A meta-converse for private communication over quantum channels

Transcription

1 A meta-coverse for private commuicatio over quatum chaels Mario Berta with Mark M. Wilde ad Marco Tomamichel IEEE Trasactios o Iformatio Theory, 63(3), (2017) Beyod IID Sigapore - July 17, 2017 Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

2 Setup I Give a quatu chael N ad a quatum key distributio (QKD) protocol that uses it times, how much key ca be geerated? A 1 N B1 A 2 N B2 A N B KA LOCC A1 B1 LOCC A2 B2 LOCC LOCC A B LOCC K B No-asymptotic private capacity: maximum rate of ε-close secret key achievable usig the chael times with two-way public classical commuicatio assistace (passive adversary) ˆP N (, ε) := sup { P : (, P, ε) is achievable for N usig LOCC }. (1) Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

3 Setup II Practical questio: how to characterize ˆP N (, ε) for all 1 ad ε (0, 1)? The aswers give the fudametal limitatios of QKD. Upper bouds o ˆP N (, ε) ca be used as bechmarks for quatum repeaters [Lütkehaus]. Today, I will preset the tightest kow upper boud o ˆP N (, ε) for may chaels of practical iterest. Iterestig special case: sigle-mode phase-isesitive bosoic Gaussia chaels. Techical level: quatum Shao theory with geeral 1 ad ε 0. Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

4 Overview 1 Mai Results i Examples 2 Proof Idea: Meta Coverse 3 Coclusio Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

5 Mai Results i Examples Mai Result: Gaussia Chaels I Coverse bouds for sigle-mode phase-isesitive bosoic Gaussia chaels, most importatly the photo loss chael L η : ˆb = ηâ + 1 ηê (2) where trasmissivity η [0, 1] ad eviromet i vacuum state. Previous asymptotic result from [Piradola et al. 2017] i the ifiite eergy limit ( ) P 1 (L η ) := lim lim ε 0 ˆP 1 N η (, ε) log, (3) 1 η ( which is actually tight i the asymptotic limit, i.e., P (N η ) = log 1 1 η Drawback: a asymptotic statemet, ad thus says little for practical protocols (called a weak coverse boud). ). Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

6 Mai Results i Examples Mai Result: Gaussia Chaels II We show the o-asymptotic coverse boud ( where C(ε) := log log ( ) 1 ˆP 1 L η (, ε) log + C(ε) 1 η, (4) 1+ε 1 ε ) (other choices possible). Ca be used to assess the performace of ay practical quatum repeater which uses a loss chael times for desired security ε. Other variatios of this boud are possible if η is ot the same for each chael use, if η is chose adversarially, etc. We give similar bouds for the quatum-limited amplifier chael (tight), thermalizig chaels, amplifier chaels, ad additive oise chaels. Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

7 Mai Results i Examples Mai Result: Dephasig Chaels I Previous asymptotic result for the qubit dephasig chael Z γ : ρ (1 γ) ρ + γzρz with γ (0, 1) is [Beett et al. 1996, Piradola et al. 2017] P 1 (Z γ ) := lim lim ε 0 ˆP Z γ (, ε) = 1 h(γ), (5) with the biary etropy h(γ) := γ log γ (1 γ) log(1 γ). By combiig with [Tomamichel et al. 2016] we show the expasio 1 ˆP v(γ) Z γ (, ε) = 1 h(γ) + Φ 1 (ε) + log ( ) O, (6) with Φ the cumulative stadard Gaussia distributio ad the biary etropy variace v(γ) := γ(log γ + h(γ)) 2 + (1 γ)(log(1 γ) + h(γ)) 2. Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

8 Mai Results i Examples Mai Result: Dephasig Chaels II For the dephasig parameter γ = 0.1 we get (figure from [Tomamichel et al. 2016]): Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

9 Mai Results i Examples Mai Result: Erasure Chaels For the qubit erasure chael E p : ρ (1 p)ρ + p e e with p (0, 1) we show by combiig with [Tomamichel et al. 2016] the expasio ( ) 1 ˆP p(1 p) E p (, ε) = 1 p + Φ 1 1 (ε) + O. (7) For the erasure parameter p = 0.25 we get for ε = 1% (figure from [Tomamichel et al. 2016]): Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

10 Proof Idea: Meta Coverse I Proof Idea: Meta Coverse Meta coverse approach from classical chael codig [Polyaskiy et al. 2010], uses coectio to hypothesis testig. I the quatum regime, e.g., for classical commuicatio [Tomamichel & Ta 2015] or quatum commuicatio [Tomamichel et al & 2016]. We exted this approach to private commuicatio. Hypothesis testig relative etropy defied for a state ρ, positive semi-defiite operator σ, ad ε [0, 1] as D ε H(ρ σ) := log if { Tr[Λσ] : 0 Λ I Tr[Λρ] 1 ε }. (8) The ε-relative etropy of etaglemet is defied as E ε R(A; B) ρ := if DH(ρ ε AB σ AB), (9) σ AB S(A:B) where S(A :B) is the set of separable states (cf. relative etropy of etaglemet). Chael s ε-relative etropy of etaglemet is the give as E ε R(N ) := sup ER(A; ε B) ρ, (10) ψ AA H AA where ρ AB := N A B(ψ AA ). Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

11 Proof Idea: Meta Coverse Proof Idea: Meta Coverse II Goal is the creatio of log K bits of key, i.e., states γ ABE with (M A M B )(γ ABE ) = 1 i i A i i B σ E (11) K i for some state σ E ad measuremet chaels M A, M B. I oe-to-oe correspodece with pure states γ AA BB E such that [Horodecki et al & 2009] γ ABA B = U ABA B (Φ AB θ A B )U ABA B, (12) where Φ AB maximally etagled, U ABA B = i,j i i A j j B U ij A B with each U ij A B a uitary, ad θ A B a state. Work i the latter, bipartite picture where the adversary is removed. Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

12 Proof Idea: Meta Coverse Proof Idea: Meta Coverse III For separable states σ AA BB (useless for private commuicatio) ad a state with log K bits of key we have [Horodecki et al. 2009] γ AA BB Tr{γ AA BB σ AA BB } 1 K. (13) The mootoicity of the chael s ε-relative etropy of etaglemet E ε R(N ) with respect to LOCC together with (13) implies the meta coverse ˆP N (1, ε) E ε R(N ) (LOCC pre- ad post-processig assistace). (14) For chael uses this gives ˆP N (, ε) E ε R ( N ). (15) Fiite block-legth versio of relative etropy of etaglemet upper boud [Horodoecki et al & 2009]. The ext step is to evaluate the meta coverse for specific chaels of iterest. Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

13 Proof Idea: Meta Coverse Proof Idea: Meta Coverse IV For teleportatio-simulable chaels N A B with associated state ω AB [Beett et al. 1996, Piradola et al. 2017] the meta coverse holds for geeral LOCC assistace ad expads as 1 ˆP V ε ( ) N E (, ε) E R(A; B) ω + R (A; B) ω Φ 1 log (ε) + O, (16) where V ε E R (A; B) ρ { maxσab Π S V (ρ AB σ AB) for ε < 1/2 mi σab Π S V (ρ AB σ AB) for ε 1/2 } (17) with Π S S(A : B) the set of separable states achievig miimum i the relative etropy of etaglemet E R(A; B) ρ := if D(ρAB σab). (18) σ AB S(A:B) Here, we have the cumulative stadard Gaussia distributio Φ, the relative etropy D(ρ σ) := Tr [ρ (log ρ log σ)], ad the relative etropy variace V (ρ σ) := Tr [ ρ (log ρ log σ D(ρ σ)) 2]. Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

14 Extra: Gaussia Formulas Proof Idea: Meta Coverse For Gaussia chaels we eed formulas for the relative etropy D(ρ σ) ad the relative etropy variace V (ρ σ). From [Che 2005, Piradola et al. 2017] ad [Wilde et al. 2016], respectively: writig zero-mea Gaussia states i expoetial form as { ρ = Zρ 1/2 exp 1 } 2 ˆxT G ρˆx with (19) Z ρ := det(v ρ + iω/2), G ρ := 2iΩ arcoth(2v ρ iω), (20) ad V ρ the Wiger fuctio covariace matrix for ρ, we have D(ρ σ) = 1 2 ( log ( Zσ Z ρ ) ) Tr [ V ρ ] (21) V (ρ σ) = 1 2 Tr{ V ρ V ρ } + 1 Tr{ Ω Ω}, (22) 8 where := G ρ G σ. Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15

15 Coclusio Coclusio Our meta coverse ˆP N (1, ε) E ε R(N ) gives bouds for the private trasmissio capabilities of quatum chaels. These give the fudametal limitatios of QKD ad thus ca be used as bechmarks for quatum repeaters. Improve our boud for the photo loss chael ( ) 1 ˆP L 1 η (, ε) log + C(ε) with C(ε) = log log 1 η ( ) to C (ε) := log? 1 1 ε ( ) 1 + ε 1 ε Correspodig matchig achievability? (Tight aalysis of radom codig i ifiite dimesios eeded.) Tight fiite-eergy bouds for sigle-mode phase-isesitive bosoic Gaussia chaels? Uderstad more chaels, for example such with P > 0 but zero quatum capacity [Horodecki et al. 2008]? (23) Mario Berta A meta-coverse for private commuicatio Beyod IID Sigapore - July 17, / 15