On the power on non-signalling and PPT-preserving codes Debbie - PowerPoint PPT Presentation

On the power on non-signalling and PPT-preserving codes Debbie Leung (IQC - University of Waterloo) Will Matthews (University of Cambridge) arXiv:1406.7142

Channel coding ˜ ˜ A A A ′ M A E φ + N ˜ AA B B ′ D τ ˜ AB ′ Size of code: K = d A = d B ′ . AB ′ φ + AB ′ = K − 1 Tr φ + Channel fidelity: F = Tr τ ˜ B ′ A M B ′ A ˜ φ + AA := | φ + � � φ + | ˜ AA , | φ + � ˜ AA := K − 1 / 2 � 0 ≤ j<K | j � ˜ A | j � A ˜ Q φ + QQ , L R ← Q X Q = Tr Q L RQ X T Choi operator: L RQ = d Q L R ← ˜ ˜ Q

Motivation Basic question: How large can F be for given K and N ? “One-shot” quantum information theory. ◮ Datta and Hsieh (1105.3321v2): general converse and achievability bounds for entanglement-assisted codes. ◮ Asymptotically correct for N ⊗ n , but not clear how to compute efficiently. ◮ Matthews and Wehner (1210.4722): ◮ Related channel coding to hypothesis testing to obtain an asymptotically correct converse for entanglement-assisted codes. ◮ SDP + channel symmetry → efficient computation for N ⊗ n ◮ Generalises (classical) results of Polyanskiy-Poor-Verd´ u (classical channels) and Wang and Renner (c-q channels).

Motivation ◮ This work: Start with a very general class of codes and apply two ‘nice’ constraints obeyed by unassisted codes to obtain upper bounds on their channel fidelity. ◮ Not asymptotically correct... ◮ ...but efficiently computable.

(0804.0180) Chiribella, D’Ariano, Perinotti Forward assisted codes (quant-ph/0104027) Eggeling, Schlingemann, and Werner ˜ ˜ A A A ′ M A E φ + N ˜ AA F B B ′ D Z τ ˜ AB ′ Most general form of linear map which takes operations to operations even when acting on part of a multipartite operation. The map only depends on Z A ′ B ′ ← AB = D B ′ ← RB F R ← Q E A ′ Q ← A , thus: M B ′ A = Tr A ′ B Z A ′ B ′ AB N T BA ′ . Z A ′ B ′ ← AB corresponds to a forward-assisted code (FAC) iff it is non-signalling from Bob to Alice.

Non-signalling quantum operations Z A ′ B ′ ← AB is non-signalling from Bob to Alice if Tr B ′ Z A ′ B ′ ← AB = Z Alice A ′ ← A Tr B . A ′ A ′ A A Z Alice = Z B ′ B B In terms of the Choi operator for Z A ′ B ′ ← AB : Tr B ′ Z A ′ B ′ AB = (Tr B ′ B Z A ′ B ′ AB /d B ) ⊗ 1 1 B Non-signalling from Alice to Bob if Tr A ′ Z A ′ B ′ AB = (Tr A ′ A Z A ′ B ′ AB /d A ) ⊗ 1 1 A

Forward assisted codes Forward-assisted codes correspond to operators Z satisfying (CP): Z A ′ B ′ AB ≥ 0 (TP): Tr A ′ B ′ Z A ′ B ′ AB = 1 1 AB (NSBA): Tr B ′ Z A ′ B ′ AB = (Tr B ′ B Z A ′ B ′ AB /d B ) ⊗ 1 1 B Channel fidelity of Z is F = K − 1 Tr φ B ′ A Z A ′ B ′ AB N T BA ′ Without further constraints, can always achieve F = 1 .

Non-signalling codes (NSAB): Tr A ′ Z A ′ B ′ AB = (Tr A ′ A Z A ′ B ′ AB /d A ) ⊗ 1 1 A Unassisted code ( UA ): A ′ A Z A ′ B ′ ← AB = E A ′ ← A D B ′ ← B E Local operations (+ shared N randomness). B B ′ D Z Entanglement-assisted codes A ′ A ( EA ): Z A ′ B ′ ← AB = E ′ E ′ A ′ ← Aa D B ′ ← Bb ψ ab N E Local operations and shared B B ′ D entanglement. Z NS ⊇ EA ⊇ UA .

PPT-preserving codes Rains (quant-ph/0008047) a Transpose map t Q : | i � � j | Q �→ | j � � i | Q . A ′ A Any separable ρ AB has positive partial-transpose (PPT): t A ρ AB ≥ 0 . Z Z A ′ B ′ ← AB is PPT-preserving (PPTp) iff t aB ρ aABb ≥ 0 . = ⇒ t aB ′ σ aA ′ B ′ b . B ′ B For d A ′ = d B = 1 : Z A ′ B ′ ← AB is called b PPT-binding or Horodecki channel. ρ aABb σ aA ′ B ′ b Zero-quantum capacity. By a PPT-preserving code, we mean any FAC whose bipartite operation is PPT-preserving. Additional constraint: (PPTp): t A ′ A Z A ′ B ′ AB ≥ 0 . We denote this class of codes by PPTp PPTp ⊇ UA , PPTp �⊇ EA .

PPT-preserving codes ˜ ˜ A A A ′ M A E φ + N ˜ AA F B B ′ D Z τ ˜ AB ′ ◮ We say a forward-assisted code is FHA if F is Horodecki. ◮ FHA ⊆ PPTp . ◮ Superactivation (Smith-Yard): Combination of Horodecki channel and (zero quantum capacity) 50 percent erasure channel can have positive capacity. ◮ Expect FHA capacity > UA capacity sometimes.

Relationships between classes Forward-assisted codes PPTp FHA UA EA NS Closed under composition and convex combination. For each class Ω we define: F Ω ( N , K ) := max K − 1 Tr φ B ′ A Z A ′ B ′ AB N T BA ′ for d A = d B ′ = K and Z A ′ B ′ AB ∈ Ω . Capacity: Q Ω ( N ) := sup { r : lim n →∞ F Ω ( N ⊗ n , ⌊ 2 rn ⌋ ) = 1 } .

Simplification of codes ˜ ˜ A A A ′ A U † E φ + N ˜ AA � dµ ( U ) F B B ′ D U ¯ Z Z τ ˜ AB ′ U ⊗ ¯ U | φ + � = | φ + � implies ¯ Z A ′ B ′ ← AB has same fidelity as ¯ Z A ′ B ′ ← AB . Z A ′ B ′ ← AB ∈ Ω = ⇒ Z A ′ B ′ ← AB ∈ Ω . If µ is Haar probability measure on U( K ) : � ¯ dµ ( U ) U B ′ ⊗ ¯ U A Z A ′ B ′ AB U † B ′ ⊗ U T Z A ′ B ′ AB := A , = K ( φ + 1 − φ + ) B ′ A ⊗ Γ A ′ B ) . B ′ A ⊗ Λ A ′ B + ( 1 State of A ′ : ρ A ′ = (Λ A ′ + ( K 2 − 1)Γ A ′ ) d − 1 B

Semidefinite programs Z is: Λ A ′ B + ( K 2 − 1)Γ A ′ B = ρ A ′ ⊗ 1 NSBA condition for ¯ 1 B , with which we can eliminate Γ A ′ B in the expression for ¯ Z . The channel fidelity simplifies to F = Tr N T A ′ B Λ A ′ B while the constraints simplify to 0 ≤ Λ A ′ B ≤ ρ A ′ ⊗ 1 1 B ρ A ′ ≥ 0 , Tr ρ A ′ = 1 1 B /K 2 NS :Λ B = 1 � t B [Λ A ′ B ] ≥ − ρ A ′ ⊗ 1 1 B /K, PPTp : t B [Λ A ′ B ] ≤ ρ A ′ ⊗ 1 1 B /K. Further simplification possible for covariant N .

Non-signalling codes and the hypothesis-testing bound For success probability over classical channels: ◮ Zero-error case: Cubitt, Leung, WM, Winter (1003.3195) ◮ General case: WM (1109.5417). Performance of NS codes equivalent to powerful hypothesis-testing based upper bound of Polyanskiy, Poor and Verd´ u. The WM-Wehner generalisation of the PPV bound gives an SDP upper-bound for performance of entanglement-assisted codes: F EA ( N , K ) ≤ B ( N , K ) = max Tr N T A ′ B Λ A ′ B 0 ≤ Λ A ′ B ≤ ρ A ′ ⊗ 1 1 B ρ A ′ ≥ 0 , Tr ρ A ′ = 1 1 B /K 2 Λ B ≤ 1

Non-signalling codes and the hypothesis-testing bound Our SDP for F NS ( N , K ) differs from B ( N , K ) only in having an 1 B /K 2 so equality in the constraint Λ B ≤ 1 F EA ( N , K ) ≤ F NS ( N , K ) ≤ B ( N , K ) . Does F NS ( N , K ) = B ( N , K ) ? True for classical channels. Since the bound B is asymptotically tight, Q NS ( N B ← A ′ ) = Q EA ( N B ← A ′ ) = 1 2 max ρ A ′ I (R : B) N B ← A ′ ρ RA ′ where ρ RA ′ is a purification of ρ A ′ . (Bennett, Shor, Smolin, Thapliyal - quant-ph/0106052)

PPTp codes and entanglement distillation Y∈ PPTp Tr φ + A ′ A F Γ ( ρ AB , K ) := max A ′ B ′ τ A ′ B ′ , d A ′ = d B ′ = K Y Rains quant-ph/0008047 B ′ B W.l.o.g. Y can be taken to be NS in both directions. ρ AB τ A ′ B ′ Z φ + F PPTp ( N , K ) ≥ F Γ ( ν BA ′ , K ) ν BA ′ = N BA ′ /d A ′ Y Bennett, DiVincenzo, Smolin, Wootters M quant-ph/9604024 N U If N can be implemented using one copy of ν BA ′ and classical communication then F PPTp ( N , K ) = F Γ ( ν BA ′ , K ) .

Werner-Holevo channels Qutrit Werner-Holevo channel: W (3) : X �→ 1 1 Tr X − X T ) . 2 ( 1 W (3) is symmetric, therefore Q ( W (3) ) = 0 , however... Q PPTp ( W (3) ) = Q PPTp ( W (3) ) = log 5 3 (using results of Rains). 0 1.0 ◆ ● ■ ◆ ● ■ ● ■ F Ω (( W (3) ) ⊗ 2 , K ) 0.8 ■ ◆ NS 0.6 ● ● ■ PPTp ■ ■ 0.4 ◆ ■ NS ⋂ PPTp ● ■ ◆ ■ ◆ ■ ● 0.2 ◆ ● ◆ ◆ ● ◆ ● ◆ ● K 2 4 6 8 10 ⇒ Q NS ∩ PPTp F NS ∩ PPTp (( W (3) ) ⊗ 2 ) = 1 = ( W (3) ) ≥ 1 / 2! 0 Can this be achieved by FHA ?

PPT-p. and NS �⊆ FHA All systems are qubits. A ′ A M M is computational basis H measurement; H is (classically controlled) Hadamard. LOCC = ⇒ PPT-preserving. B ′ B Non-signalling in both M Z directions. G := F ⊗ C ◦ E G A ′ A Tr G ( | 0 � � 0 | ) G ( | 1 � � 1 | ) = 0 , E Tr G ( | + � � + | ) G ( |−� �−| ) = 0 F C B ′ B D Z Cubitt and Smith (0912.2737): G has quantum zero-error capacity. Therefore, so must F .

Q NS ∩ PPTp ( W (3) ) < Q PPTp ( W (3) ) ? ( W (3) ) = log 5 Q PPTp ( W (3) ) = Q PPTp 0 3 ● ● n ● 20 40 60 80 100 120 ● ● ● ● ● - 0.5 ■ - 1.0 ■ R = log ( 5 / 3 - 1 / 30 ) ● - 1.5 ■ ■ R = log ( 5 / 3 - 1 / 40 ) ■ ■ - 2.0 ■ ■ - 2.5 ■ ■ ■ - 3.0 ■ F NS ∩ PPTp (( W (3) ) ⊗ n , ⌊ 2 Rn ⌋ ) ■

Example: Qubit depolarising channel F Ω ( D ⊗ 5 α , 2) F 1.0 0.9 PPTp NS 0.8 PPTp and NS 0.7 α 0.7 0.8 0.9 1.0

Outlook ◮ Investigate further constraints e.g. k-extendibility. ◮ What is the asymptotic capacity of PPTp / PPTp-NS codes? ◮ Is true zero-error superactivation possible?

Outlook ◮ Investigate further constraints e.g. k-extendibility. ◮ What is the asymptotic capacity of PPTp / PPTp-NS codes? ◮ Is true zero-error superactivation possible? Thanks!