Semidefinite programming converse bounds for quantum communication - PowerPoint PPT Presentation

Semidefinite programming converse bounds for quantum communication arXiv:1709.00200 Kun Fang Joint work with Xin Wang, Runyao Duan Centre for Quantum Software and Information U niversity of T echnology S ydney

Quantum communication A 1 B 1 A E N D B ≈ id k A B How well the simulation is? [Kretschmann, Werner, 2004] ⊚ Channel distance � D ◦ N ◦ E − id k � ♦ . ⊚ Channel fidelity F ( Φ k , D ◦ N ◦ E ( Φ k )) . , where Φ k is k -dimensional maximally entangled state. ⊚ ... Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Quantum capacity R � Φ k Φ k A 1 B 1 ⊚ r : qubits transmitted per channel use. A N B B 2 A 2 ⊚ n : number of channel copies. N E n D n . ⊚ ε : error tolerance. . . An Bn N id k ⊚ A triplet ( r , n , ε ) is achievable if ∃ Φ k , E n and D n such that 1 � Φ k , � � n log k ≥ r , F Φ k ≥ 1 − ε. ⊚ Optimal achievable rate given n , ε r ∗ ( n , ε ) : � max { r : ( r , n , ε ) achievable } . ⊚ Quantum capacity n →∞ r ∗ ( n , ε ) . Q ( N ) : � lim ε → 0 lim Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Theorem ( Barnum, Nielsen, Schumacher, 1996-2000; Lloyd, Shor, Devetak, 1997-2005) For any quantum channel N , it quantum capacity is equal to the regularized coherent information of the channel: � N ⊗ n � 1 Q ( N ) � lim n I c , n →∞ where I c ( N ) � max φ AA ′ I ( A � B ) N A ′→ B ( φ AA ′ ) and φ AA ′ pure state. ⊚ Not a single-letter formula. ⊚ I c ( N ) not additive in general. Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Known converse bounds Strong converse Efficiently computable For general channels R ✓ ? (max-min) ✓ ε -DEG ? ✓ ✗ E C ✓ ? (regularization) ✓ Q E ✓ ✓ ✓ Q ss ? ? (unbounded dimension) ✓ Q Θ ✓ ✓ ✓ ⊚ R : Rains information [Tomamichel, Wilde, Winter, 2017] ⊚ ε -DEG: Epsilon degradable bound [Sutter, Scholz, Winter, Renner, 2014] ⊚ E C : Channel’s entanglement cost [Berta, Brandão, Christandl,Wehner, 2013] ⊚ Q E : Entanglement assisted quantum capacity [Bennett, Devetak, Harrow, Shor, Winter,2014; Berta, Christandl, Renner,2011] ⊚ Q ss : Quantum capacity with symmetric side channels [Smith, Smolin, Winter, 2008] ⊚ Q Θ : Partial transposition bound [Holevo,Werner, 2001] Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

One-shot quantum capacity

One-shot quantum capacity ⊚ Unassisted code (UA): R Π A i B i → A o B o � E A i → A o ⊗ D B i → B o . B o A i E D ⊚ Positive partial transpose preserving Π (PPT) code: [Rains, 1999; Rains, 2001] T Bi Bo N Π A i B i → A o B o PPT operation J ≥ 0 . A o B i Π id k ⊚ Non-signalling (NS) code: [Leung, Matthews, 2015; Duan, Winter, 2016] 1 A i A i B o Tr A o J Π � ⊗ Tr A i A o J Π , ( A � B ) d A i 1 B i E Π D Tr B o J Π � ⊗ Tr B i B o J Π , ( B � A ) d B i A o B i � � PPT UA NS J Π � Π A i B i → A o B o Φ A i B i : A ′ i B ′ i Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

R Φ k B o A i E D Π N A o B i id k Maximum channel fidelity Tr � � F Ω ( N , k ) : � sup Φ k · Π ◦ N ( Φ k ) . Π ∈ Ω input output One-shot quantum capacity error tolerance Q ( 1 ) Ω ( N , ε ) : � log max { k : F Ω ( N , k ) ≥ 1 − ε } . (Asymptotic) quantum capacity � N ⊗ n , ε � 1 n Q ( 1 ) Q Ω ( N ) � lim ε → 0 lim . Ω n →∞ Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

SDP converse bounds for one-shot quantum capacity [Leung, Matthews, 2015] F Ω ( N , k ) � max Tr J N W AB s.t. 0 ≤ W AB ≤ ρ A ⊗ 1 B , Tr ρ A � 1 , PPT: − k − 1 ρ A ⊗ 1 B ≤ W T B AB ≤ k − 1 ρ A ⊗ 1 B , NS: Tr A W AB � k − 2 1 B . Optimization characterization Q ( 1 ) PPT ( N , ε ) � − log min m s.t. Tr J N W AB ≥ 1 − ε, 0 ≤ W AB ≤ ρ A ⊗ 1 B , Tr ρ A � 1 , − m ρ A ⊗ 1 B ≤ W T B AB ≤ m ρ A ⊗ 1 B , � � Tr A W AB � m 2 1 B . NS condition Non-linear terms Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Q ( 1 ) PPT ( N , ε ) � − log min m s.t. Tr J N W AB ≥ 1 − ε, 0 ≤ W AB ≤ ρ A ⊗ 1 B , (1) TB Tr ρ A � 1 , − m ρ A ⊗ 1 B ≤ W AB ≤ m ρ A ⊗ 1 B . � Tr A W AB � m 2 1 B . NS condition � g ( N , ε ) : � min Tr S A s.t. Tr J N W AB ≥ 1 − ε, 0 ≤ W AB ≤ ρ A ⊗ 1 B , (2) TB Tr ρ A � 1 , − S A ⊗ 1 B ≤ W AB ≤ S A ⊗ 1 B . � g ( N , ε ) : � min Tr S A s.t. Tr J N W AB ≥ 1 − ε, 0 ≤ W AB ≤ ρ A ⊗ 1 B , (3) TB Tr ρ A � 1 , − S A ⊗ 1 B ≤ W AB ≤ S A ⊗ 1 B , Tr A W AB � t 1 B . g ( N , ε ) : � min Tr S A � s.t. Tr J N W AB ≥ 1 − ε, 0 ≤ W AB ≤ ρ A ⊗ 1 B , TB Tr ρ A � 1 , − S A ⊗ 1 B ≤ W AB ≤ S A ⊗ 1 B , (4) m 2 , Tr A W AB � t 1 B , t ≥ � Q ( 1 ) � � PPT ∩ NS ( N , ε ) ≤ − log � m . Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Main result 1: SDP converse bounds for one-shot quantum capacity [Tomamichel, Berta, Renes, 2016] f ( N , ε ) � min Tr S A s.t. Tr J N W AB ≥ 1 − ε, S A , Θ AB ≥ 0 , Tr ρ A � 1 , (5) 0 ≤ W AB ≤ ρ A ⊗ 1 B , S A ⊗ 1 B ≥ W AB + Θ T B AB . Theorem For any quantum channel N and error tolerance ε , the inequality chain holds Q ( 1 ) ( N , ε ) ≤ Q ( 1 ) PPT ∩ NS ( N , ε ) (6) ≤ − log � g ( N , ε ) ≤ − log � g ( N , ε ) ≤ − log g ( N , ε ) ≤ − log f ( N , ε ) . Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Example: Amplitude damping channel Amplitude damping channel N AD � � 1 i � 0 E i · E † i with √ √ E 0 � | 0 � � 0 | + 1 − r | 1 � � 1 | r | 0 � � 1 | , E 1 � 0 ≤ r ≤ 1 1.15 1.1 1.05 Qubit 1 0.95 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.094 0.094 0.094 0.094 0.094 0.094 0.094 0.094 0.094 0.094 0.094 0.9 0.06 0.07 0.08 0.09 0.1 Channel parameter r Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Example: Qubit depolarizing channel � ρ � � � 1 − p � � X ρ X + Y ρ Y + Z ρ Z � ρ + p Qubit depolarizing channel N D , 3 where X , Y , Z are Pauli matrices . 2.5 2 Qubit 1.5 1 0.5 17 27 0 5 10 15 20 25 30 Number of channel copies, n Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Asymptotic quantum capacity

SDP strong converse bound for quantum capacity Q ( 1 ) PPT ( N , ε ) � − log min m s.t. Tr J N W AB ≥ 1 − ε, 0 ≤ W AB ≤ ρ A ⊗ 1 B , Tr ρ A � 1 , − m ρ A ⊗ 1 B ≤ W T B AB ≤ m ρ A ⊗ 1 B . Take R AB � W AB / m and throw away the condition W AB ≤ ρ A ⊗ 1 B , we obtain an additive SDP upper bound Q ( 1 ) PPT ( N , ε ) ≤ Q Γ ( N ) − log ( 1 − ε ) , where Q Γ ( N ) � log max Tr J N R AB s.t. R AB , ρ A ≥ 0 , Tr ρ A � 1 , (7) − ρ A ⊗ 1 B ≤ R T B AB ≤ ρ A ⊗ 1 B . ⊚ Additivity: Q Γ ( N ⊗ M ) � Q Γ ( N ) + Q Γ ( M ) (by utilizing SDP duality). ⊚ Converse bound for Q ( N ) : Q ( N ) ≤ Q PPT ( N ) ≤ Q Γ ( N ) . ⊚ For noiseless quantum channel I d , Q ( I d ) � Q Γ ( I d ) � log 2 d . ⊚ Strong converse: denote the n-shot optimal rate as r , then ( r , n , ε ) satisfies nr ≤ nQ Γ ( N ) − log ( 1 − ε ) , which implies ε ≥ 1 − 2 n ( Q Γ ( N ) − r ) . Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan

Main result 2: 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 ) . How to understand Q Γ ( N ) ? Entanglement measure � � φ AA ′ �� Q Γ ( N ) � ρ A ∈ S ( A ) E W max N A ′ → B � � φ AA ′ � � σ � � max ρ ∈ S ( A ) min σ ∈ PPT ′ D max N A ′ → B � � � ρ � Tr ρ R AB : − 1 AB ≤ R T B where E W : � log max AB ≤ 1 AB , R AB ≥ 0 , [Wang, Duan, 2016], φ AA ′ is a purification of ρ A and PPT’ � � σ ≥ 0 : � σ T B � 1 ≤ 1 � . Remark: For any EB channel N , Q Γ ( N ) � 0 . If Q E ( N ) � 0 , Q Γ ( N ) < Q E ( N ) . Semidefinite programming converse bounds for quantum communication(1709.00200) X. Wang, K. Fang , R. Duan