Q UANTUM C OMMUNICATION Aditi Sen(De) Harish-Chandra Research Institute, India

O UTLINE Communication Secure Communication Quantum Cryptography Communication

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

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

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

C OMMUNICATION

W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob Communication is a process by which information is sent by a sender to a receiver via some medium.

W HAT IS C OMMUNICATION ? At least 2 parties Sender Receiver Alice Bob a process by which information is sent by a sender to a receiver via some medium.

W HAT IS C OMMUNICATION ? Alice (Encoder) Sends encodes Bob (Decoder) receives & decodes

W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/Decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/Decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer

W HAT IS C OMMUNICATION ? information must be encoded in, and decoded from a physical system. encoding/decoding red-green balls, sign of charge of a particle. Only orthogonal states Quantum World: Nonorthogonal states Classical World “Information is physical” ---Landauer Do quantum states advantageous?

Classical Information Transmission via Quantum States Part 1

Quantum Dense Coding Bennett & Wiesner, PRL 1992

C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

C LASSICAL P ROTOCOL Sunny Windy

C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

C LASSICAL P ROTOCOL Sunny Snowing Windy Raining

C LASSICAL P ROTOCOL Sunny Snowing Windy Raining 2 bits

C LASSICAL P ROTOCOL Sunny Snowing Windy Raining 2 bits Classical computer unit: Bit = one of {0, 1}

C LASSICAL P ROTOCOL Message Sunny Snowing Windy Raining EncodingDecoding Distinguishable by color Alice Bob Sending

C LASSICAL P ROTOCOL Message Sunny Snowing Windy Raining EncodingDecoding Distinguishable by color Alice Bob 2 bits 4 dimension

What abt Quantum?

Q UANTUM P ROTOCOL Message Sunny Snowing Windy Raining Alice Bob B A Singlet state

Message Sunny Snowing Windy Raining Alice Bob B A I U Alice performs unitary on her particle

Message Sunny Snowing Windy Raining Alice Bob B A I U Creates 4 orthogonal states Singlet, Triplets Alice performs unitary on her particle

Message Sunny Snowing Windy Raining Alice Bob B A I U Alice sends her particle to Bob

Message Sunny Snowing Windy Raining Alice Bob I A B Bob has 2 particles: one of the triplets or singlet

Message Sunny Snowing Windy Raining Alice Bob I A B Decoding 4 orthogonal states Possible to distinguish 4 orthogonal states Possible to distinguish

Message Sunny Snowing Windy Raining Alice Bob I A B Decoding 4 orthogonal states Possible to distinguish 4 orthogonal states Possible to distinguish Decodes message

Message Sunny Snowing Windy Raining Alice Bob I A B Decoding 4 orthogonal states Possible to distinguish 4 orthogonal states Possible to distinguish 2 bits 2 dimension

M ORAL Classical Quantum Vs. Task: sending 2 bits Encoding: 4 Dimensions Encoding: 2 Dimensions

M ORAL Classical Quantum Vs. Task: sending 2 bits Encoding: 4 Dimensions Encoding: 2 Dimensions Bennett & Weisner, PRL 69, 2881 (’92).

D ENSE C ODING FOR ARBITRARY STATE Hiroshima, J. Phys. A ’01; Ziman & Buzek, PRA ’03, Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 Hiroshima, J. Phys. A ’01; Ziman & Buzek, PRA ’03, Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04

B A Alice & Bob share a state

B A Alice’s aim: to send classical info i Alice’s aim: to send classical info i Encoding

B A Alice’s aim: to send classical info i which occurs with probability p i Alice’s aim: to send classical info i which occurs with probability p i Encoding

UiUi UiUi B A Alice performs p i, U i Encoding

UiUi UiUi B A Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice performs p i, U i she produces the ensemble E = {p i,  i } Encoding

UiUi UiUi B A Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice performs p i, U i she produces the ensemble E = {p i,  i } Encoding

UiUi UiUi B A Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice performs p i, U i she produces the ensemble E = {p i,  i } Alice sends her particle to Bob Sending

AB Alice Bob Decoding

AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding

AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Bob measures and obtains outcome j with prob q j

AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Post measurement ensemble: E |j= {p i|j,  i|j }

AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Post measurement ensemble: E |j= {p i|j,  i|j } Mutual information: i

AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Decoding Mutual information: i I acc = max I(i:M)

AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i = Maximal classical information from E= {p i,  i }. = Maximal classical information from E= {p i,  i }. Decoding I acc = max I (i:M)

H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i }

H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i }

H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i } d: dimension of  i

H OLEVO T HEOREM 1973 Initial ensemble E = {p i,  i } Bit per qubit

AB Alice Bob’s task: Gather info abt i Bob’s task: Gather info abt i Accessible information = Maximal classical information from E = {p i,  i }. Accessible information = Maximal classical information from E = {p i,  i }. Decoding

DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc

DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max Holevo quantity obtained by Bob

DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max Holevo quantity obtained by Bob Holevo can be achieved asymptotically Schumacher, Westmoreland, PRA 56, 131 (’97)

DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max

DC C APACITY Dense coding capacity: maximization over all encodings i.e. over all { p i, U i } C = Max I acc = Max

C = Max DC C APACITY

C = Max DC C APACITY

C = Max DC C APACITY

C = log 2 d A + S(ρ B ) - S(ρ AB )

DC C APACITY C = log 2 d A + S(ρ B ) - S(ρ AB ) I B = S(ρ B ) - S(ρ AB ) > 0 A state is dense codeable

C LASSIFICATION OF STATES Entangled S DC In 2  2, 2  3

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

D ENSE C ODING N ETWORK

W HY QUANTUM DENSE CODING NETWORK ? Point to point communication has limited commercial use

W HY QUANTUM DENSE CODING NETWORK ? To build a quantum computer, or communication network To build a quantum computer, or communication network

W HY QUANTUM DENSE CODING NETWORK ? To build a quantum computer, or communication network, classical info transmission To build a quantum computer, or communication network, classical info transmission

W HY QUANTUM DENSE CODING NETWORK ? To build a quantum computer, or communication network, classical info transmission via quantum state in network To build a quantum computer, or communication network, classical info transmission via quantum state in network

Dense Coding Network 1

D ENSE C ODING N ETWORK Bob Debu Charu Nitu Alice Receivers Sender

D ENSE C ODING N ETWORK Bob Debu Charu Nitu Alice Receivers Sender Task: Alice individually sends classical info to all the receivers Task: Alice individually sends classical info to all the receivers

D ENSE C ODING N ETWORK Bob Debu Charu Nitu Alice Receivers R. Prabhu, A. K. Pati, ASD, U. Sen, PRA ’ 2013 R. Prabhu, ASD, U. Sen, PRA’ 2013 R. Nepal, R. Prabhu, ASD, U. Sen, PRA’ 2013 Sender

D ENSE C ODING N ETWORK Bob Debu Charu Nitu Alice Receivers R. Prabhu, A. K. Pati, ASD, U. Sen, PRA ’ 2013 R. Prabhu, ASD, U. Sen, PRA’ 2013 R. Nepal, R. Prabhu, ASD, U. Sen, PRA’ 2013 Sender Ujjwal’s Talk Prabhu’s Talk Ujjwal’s Talk Prabhu’s Talk

Dense Coding Network 2

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver Several senders & a single receiver

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver Several senders & a single receiver

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver Task: All senders send classical info {i k, k=1,2,..N} to a receiver

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver senders perform U i k, k=1,2,..N on her parts

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver Senders create ensemble

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver Senders create ensemble

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver Senders send ensemble to Bob

D ENSE C ODING N ETWORK Alice Debu Charu Nitu Bob Senders Receiver Bob’s task: gather info abt

DC C APACITY NETWORK DC capacity network maximization over all encodings i.e. over all { p { i}, U { i } } C = Max I acc = Max Holevo quantity obtained by Bob

DC C APACITY N ETWORK C =C =C =C = Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 Bruss, Lewenstein, ASD, Sen, D’Ariano, Macchiavello, Int. J. Quant. Info. ’05

DC C APACITY N ETWORK C =C =C =C = Bruss, D’Ariano, Lewenstein, Macchiavello, ASD, Sen, PRL’ 04 Bruss, Lewenstein, ASD, Sen, D’Ariano, Macchiavello, Int. J. Quant. Info. ’05 Tamoghna’s Poster

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 