Q UANTUM C OMMUNICATION P ART 2 Aditi Harish-Chandra Research Institute, India

O UTLINE Communication Secure Communication Quantum Cryptography Communication Without security Classical info transmission Classical info transmission Quantum state transmission

DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved

Dense Coding Network 3

D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 )

D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 ) LOCC i1i1 i2i2

D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 )

D ISTRIBUTED DC: T WO RECEIVERS Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 ) Alices send her particles to Bobs

D ISTRIBUTED DC: T WO RECEIVERS Bob (B 1 ) Bob (B 2 ) Bobs task: gather info abt i k by LOCC

D ISTRIBUTED DC: T WO RECEIVERS Bob (B 1 ) Bob (B 2 ) Bobs task: gather info abt i k by LOCC LOCC

C = Max D ISTRIBUTED DC: T WO RECEIVERS

C = Max Max LOCC Holevo bound Maximization over all encodings i.e. over all {p i, U i } D ISTRIBUTED DC: T WO RECEIVERS

C = Max Max LOCC Holevo bound Maximization over all encodings i.e. over all {p i, U i } Badziag, Horodecki, ASD, Sen, PRL’03 D ISTRIBUTED DC: T WO RECEIVERS

C = Max Max LOCC Holevo bound Maximization over all encodings i.e. over all {p i, U i } Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 D ISTRIBUTED DC: T WO RECEIVERS

DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved

DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved Many Senders – Two Receivers

DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved Many Senders – Two Receivers Partially Solved

DC C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver Many Senders – Single Receiver Solved Many Senders – Two Receivers Partially Solved Many Senders – Many Receivers Not Solved 

O UTLINE Communication Secure Communication Quantum Cryptography Communication Without security Classical info transmission Classical info transmission Quantum state transmission

Quantum Dense Coding Task: Classical info transmission Quantum Dense Coding Task: Classical info transmission Quantum Dense Coding

Task: Classical info transmission Medium: Quantum State Quantum Dense Coding Task: Classical info transmission Medium: Quantum State Quantum Dense Coding

Task: Classical info transmission Medium: Quantum State Quantum Dense Coding Task: Classical info transmission Medium: Quantum State Task: quantum state/info transmission Task: quantum state/info transmission Task: Quantum state/info transmission Task: Quantum state/info transmission Quantum Dense Coding

Task: Classical info transmission Medium: Quantum State Quantum Dense Coding Task: Classical info transmission Medium: Quantum State Task: Quantum state/info transmission Medium: Quantum State Task: Quantum state/info transmission Medium: Quantum State Quantum Dense Coding

Task: Classical info transmission Medium: Quantum State Task: Classical info transmission Medium: Quantum State Task: Quantum state/info transmission Medium: Quantum State Task: Quantum state/info transmission Medium: Quantum State Quantum Dense Coding Quantum Teleportation

Bennett, Brassard, Crepeau, Jozsa, Peres, Wootters, PRL 1993

Q UANTUM T ELEPORTATION Task: Sending arbitrary quantum state

Q UANTUM T ELEPORTATION Task: Sending

Q UANTUM T ELEPORTATION Task: Sending

Q UANTUM T ELEPORTATION Task: Sending Classical: Infinite communication

Q UANTUM T ELEPORTATION Task: Sending Classical: Infinite communication

Q UANTUM T ELEPORTATION A in B Alice Bob

Q UANTUM T ELEPORTATION A in B Alice Bob Alice performs measurements on “in’’ and A

Q UANTUM T ELEPORTATION A in B Alice Bob After measurement

Q UANTUM T ELEPORTATION A in B Alice Bob After measurement 2 bits of classical comm. sent by Alice to Bob

Q UANTUM T ELEPORTATION A in B Alice Bob After measurement Bob performs unitary

Q UANTUM T ELEPORTATION A in B Alice Bob After measurement State is with Bob

M ORAL Classical Quantum Vs. Task: sending arbitrary quantum state Infinite classical comm 2 bits of classical comm

M ORAL Classical Quantum Task: sending arbitrary quantum state Infinite classical comm 2 bits of classical comm Vs.

I S IT MAGIC ?

Of course not!

I S IT MAGIC ? Ingredient: Quantum Mechanics

I S IT MAGIC ? Entangled states

W HAT IS ENTANGLEMENT ? Unentangled/Useless states : Entangled/Useful states :

W HAT IS ENTANGLEMENT ? Unentangled/Useless states : Entangled/Useful states :

W HAT IS ENTANGLEMENT ? Unentangled/Useless states : Entangled/Useful states :

Is it just theory ?

Experiments

P HOTONS

143 Km Teleportation

~100KM

E NTANGLEMENT S WAPPING Zukowski, Zeilinge, Horne, Ekert, PRL 71, 4287 (’93)

E NTANGLEMENT S WAPPING Zukowski, Zeilinge, Horne, Ekert, PRL 71, 4287 (’93)

P HOTONS Photons

P HOTONS Photons

IONS

IONS 14 ions entangled

ION E NTANGLED S TATES Phys. Rep. 2008

Quantum Teleportation between Light and Matter

Quantum Teleportation between Light and Matter

Quantum Teleportation between Light and Matter Polzik’s group, Nature 443, 557 (’06)

Quantum Teleportation between Light and Matter Polzik’s group, Nature 443, 557 (’06)

IONS: T ELEPORTATION

NMR Entangled states in NMR

T ELEPORTATION BY NMR

Nielsen, Knill, Laflamme, Nature 395 (’98)

O PTICAL L ATTICES

Entangled states in Optical lattices

O PTICAL L ATTICES Entangled states in Optical lattices

O PTICAL L ATTICES Resonating valence bond states in Optical lattices

T ELEPORTATION : N EUTRAL A TOMS Wu, Yang, Shen, Zheng, J. Phys. B: At. Mol. Opt. Phys. 46, (’13)

T ELEPORTATION : S PIN CHAIN 1. Initially the spin chain is in the ground state:

T ELEPORTATION : S PIN CHAIN 1. Initially the spin chain is in the ground state: 2. Alice places an arbitrary state at her end:

T ELEPORTATION : S PIN CHAIN 1. Initially the spin chain is in the ground state: 2. Alice places an arbitrary state at her end: 3. The state evolves according to some Hamiltonian:

T ELEPORTATION : S PIN CHAIN 1. Initially the spin chain is in the ground state: 2. Alice places an arbitrary state at her end: 3. The state evolves according to some Hamiltonian: 4. Bob receives the state after some time. Bose, PRL(’03), Subrahmanyam, PRA (’03)

Many other systems …….

Teleportation for arbitrary states

AB Alice & Bob share a state

AB

A in B Task: To send to Bob single copy of is available CdCd

A in B Allowed operations: LOCC

A in B Not allowed operations: exchange qubits

A in B Alice & Bob perform some LOCC, T,

A in B Alice & Bob perform some LOCC, T, and create at Bob’s side

A in B Alice & Bob perform some LOCC, T, and create at Bob’s side Check its closeness with

A in B Quantify closeness: integration over all inputs Teleportation fidelity

Singlet Fraction: : max singlet fraction from by LOCC Bennett, Divincenzo, Smolin, Wootters, PRA 54, 3824 (’97) MPR Horodeccy, PRA 60, 1888 (’99) M.A. Nielsen, quant-ph/

Singlet Fraction: : max singlet fraction from by LOCC Horodeccy, PRA 60, 1888 (’99) Nielsen, arXiv: Verstreate, Verschele, PRL 90, (’03)

Popescu, PRL 72, 797 (’94) Alice & Bob share separable state, then f max =2/3

T ELE C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver d  d Solved

Teleportation Network 1

T ELEPORTATION : M ONOGAMY Alice Debu Charu Nitu Bob SendersReceiver

T ELEPORTATION : M ONOGAMY Alice Debu Charu Nitu Bob SendersReceiver Teleportation monogamy Faithful teleportation cannot be freely performed

T ELEPORTATION : M ONOGAMY Tele mono ineq: Lee and Park, PRA 79, (’09)

T ELEPORTATION : M ONOGAMY Holds for pure states: Lee and Park, PRA 79, (’09)

T ELEPORTATION : M ONOGAMY Holds for pure states: Lee and Park, PRA 79, (’09) Follows from monogamy of concurrence squared in 2  d

T ELEPORTATION : M ONOGAMY Does Not hold for mixed states: Lee and Park, PRA 79, (’09)

Teleportation Network 2

T ELEPORTATION N ETWORK Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 ) ASD, U. Sen, PRA 81, (’01)

T ELEPORTATION N ETWORK Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 ) ASD, U. Sen, PRA 81, (’01)

T ELEPORTATION N ETWORK Alice (A 1 ) Alice (A 2 ) Bob (B 1 ) Bob (B 2 ) ASD, U. Sen, PRA 81, (’01) Establish relation between capacity & entanglement

T ELE C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver d  d Solved

T ELE C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver d  d Solved Many Senders – single Receiver Not Solved 

T ELE C APACITY : K NOWN /U NKNOWN Single Sender – Single Receiver d  d Solved Many Senders – single Receiver Not Solved  Many Senders – many Receiver Not Solved 

Single Sender – Single Receiver Many Senders – Single Receiver Solved Many Senders – Two Receivers Many Senders – Many Receivers Partially Solved Not Solved  Classical info transmit known/Unknown known/Unknown Quantum info transmission Single Sender – Single Receiver d  d Solved Many Senders – Single Receiver Not Solved  Many Senders – Many Receivers Not Solved 