Infinite-fold enhancement in communications capacity using pre-shared - PowerPoint PPT Presentation

Infinite-fold enhancement in communications capacity using pre-shared entanglement Saikat Guha July 21, 2020 College of Optical Sciences, Department of Electrical and Computer Engineering University of Arizona, Tucson AZ (USA) arXiv:2001.03934 (2020)

Outline • Background – Quantum limit of classical communications – Joint detection receivers for superadditive capacity • Entanglement assisted communications over the bosonic channel – Transmitter-receiver design – Infinite-fold capacity enhancement with pre-shared entanglement • Conclusions and ongoing work

Outline • Background – Quantum limit of classical communications – Joint detection receivers for superadditive capacity • Entanglement assisted communications over the bosonic channel – Transmitter-receiver design – Infinite-fold capacity enhancement with pre-shared entanglement • Conclusions and ongoing work

Quantum limit of classical communications Shannon’s “channel” p Y | X ( y | x ) Physical (quantum) Optical (quantum) Optical (quantum) channel X Y Alice Bob receiver modulation (e.g., loss, noise) Shannon, 1948 Shannon capacity, C Shannon = max p X I ( X ; Y ) Holevo, 1996 Schumacher, Westmoreland, 1997 Physical (quantum) ρ B n ( x ) ρ A n ( x ) A Y n B Bob channel receiver X n modulation Alice (e.g., loss, noise) (may use quantum joint (may use entangled state over detection over n channel uses) N A → B n channel uses) S ( ρ ) = − Tr( ρ log ρ ) max p Xn ( x ) , ρ An ( x ) S ( P p X n ( x ) ρ B n ( x )) − P p X n ( x ) S ( ρ B n ( x )) Holevo capacity, C = sup n n

“Superadditivity” on the transmitter and receiver side • Superadditivite capacity – Transmitter side: Using entangled states help achieve higher capacity – Receiver side: Quantum joint detection receiver helps achieve higher capacity • Channel not transmitter-side superadditive, it can still be receiver-side – Product state modulation is optimal (entanglement at transmitter doesn’t help) ⇣X ⌘ X – Holevo Capacity, C = max p X ( x ) ρ B ( x ) p X ( x ) S ( ρ B ( x )) p X S − Physical (quantum) receiver ρ B ( x ) Π ( y ) Y ρ A ( x ) X A B modulation channel Alice Bob (e.g., loss, noise) � � p Y | X ( y | x ) = Tr ρ B ( x ) Π ( y ) – Shown : receiver uses symbol by symbol detection; Shannon capacity C 1 < C – Joint detection receiver (JDR) required, unless orthogonal pure states ρ B ( x ) – Calculation of does not need receiver. Finding receiver achieving is hard C C

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<latexit sha1_base64="ds1Sc5HUGxchqQ9VKmYvHTpQkVA=">AB+HicbVDLSgNBEJyNrxgfWfXoZTAIOUjYjYIeg148RjAxkF3C7KSTDJmdXWZ6hRjyJV48KOLVT/Hm3zh5HDSxoKGo6qa7K0qlMOh5305ubX1jcyu/XdjZ3dsvugeHTZNkmkODJzLRrYgZkEJBAwVKaKUaWBxJeIiGN1P/4RG0EYm6x1EKYcz6SvQEZ2iljlsMABkNhKJl74z6YcteRVvBrpK/AUpkQXqHfcr6CY8i0Ehl8yYtu+lGI6ZRsElTApBZiBlfMj60LZUsRhMOJ4dPqGnVunSXqJtKaQz9fEmMXGjOLIdsYMB2bZm4r/e0Me1fhWKg0Q1B8vqiXSYoJnaZAu0IDRzmyhHEt7K2UD5hmHG1WBRuCv/zyKmlWK/5pXp3UapdL+LIk2NyQsrEJ5ekRm5JnTQIJxl5Jq/kzXlyXpx352PemnMWM0fkD5zPHzpEkYA=</latexit> <latexit sha1_base64="nFKUj2LP49QZgKThCS4yx+/rK8=">ACGHicbVDLSgMxFM34rPU16tJNsAgt0jpTBd0Ipd24KhXtA9phyKSZNjTzIMkIZehnuPFX3LhQxG13/o2Zdha19cCFk3PuJfceJ2RUSMP40dbWNza3tjM72d29/YND/ei4JYKIY9LEAQt4x0GCMOqTpqSkU7ICfIcRtrOqJb47WfCBQ38JzkOieWhgU9dipFUkq1f1uAdHOR7RCJYtx/hBcybxeRVqNvVAiwqb1Eo2HrOKBkzwFVipiQHUjRsfdrBzjyiC8xQ0J0TSOUVoy4pJiRSbYXCRIiPEID0lXURx4RVjw7bALPldKHbsBV+RLO1MWJGHlCjD1HdXpIDsWyl4j/ed1IurdWTP0wksTH84/ciEZwCQl2KecYMnGiDMqdoV4iHiCEuVZVaFYC6fvEpa5ZJ5VSo/XOcq1TSODgFZyAPTHADKuAeNEATYPAC3sAH+NRetXftS/uet65p6cwJ+ANt+gsbSprU</latexit> Holevo capacity of the single-mode bosonic channel C = g ( η N S + (1 − η ) N B ) − g ((1 − η ) N B )) bits per mode • C > Shannon capacity of all known standard optical receivers • Product coherent state modulation achieves capacity • Need joint detection receiver (collective measurement over codeword) Z 1 e − | β | 2 /N B | β ih β | d 2 β ρ B = 1 π N B e − | α | 2 /N S p ( α ) = 1 Z π N S | α i e − | β −√ ηα | 2 / (1 − η ) N B | β ih β | d 2 β ρ R = π (1 � η ) N B η ∈ (0 , 1] g ( N ) = (1 + N ) log(1 + N ) − N log( N ) V. Giovannetti, SG, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, Phys. Rev. Lett. 92, 027902 (2004) V. Giovannetti, SG, S. Lloyd, L. Maccone, and J. H. Shapiro, Phys. Rev. A (2004) V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, Nat. Photonics 8, 796 (2014)

Quantum limit of classical (optical) communications over a lossy channel, with an input mean photon number constraint N B = 0 ��� N g ( N ) = (1 + N ) log(1 + N ) − N log( N ) C ( N ) C N 1 2 log(1 + 4 N ) log(1 + N ) ���� N N = η N S η ∼ e − α L fiber η ∼ 1 /L 2 free-space C ( N ) Transmissivity η : Mean photon number Giovannetti, SG, Lloyd, Maccone, Shapiro, & Yuen, PRL (2004) N S : transmitted per mode Takeoka and SG, Physical Review A 89, 042309 (2014)

Outline • Background – Quantum limit of classical communications – Joint detection receivers for superadditive capacity • Entanglement assisted communications over the bosonic channel – Transmitter-receiver design – Infinite-fold capacity enhancement with pre-shared entanglement • Conclusions and ongoing work