De Finetti Theorems for Quantum Channels arXiv:1810.12197 Mario - PowerPoint PPT Presentation

De Finetti Theorems for Quantum Channels arXiv:1810.12197 Mario Berta with Borderi, Fawzi, Scholz Banfg 07/25/2019 1 / 24

Outline Motivation: Noisy Channel Coding De Finetti Theorems Application: Noisy Channel Coding Conclusion Add-on: De Finetti with Linear Constraints 2 / 24

Motivation: Noisy Channel Coding 3 / 24

Error Correction m bits are subject to noise modelled by N y x , find encoder e and decoder d to maximize probability p N m of retrieving m bits Noisy Channel Coding Noise Information Information Encoder Decoder 4 / 24

Noisy Channel Coding Noise Information Information Encoder Decoder Error Correction m bits are subject to noise modelled by N ( y ∣ x ) , find encoder e and decoder d to maximize probability p ( N , m ) of retrieving m bits 4 / 24

Approximating p N m up to multiplicative factor better than e 1 is NP-hard in the worst case [Barman & Fawzi 18] 1 x 1 i Noisy Channel Coding (continued) ▸ Fixed number of bits m and noise model N gives bilinear ▸ optimization p ( N , m ) = max N ( y ∣ x ) d ( i ∣ y ) e ( x ∣ i ) 2 m ∑ ( e , d ) x , y , i e ( x ∣ i ) = 1 , 0 ≤ e ( x ∣ i ) ≤ 1 s.t. ∑ d ( i ∣ y ) = 1 , 0 ≤ d ( i ∣ y ) ≤ 1 ∑ 5 / 24

1 i x Noisy Channel Coding (continued) ▸ Fixed number of bits m and noise model N gives bilinear ▸ optimization p ( N , m ) = max N ( y ∣ x ) d ( i ∣ y ) e ( x ∣ i ) 2 m ∑ ( e , d ) x , y , i e ( x ∣ i ) = 1 , 0 ≤ e ( x ∣ i ) ≤ 1 s.t. ∑ d ( i ∣ y ) = 1 , 0 ≤ d ( i ∣ y ) ≤ 1 ∑ ▸ Approximating p ( N , m ) up to multiplicative factor better than ▸ ( 1 − e − 1 ) is NP-hard in the worst case [Barman & Fawzi 18] 5 / 24

e 1 -multiplicative approximation algorithms Polynomial 1 x x 1 1 Noisy Channel Coding (continued) ▸ For the linear program [Hayashi 09, Polyanski et al. 10] ▸ lp ( N , m ) = max N ( y ∣ x ) r xy 2 m ∑ ( r , p ) x , y s.t. r xy ≤ 1 , ∑ ∑ p x = k r xy ≤ p x , 0 ≤ r xy , p x ≤ 1 we have the approximation [Barman & Fawzi 18] p ( N , m ) ≤ lp ( N , m ) ≤ 1 − e − 1 ⋅ p ( N , m ) 6 / 24

x 1 1 x Noisy Channel Coding (continued) ▸ For the linear program [Hayashi 09, Polyanski et al. 10] ▸ lp ( N , m ) = max N ( y ∣ x ) r xy 2 m ∑ ( r , p ) x , y s.t. r xy ≤ 1 , ∑ ∑ p x = k r xy ≤ p x , 0 ≤ r xy , p x ≤ 1 we have the approximation [Barman & Fawzi 18] p ( N , m ) ≤ lp ( N , m ) ≤ 1 − e − 1 ⋅ p ( N , m ) ▸ Polynomial ( 1 − e − 1 ) -multiplicative approximation algorithms ▸ 6 / 24

Quantum Error Correction Find encoder E and decoder D to maximize quantum probability m of retrieving m qubits F Quantum Noisy Channel Coding ▸ Main question: Similar results for quantum error correction? ▸ [Matthews 12, Leung & Matthews 15] Quantum Noise Quantum Quantum Information Information Encoder Decoder 7 / 24

Quantum Noisy Channel Coding ▸ Main question: Similar results for quantum error correction? ▸ [Matthews 12, Leung & Matthews 15] Quantum Noise Quantum Quantum Information Information Encoder Decoder Quantum Error Correction Find encoder E and decoder D to maximize quantum probability F ( N , m ) of retrieving m qubits 7 / 24

Optimize Quantum Information Processing Develop mathematical toolbox rooted in optimization theory Φ n Φ n Quantum Noisy Channel Coding (continued) ▸ Near-term quantum devices are of intermediate scale and noisy ▸ E N D ▸ Tailor-made approximation algorithms for encoder/ decoder? ▸ 8 / 24

Φ n Φ n Quantum Noisy Channel Coding (continued) ▸ Near-term quantum devices are of intermediate scale and noisy ▸ E N D ▸ Tailor-made approximation algorithms for encoder/ decoder? ▸ Optimize Quantum Information Processing Develop mathematical toolbox rooted in optimization theory 8 / 24

Φ n Φ n Quantum Noisy Channel Coding (continued) E N D ▸ m qubits with quantum noise model N leads to quantum ▸ channel fidelity F ( N , n ) ∶ = max F ( Φ n , (( D ○ N ○ E ) ⊗ I )( Φ n )) s.t. E , D quantum operations (+ physical constraints) with fidelity F ( ρ , σ ) ∶ = ∥√ ρ √ σ ∥ 2 1 . 9 / 24

To characterize is set SEP AA BB of separable channels strong hardness for quantum separability [Barak et al. 12] Lower bounds on figure of merit via, e.g., physical intuition or iterative see-saw methods upper bounds ? B B i A A i p i i I Quantum Noisy Channel Coding (continued) ▸ For d ∶ = dim ( N ) becomes bilinear optimization ▸ F ( N , n ) = max d ⋅ Tr [( N ¯ A → B ( Φ ¯ A ) ⊗ Φ A ¯ B )( ∑ B ) ( Φ AA ⊗ Φ BB ) ] p i E i A ⊗ D i A ¯ A → ¯ B → ¯ i ∈ I E i , D i quantum operations , p i ≥ 0 , ∑ s.t. p i = 1 i ∈ I 10 / 24

Lower bounds on figure of merit via, e.g., physical intuition or iterative see-saw methods upper bounds ? B Quantum Noisy Channel Coding (continued) ▸ For d ∶ = dim ( N ) becomes bilinear optimization ▸ F ( N , n ) = max d ⋅ Tr [( N ¯ A → B ( Φ ¯ A ) ⊗ Φ A ¯ B )( ∑ B ) ( Φ AA ⊗ Φ BB ) ] p i E i A ⊗ D i A ¯ A → ¯ B → ¯ i ∈ I E i , D i quantum operations , p i ≥ 0 , ∑ s.t. p i = 1 i ∈ I ▸ To characterize is set SEP N ( A ¯ A ∣ B ¯ B ) of separable channels ▸ ∑ p i E i A ⊗ D i A → ¯ B → ¯ i ∈ I ⇒ strong hardness for quantum separability [Barak et al. 12] 10 / 24

B Quantum Noisy Channel Coding (continued) ▸ For d ∶ = dim ( N ) becomes bilinear optimization ▸ F ( N , n ) = max d ⋅ Tr [( N ¯ A → B ( Φ ¯ A ) ⊗ Φ A ¯ B )( ∑ B ) ( Φ AA ⊗ Φ BB ) ] p i E i A ⊗ D i A ¯ A → ¯ B → ¯ i ∈ I E i , D i quantum operations , p i ≥ 0 , ∑ s.t. p i = 1 i ∈ I ▸ To characterize is set SEP N ( A ¯ A ∣ B ¯ B ) of separable channels ▸ ∑ p i E i A ⊗ D i A → ¯ B → ¯ i ∈ I ⇒ strong hardness for quantum separability [Barak et al. 12] ▸ Lower bounds on figure of merit via, e.g., physical intuition or ▸ iterative see-saw methods ⇒ upper bounds ? 10 / 24

De Finetti Theorems 11 / 24

De Finetti for Quantum States For states ρ AB k 1 we have that [Christandl et al. 07] 1 ρ AB k p i σ i A p i σ i i ρ AB 1 π B k B 1 d 2 B k σ i Monogamous Entanglement ▸ Quantum states ρ AB is called k -shareable if ▸ ρ AB 1 ⋯ B k with ρ AB j = ρ AB ∀ j ∈ [ k ] ⇒ characterizes separable states [Stoermer 69, Doherty et al. 02] 12 / 24

1 B k p i σ i i Monogamous Entanglement ▸ Quantum states ρ AB is called k -shareable if ▸ ρ AB 1 ⋯ B k with ρ AB j = ρ AB ∀ j ∈ [ k ] ⇒ characterizes separable states [Stoermer 69, Doherty et al. 02] De Finetti for Quantum States 1 ( ρ AB k 1 ) we have that [Christandl et al. 07] For states ρ AB k 1 = π B k { p i , σ i } ∥ ρ AB − ∑ B ∥ min A ⊗ σ i ≤ d 2 12 / 24

d A B k d B B k B k B k k B k B k B k k -shareable Quantum Channels De Finetti for Quantum Channels 1 ( π B k 1 ( ⋅ )) = π ¯ 1 ( N AB k 1 ( ⋅ )) with For channels N AB k 1 → ¯ A ¯ 1 → ¯ A ¯ A [ N AB k 1 ( ⋅ )] = Tr ¯ A [ N AB k 1 ( 1 A ⊗ Tr A [ ⋅ ])] Tr ¯ 1 → ¯ A ¯ 1 → ¯ A ¯ B k [ N AB k 1 ( ⋅ )] = Tr ¯ B k [ N AB k 1 ( Tr B k [ ⋅ ] ⊗ 1 B k )] Tr ¯ 1 → ¯ A ¯ 1 → ¯ A ¯ we have that (cf. asymptotic bounds [Fuchs et al. 04]) √ poly ( d A d ¯ B ) { p i , E i , D i } ∥ N AB → ¯ B ∥ A d B d ¯ B − ∑ min p i E i A ⊗ D i ≤ A ¯ A → ¯ B → ¯ i ∈ I ◇ ⇒ characterizes separable quantum channels 13 / 24

Directly de Finetti theorems with linear constraints (add-on) Classical de Finetti + informationally complete measurements — relative to quantum side information Sum-of-Squares Hierarchies [Lasserre 00, Parrilo 03] via information-theoretic approach based on entropy inequalities [Brandão & Harrow 16] Various extensions possible — basic open questions for classical/quantum settings Proof Ideas ▸ Choi-Jamiolkowski isomorphism gives Choi constraints for states ▸ that represent channels 14 / 24

Classical de Finetti + informationally complete measurements — relative to quantum side information Sum-of-Squares Hierarchies [Lasserre 00, Parrilo 03] via information-theoretic approach based on entropy inequalities [Brandão & Harrow 16] Various extensions possible — basic open questions for classical/quantum settings Proof Ideas ▸ Choi-Jamiolkowski isomorphism gives Choi constraints for states ▸ that represent channels ▸ Directly de Finetti theorems with linear constraints (add-on) ▸ 14 / 24

Sum-of-Squares Hierarchies [Lasserre 00, Parrilo 03] via information-theoretic approach based on entropy inequalities [Brandão & Harrow 16] Various extensions possible — basic open questions for classical/quantum settings Proof Ideas ▸ Choi-Jamiolkowski isomorphism gives Choi constraints for states ▸ that represent channels ▸ Directly de Finetti theorems with linear constraints (add-on) ▸ ▸ Classical de Finetti + informationally complete ▸ measurements — relative to quantum side information 14 / 24