On converse bounds for classical communication over quantum channels - PowerPoint PPT Presentation

On converse bounds for classical communication over quantum channels Xin Wang UTS: Centre for Quantum Software and Information Joint work with Kun Fang, Marco Tomamichel (arXiv:1709.05258) QIP 2018, QuTech, Delft

Background Activated NS codes New meta-converse Υ -information Summary Classical communication over quantum channels ▸ [ Shannon’48] Communication is that of reproducing at one point, either exactly or approximately, a message selected at another point. ▸ Quantum Shannon Theory ▸ Ultimate limits of communication with quantum systems. ▸ Various kinds of capacities (classical, quantum, private, alphabit), different kinds of assistance. | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Communication with general codes A ′ B ′ A B N E D Π ▸ An unassisted code reduces to the product of encoder and decoder, i.e., Π = D B → B ′ E A → A ′ ; ▸ An entanglement-assisted code (EA) corresponds to a bipartite operation of the form Π = D B ̂ B → B ′ E A ̂ A → A ′ Ψ ̂ A ̂ B ▸ A no-signalling-assisted code (NS) corresponds to a bipartite operation which is no-signalling from Alice to Bob and vice-versa [ Leung, Matthews’16; Duan, Winter’16] . ▸ We use Ω to denote the general code. | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary How well can we transmit classical information over N ? ▸ Finite blocklength (non-asymptotic) regime studies the practical senario of optimizing the trade-off between: A 1 N B 1 ▸ r : bits sent per channel use. A 2 N B 2 E n D n ⋮ ▸ n : number of channel uses. A n N B n ▸ ε : error tolerance. ▸ Capacity is the maximum rate for asymptotically error-free data transmission using the channel many times. ▸ Considering that the resource is finite, we also want a finite blocklength analysis. ▸ One-shot analysis yields results in the asymptotic limit. | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Communication capability ▸ Given N and Ω -assisted code Π with size m , the optimal coding success probability is m p succ , Ω (N , m ) ∶= 1 Tr M(∣ k ⟩⟨ k ∣)∣ k ⟩⟨ k ∣ , ∑ m sup k = 1 s.t. M = Π ○ N is the effective channel. ▸ One-shot ε -error capacity: C ( 1 ) Ω (N ,ε ) ∶= sup { log m ∶ p succ , Ω (N , m ) ≥ 1 − ε } . ▸ The Ω -assisted capacity: 1 nC ( 1 ) C Ω (N) = lim Ω (N ⊗ n ,ε ) . ε → 0 lim n →∞ | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary HSW theorem ▸ [ Holevo’73, 98; Schumacher & Westmoreland’97] : the classical capacity of a quantum channel N is given by 1 C (N) = sup k χ (N ⊗ k ) , k →∞ with χ (N) = max {( p i ,ρ i )} H (∑ i p i N( ρ i ))−∑ i p i H (N( ρ i )) . ▸ For certain classes of channels, C (N) = χ (N) , e.g., ▸ Classical-quantum channel, N ∶ ∣ j ⟩⟨ j ∣ → ρ j . ▸ Quantum erasure channel [ Bennett, DiVincenzo, Smolin’97] . ▸ Depolarizing channel [ King’03] . ▸ However, χ (N) is not additive for general N [ Hastings’09] . | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Challenges Asymptotic regime ▸ The capacity C ( N ) is extremely difficult to compute. ▸ Few known efficiently computable bounds: ▸ Entanglement-assisted capacity [ Bennett et al.’99] , ▸ Upper bound from entanglement measure [ Brandao et al.’11,] ▸ SDP converse bound [ XW, Xie, Duan.’17] , ▸ Bounds via approximate additivity [ Leditzky et al.’17] . ▸ Even for the amplitude damping channel, we do not know. Finite blocklength regime ▸ We know a lot about classical-quantum channel coding, e.g., second-order asymptotics [ Tan, Tomamichel’15] . ▸ But we know little beyond classical-quantum channels. Quantum erasure channel All channels Amplitude damping channel C-Q Depolarizing channel | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Outline of this talk ▸ Activated no-signalling-assisted codes. ▸ New meta-converse for unassisted codes via constant-bounded subchannels. ▸ Converse on asymptotic capacity. | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Activated no-signalling-assisted codes | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Hypothesis testing converse and NS-assisted capacity ▸ Classical channels ▸ Polyanskiy-Poor-Verdu hypothesis testing converse. ▸ Achieving PPV converse via NS codes [ Matthews’12] ▸ Quantum channels ▸ PPV converse for unassisted capacity [ Wang, Renner’12] ▸ PPV converse for EA capacity: [ Matthews, Wehner’14] , H ( N A → B ( φ A ′ A )∣∣ ρ A ′ ⊗ σ B ) . R MW ( N ,ε ) = max σ B D ε ρ A ′ min where D ε H is the hypothesis testing relative entropy and φ A ′ A is the purification of ρ A . ▸ One-shot NS-assisted capacity [ Wang, Xie, Duan’17] : C ( 1 ) NS ( N ,ε ) ≤ R MW ( N ,ε ) . ▸ However, the inequality can be strict for quantum channels! ▸ Q: Why the gap appears or how to fix the gap? | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Activated capacity ▸ Potential capacity [ Winter, Yang’16] C p ( N ) = sup ( C ( N ⊗ M ) − C ( M )) . M ▸ Activated NS-assisted capacity ▸ Restrict the catalytic channel to noiseless channel; ▸ One-shot ε -error activated NS-assisted capacity [ C ( 1 ) NS ( N ⊗ I m ,ε ) − log m ] , C ( 1 ) NS , a ( N ,ε ) ∶ = sup (1) m ≥ 1 I m k ∈ { 1 ,..., M } k ∈ { 1 ,..., M } ˆ E D N ▸ Zero-error inforation theory [ Acín, Duan, Roberson, Sainz, Winter’17; Duan, Wang’15] . | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Result 1: Achieving MW converse via activated NS codes Theorem For any quantum channel N A → B , we have C ( 1 ) H (N A → B ( φ A ′ A )∣∣ ρ A ′ ⊗ σ B ) . NS , a (N ,ε ) = max σ B D ε ρ A ′ min ▸ It generalizes the case of classical channels [ Matthews’12] . ▸ For quantum channels, the NS codes require a classical noiseless channel as a catalyst to achieve the hypothesis testing converse. ▸ Ituition of achievability: the catalytic noiseless channel provides a larger solution space to activate the capacity. ▸ Converse part: duality theory of SDP. | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Constant-bounded subchannels and a new meta-converse | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Brief idea: constant-bounded subchannel ▸ Rough intuition: The “divergence” between N and “useless channels” measures the communication capability of N . (E.g., entanglement theory, E D ( ρ ) ≤ min σ ∈ SEP D ( ρ ∣∣ σ ) .) ▸ The useless channel for c.c. is the constant channel: N( ρ ) = σ B , ∀ ρ ∈ S ( A ) ▸ As a natural extension, we say a CP map M is constant-bounded if there exists a state σ B such that M ( ρ )≤ σ B , ∀ ρ ∈ S ( A ) . Bounded by constant σ B ▸ Constant-bounded (CB) CP map = CB subchannel. ▸ We denote the set of constant-bounded subchannels as V . | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)

Background Activated NS codes New meta-converse Υ -information Summary Result 2: converse bounds on one-shot capacities Theorem For any quantum channel N A ′ → B , we have H (N A ′ → B ( φ A ′ A )∥M A ′ → B ( φ A ′ A )) C ( 1 ) (N ,ε ) ≤ max M∈V D ε ρ A ′ min where φ A ′ A is a purification of ρ A ′ . ▸ Hypothesis test between N and the useless channel M D ε H ( ρ 1 ∣∣ ρ 2 ) = − log min Tr M 1 ρ 2 Type-II error s.t. Tr M 2 ρ 1 ≤ ε, Type-I error M 1 , M 2 ≥ 0 , M 1 + M 2 = 1 . ▸ We have a necessary SDP condition for M ∈ V . | Converse bounds for classical communication over quantum channels | QIP 2018, QuTech, Delft Xin Wang (UTS:QSI)