup{r lim n p s (N n, 2 rn ) = 1}.

3 Quantum channel Quantum Channel: completely positive (CP) trace-preserving (TP) linear map N. Choi-Kraus rep: N ( ) = k E k E k Choi-Jamiołkowski rep. of N : J N = i j A N ( i j A ) ij = (id A N ) Φ A A Φ A A, with Φ A A = k k A k A. Note that J N 0, Tr B J N = 1. A N B

4 Classical communication via quantum channels Classical capacity (Holevo 73, 98; Schumacher & Westmoreland 97): 1 C(N ) = sup k k χ(n k ), with χ(n ) = max {(pi,ρ i )} H ( i p i N (ρ i )) i p i H(N (ρ i )). Difficulties of evaluating C(N ) χ(n ): NP-hard (Beigi & Shor 07) Worse: χ(n ) is not additive (Hastings 09) Classical capacity of amplitude damping channel is unknown.

5 Main question and outline Non-asymptotic communication capability Given n uses of the channel, we hope to optimize the trade-off between ɛ and R. p succ (N, R) - the maximum success probability of transmitting classical information at rate R C (1) (N, ɛ) - the maximum rate for transmission with error tolerance ɛ (or the one-shot ɛ-error capacity) Asymptotic communication capability Efficiently computable strong converse bounds for classical and quantum capacities Estimation of the capacities for basic channels

6 Non-asymptotic communication capability

7 Brief Idea A B k {1,..., m} E N D ˆk {1,..., m} M = D N E Central question: how to solve or estimate p s (N, m)? p s (N, m) = sup E,D 1 m m k=1 Tr[D N E( k k ) k k ]. Idea: Relax this to a convex optimization problem (SDP) Jump out the entropy zoo.

8 General codes No-signalling code (Leung, Matthews 16; Duan, Winter 16): a bipartite channel Π L(A i ) L(B i ) L(A o ) L(B o ) with NS constraints, i.e., A and B cannot use Π to communicate classical information. Entanglement assisted codes (EA) NS. Also see causal operations (e.g., Beckman, Gottesman, Nielsen, Preskill 01; Eggeling, Schlingemann, Werner 02). A hierarchy of codes by adding constraints on Π, e.g., Positive-partial-transpose preserving (PPT) constraint (Rains 01; Leung & Matthews 16). Unassisted codes (UA) NS PPT. A i E Π N D B o M A o B i M(A i B o ): k ˆk

9 Optimal success probability A i B o E Π D M A o N B i M(A i B o ): k ˆk Optimal success probability of Ω codes (Ω = NS or NS PPT in this talk) p s,ω (N, m) = sup Π Ω 1 m m Tr[M( k k ) k k ], M given by N, Π. k=1

10 Result 1: Optimal success probability for NS/PPT codes Theorem For any N, the optimal success probability to transmit m messages assisted by NS PPT codes is given by the following SDP: p s,ns PPT (N, m) = max Tr J N F s.t. 0 F ρ A 1 B, Tr ρ A = 1, Tr A F = 1/m, 0 F T B ρ A 1 B (PPT), where J N is the Choi-Jamiołkowski matrix of N. When assisted by NS codes, p s,ns (N, m) = max Tr J N F s.t. 0 F ρ A 1 B, Tr ρ A = 1, Tr A F = 1/m. T B : the partial transpose with respect to B, i.e., ( ij kl ) T B = il kj. SDP is an analytical tool in proof (see Watrous Book) and it can be solved by efficient algorithms (e.g., implementation via CVX).

11 Sketch of proof Target: p s,ω (N, m) = sup Π Ω 1 m m Tr[M( k k ) k k ], (1) k=1 Recall J M = ij i j A i M( i j Ai ) and let V = m k=1 kk kk Key: m 1 Tr[M( k k ) k k ] = 1 m k=1 m Tr[J MV ]. (2) Moreover, J M can be represented by J N and J Π (Leung & Matthews 16; based on Chiribella, D Ariano, Perinotti 08) J M = Tr AoB i (J T N 1 Ai B o )J Π. (3) Combining Eqs. (1), (2), (3), we have p s,ω (N, m) = max Π Ω Tr[(JT N 1 Ai B o )J Π (1 V )]/m,

12 Sketch of proof cont. We already obtain the linear objective function, then just impose the NS and PPT constraints of Π to obtain the SDP: p s,ns PPT (N, m) = max Tr[(J T N 1)J Π (1 V )]/m, s.t. Π is CP, TP, NS, PPT. Exploit the permutation invariance of V = m k=1 kk kk to simplify the SDP: p s,ns PPT (N, m) = max Tr J N F s.t. 0 F ρ A 1 B, Tr ρ A = 1, Tr A F = 1 B /m, 0 F T B ρ A 1 B (PPT),

13 Example: assess the preformance of AD channel For amplitude damping channel Nγ AD (ρ) = 1 i=0 E iρe i with E 0 = γ 1 1 and E 1 = γ 0 1, if we use the channel 3 times, the optimal success probabilty to transmit 1 bit is given as follows:

14 Result 2: One-shot capacities One-shot ɛ-error capacity assisted with Ω-codes: C (1) Ω (N, ɛ) = sup{log λ 1 p s,ω(n, λ) ɛ}. Theorem For given channel N and error threshold ɛ, C (1) NS PPT (N, ɛ) = log min η s.t. 0 F ρ A 1 B, Tr ρ A = 1, Tr A F = η1 B, Tr J N F 1 ɛ, 0 F T B ρ A 1 B (PPT), To obtain C (1) NS (N, ɛ), one only needs to remove the PPT constraint. Study zero-error capacity by setting ɛ = 0, e.g., C (1) NS (N, 0) recovers the one-shot NS assisted zero-error capacity in (Duan & Winter 16).

15 Previous converse bounds Converse for classical channel (Polyanskiy, Poor, Verdú 2010) and classical-quantum channel (Wang & Renner 2010, also Hayashi s book). (Datta & Hsieh 13) gives converse for C (1) E (N, ɛ) (hard to compute). (Matthews & Wehner 2014) shows SDP converse bounds C (1) E (N, ɛ) max ρ A min σ B D ɛ H((id A N )(ρ A A) ρ A σ B ), C (1) (N, ɛ) max min D ɛ H,PPT ((id A N )(ρ A ρ A σ A) ρ A σ B ), B where D ɛ H and D ɛ H,PPT are hypothesis testing relative entropies. See (Anshu, Jain, Warsi 17) for a new proof. One-shot ɛ-error capacities can provide better efficiently computable converse bounds: C (1) E (N, ɛ) C (1) NS (N, ɛ) max min DH((id ɛ A N )(ρ A ρ A σ A) ρ A σ B ), B C (1) (N, ɛ) C (1) NS PPT (N, ɛ) max min D ɛ H,PPT ((id A N )(ρ A ρ A σ A) ρ A σ B ). B The blue inequalities can be strict for amplitude damping channels.

16 Comparsion - Examples For amplitude damping channel Nγ AD (ρ) and error tolerance ε = 0.01, solid line depicts our bound C (1) AD NS (Nγ, 0.01); dashed line depicts the hypothesis testing bound, i.e., max ρa min σb DH ɛ ((id A N γ AD )(ρ A A) ρ A σ B ).

17 Asymptotic communication capability

18 Weak vs Strong Converse The converse part of the HSW theorem due to Holevo (1973) only establishes a weak converse. A strong converse bound: p succ 0 as n increases if the rate exceeds this bound. If the capacity of a channel is also its strong converse bound, then the strong converse property holds.

19 Result 3: Strong converse bound for classical capacity Known strong converse bound: the entanglement-assisted capacity (Bennett, Shor, Smolin, Thapliyal 1999, 2002) Theorem (SDP strong converse bound for C) For any quantum channel N, C(N ) C β (N ) = log min Tr S B s.t. R AB J T B N R AB, 1 A S B R T B AB 1 A S B. And p succ 0 when the rate exceeds C β (N ). Properties: C β (N ) > 0 if and only if C(N ) > 0. For qudit noiseless channel I d, C(I d ) = C β (I d ) = log d. C β (N 1 N 2 ) = C β (N 1 ) + C β (N 2 ) for any N 1 and N 2.

20 Sketch of proof Subadditive bounds on p s (Tool: duality of SDP) p s,ns PPT (N n, 2 rn ) p + s (N n, 2 rn ) p + s (N, 2 r ) n, (4) where p + s (N, m) = min Tr Z B s.t. R AB J T B N R AB, m1 A Z B R T B AB m1 A Z B. For any r > C β (N ), one can prove that p + s (N, 2 r ) < 1. Thus, p s,ns PPT (N n, 2 rn ) p + s (N, 2 r ) n 0, (when n increases)

21 Application: Amplitude damping channel For amplitude damping channel, N AD γ (ρ) = 1 i=0 E iρe i with E 0 = γ 1 1 and E 1 = γ 0 1, C(N AD γ ) C β (Nγ AD ) = log(1 + 1 γ). Solid line depicts our bound; Dashed line depicts the previously best upper bound (Brandão, Eisert, Horodecki, Yang 2011). Dotted line depicts the lower bound (Giovannetti and Fazio 2005). Problem: how to further improve the lower bound or upper bound?

22 Quantum capacity The quantum capacity Q(N ), is the rate at which qubits can be transmitted with fidelity approaching one asymptotically. Quantum capacity is established by (Lloyd, Shor, Devetak 97-05) & (Barnum, Nielsen, Schumacher 96-00) 1 Q(N ) = lim m m I c(n m ). Coherent information I c (N ) = max ρ [H(N (ρ)) H(N c (ρ))] Q(N ) is also difficult to evaluate. Known strong converse bounds: Partial Transposition bound (Holevo, Werner 2001) Rains information (Tomamichel, Wilde, Winter 2015) Channel s entanglement cost (Berta, Brandao, Christandl, Wehner 2013)

23 SDP strong converse bound for quantum capacity Theorem (SDP strong converse bound for Q) For any quantum channel N, Q(N ) Q Γ (N ) = log max Tr J N R AB s.t. R AB, ρ A 0, Tr ρ A = 1, ρ A 1 B R T B AB ρ A 1 B. The fidelity of transmission goes to zero if the rate exceeds Q Γ (N ). This is based on the optimal fidelity of transmitting quantum information assisted with PPT codes (Leung and Matthews 16). The proof idea is similar to previous bound for classical capacity. For noiseless quantum channel I d, Q(I d ) = Q Γ (I d ) = log 2 d. Q Γ (N M) = Q Γ (M) + Q Γ (N ) (by utilizing SDP duality).

24 Comparison with other bounds Partial Transposition bound (Holevo & Werner 01) Q(N ) Q Θ (N ) = log 2 J T B N cb, where cb uses an alternative expression from (Watrous 12). Improved efficiently computable bound For any quantum channel N, Q Γ (N ) Q Θ (N ). Example: N r = i E i E i with E 0 = r 1 1 and E 1 = 1 r Solid line: SDP bound Q Γ Dashed line: PT bound Q Θ QΓ(Nr) QΘ(Nr) r from 0 to 0.5

25 Comparison with other bounds cont. The SDP strong converse bound can be refined as Q Γ (N ) = max ρ S(A) min D max(n A B(φ AA ) σ), σ PPT where φ AA is a purification of ρ A and the set PPT = {σ P(A B) σ T B 1 1}. Rains information [Tomamichel, Wilde, Winter 16] R(N ) = max ρ A S(A) min D(N A B(φ AA ) σ), (5) σ PPT But it is not efficiently computable in general. For any quantum channel N, it holds Q(N ) R(N ) Q Γ (N ) Q Θ (N ).

26 Summary and Outlook Non-asymptotic classical communication (NS/NS PPT codes) Optimal success probability of communication is given by SDP One-shot ɛ-error capacity is given by SDP Limits for asymptotic classical/quantum communication SDP strong converse bound for C(N ) Application: Improved upper bound for C(N AD ) SDP strong converse bound for Q(N ) (improve Partial Transposition bound, but no better than Rains information). Outlook Tighter strong converse bounds without using additive SDP? Classical capacity of basic channels such as AD channel? Second-order of classical capacity of fully quantum channels?

27 arxiv: & Wei Xie Runyao Duan Thank you for your attention!