Recent Trends in Quantum Information

Quantum CommunicationAditi Sen(De)Harish-Chandra Research Institute, India

1OutlineCommunicationSecure CommunicationQuantum Cryptography

CommunicationOutlineCommunicationSecure CommunicationQuantum Cryptography

CommunicationWithout securityClassical infotransmissionQuantum statetransmissionOutlineCommunicationSecure CommunicationQuantum Cryptography

CommunicationWithout securityClassical infotransmissionQuantum statetransmissionOutlineCommunicationSecure CommunicationQuantum Cryptography

CommunicationWithout securityClassical infotransmissionQuantum statetransmissionCommunication

We r in an age of communication.6Communication

Recently, the telegram was phased out.7What is Communication?At least 2 parties

SenderReceiverAliceBobCommunication is a process by which information is sent by a sender to a receiver via some medium.What is Communication?At least 2 parties

SenderReceiverAliceBobCommunication is a process by which information is sent by a sender to a receiver via some medium.What is Communication?At least 2 parties

SenderReceiverAliceBobCommunication is a process by which information is sent by a sender to a receiver via some medium.What is Communication?At least 2 parties

SenderReceiverAliceBobCommunication is a process by which information is sent by a sender to a receiver via some medium.What is Communication?At least 2 parties

SenderReceiverAliceBoba process by which information is sent by a sender to a receiver via some medium.What is Communication?Alice (Encoder)

SendsencodesBob (Decoder) receives & decodesWhat is Communication?information must be encoded in, and decoded from a physical system.

encoding/Decodingred-green balls,sign of charge of a particle.

Only orthogonal states Quantum World: Nonorthogonal statesClassical WorldInformation is physical ---Landauer What is Communication?information must be encoded in, and decoded from a physical system.

encoding/Decodingred-green balls,sign of charge of a particle.

Only orthogonal states Quantum World: Nonorthogonal statesClassical WorldInformation is physical ---Landauer What is Communication?information must be encoded in, and decoded from a physical system.

encoding/decodingred-green balls,sign of charge of a particle.

Only orthogonal states Quantum World: Nonorthogonal statesClassical WorldInformation is physical ---Landauer What is Communication?information must be encoded in, and decoded from a physical system.

encoding/decodingred-green balls,sign of charge of a particle.

Only orthogonal states Quantum World: Nonorthogonal statesClassical WorldInformation is physical ---Landauer What is Communication?information must be encoded in, and decoded from a physical system.

encoding/decodingred-green balls,sign of charge of a particle.

Only orthogonal states Quantum World: Nonorthogonal statesClassical WorldInformation is physical ---Landauer Do quantum states advantageous?

Classical Information Transmission via Quantum States

Part 1

Quantum Dense Coding

Bennett & Wiesner, PRL 1992Unfortunately, I could not find a picture of Wiesner.20Classical Protocol

Sunny

SnowingWindy

Raining

Here, Indian could be native Americans.21Classical Protocol

Sunny

SnowingWindy

Raining

Classical Protocol

Sunny

WindyClassical Protocol

Sunny

SnowingWindy

Raining

Classical Protocol

Sunny

SnowingWindy

Raining

Classical Protocol

Sunny

SnowingWindy

Raining

2 bitsClassical Protocol

Sunny

SnowingWindy

Raining

2 bitsClassical computer unit: Bit = one of {0, 1}

Classical Protocol

MessageSunnySnowingWindyRainingEncodingDecodingDistinguishable by colorAliceBobSending00011011Classical Protocol

MessageSunnySnowingWindyRainingEncodingDecodingDistinguishable by colorAliceBob 2 bits 4 dimension

What abt Quantum?Quantum Protocol

MessageSunnySnowingWindyRainingAliceBobBASinglet state

MessageSunnySnowingWindyRainingAliceBobBA

IU

Alice performs unitary on her particleMessageSunnySnowingWindyRainingAliceBobBA

IU

Creates 4 orthogonal statesSinglet, TripletsAlice performs unitary on her particleMessageSunnySnowingWindyRainingAliceBobBA

IU

Alice sends her particle to BobMessageSunnySnowingWindyRainingAliceBob

I

ABBob has 2 particles: one of the triplets or singletMessageSunnySnowingWindyRainingAliceBob

I

ABDecoding4 orthogonal statesPossible to distinguishMessageSunnySnowingWindyRainingAliceBob

I

ABDecoding4 orthogonal statesPossible to distinguishDecodes messageMessageSunnySnowingWindyRainingAliceBob

I

ABDecoding4 orthogonal statesPossible to distinguish 2 bits 2 dimension Moral

Classical

Quantum

Vs. Task: sending 2 bitsEncoding: 4 Dimensions Encoding: 2 Dimensions Moral

Classical

Quantum

Vs. Task: sending 2 bitsEncoding: 4 Dimensions Encoding: 2 Dimensions

Bennett & Weisner, PRL 69, 2881 (92).Dense Codingfor arbitrary stateHiroshima, J. Phys. A 01; Ziman & Buzek, PRA 03, Bruss, DAriano, Lewenstein, Macchiavello, ASD, Sen, PRL 04

41BA

Alice & Bob share a state

BA

Alices aim: to send classical info i

EncodingBA

Alices aim: to send classical info iwhich occurs with probability pi

EncodingUiBA

Alice performs pi , Ui

EncodingUiBA

Alice performs pi , Ui she produces the ensemble E = {pi, ri}

EncodingUiBA

Alice performs pi , Ui she produces the ensemble E = {pi, ri}

EncodingUiBA

Alice performs pi , Ui she produces the ensemble E = {pi, ri}

Alice sends her particle to BobSending

ABAliceBobDecoding

ABAliceBobs task:Gather info abt iDecoding

ABAliceBobs task:Gather info abt iDecodingBob measures and obtains outcome j with prob qj

ABAliceBobs task:Gather info abt iDecodingPost measurement ensemble: E|j= {pi|j, i|j}

ABAliceBobs task:Gather info abt iDecodingPost measurement ensemble: E|j= {pi|j, i|j}

Mutual information: i

ABAliceBobs task:Gather info abt iDecoding

Mutual information: iIacc = max I(i:M)

ABAliceBobs task:Gather info abt i= Maximal classical information from E= {pi, ri}.DecodingIacc = max I (i:M)Holevo Theorem 1973

Initial ensemble E = {pi, ri}

Holevo Theorem 1973

Initial ensemble E = {pi, ri}

Holevo Theorem 1973

Initial ensemble E = {pi, ri}

d: dimension of ri

Holevo Theorem 1973

Initial ensemble E = {pi, ri}

Bit per qubit

ABAliceBobs task:Gather info abt iAccessible information = Maximal classical information from E = {pi, ri}.DecodingDC CapacityDense coding capacity:

maximization over all encodings i.e. over all {pi, Ui }C = Max Iacc

DC CapacityDense coding capacity:

maximization over all encodings i.e. over all {pi, Ui }C = Max Iacc = Max Holevo quantity obtained by Bob

DC CapacityDense coding capacity:

maximization over all encodings i.e. over all {pi, Ui }C = Max Iacc = Max Holevo quantity obtained by Bob

Holevo can be achieved asymptoticallySchumacher, Westmoreland, PRA 56, 131 (97)DC CapacityDense coding capacity:

maximization over all encodings i.e. over all {pi, Ui }C = Max Iacc = Max

DC CapacityDense coding capacity:

maximization over all encodings i.e. over all {pi, Ui }C = Max Iacc = Max

C = Max

DC Capacity

C = Max

DC Capacity

C = Max

DC Capacity

DC CapacityC = log2 dA + S(B) - S(AB)

DC CapacityC = log2 dA + S(B) - S(AB) IB = S(B) - S(AB) > 0A state is dense codeable

Classification of states Entangled SDC

In 22, 23 DC Capacity: Known/UnknownSingle Sender Single Receiver

Solved

Dense CodingNetwork

73Why quantum dense coding network?Point to point communication has limited commercial useWhy quantum dense coding network?To build a quantum computer,or communication network

Why quantum dense coding network?To build a quantum computer,or communication network, classical info transmission

Why quantum dense coding network?To build a quantum computer,or communication network, classical info transmissionvia quantum state in networkDense Coding Network 1

Dense Coding Network

BobDebuCharuNitu....AliceReceiversSender

Dense Coding Network

BobDebuCharuNitu....AliceReceiversSenderTask: Alice individually sends classical info to all the receivers

Dense Coding Network

BobDebuCharuNitu....AliceReceiversR. Prabhu, A. K. Pati, ASD, U. Sen, PRA 2013R. Prabhu, ASD, U. Sen, PRA 2013R. Nepal, R. Prabhu, ASD, U. Sen, PRA 2013Sender

Dense Coding Network

BobDebuCharuNitu....AliceReceiversR. Prabhu, A. K. Pati, ASD, U. Sen, PRA 2013R. Prabhu, ASD, U. Sen, PRA 2013R. Nepal, R. Prabhu, ASD, U. Sen, PRA 2013SenderUjjwals TalkPrabhus Talk

Dense Coding Network 2

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiver

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiverSeveral senders & a single receiver

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiverTask: All senders send classical info {ik, k=1,2, ..N} to a receiverSeveral senders & a single receiver

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiverTask: All senders send classical info {ik, k=1,2, ..N} to a receiver

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiver senders perform Uik, k=1,2, ..N on her parts

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiverSenders create ensemble

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiverSenders create ensemble

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiverSenders send ensemble to Bob

Dense Coding Network

AliceDebuCharuNitu....BobSendersReceiver Bobs task: gather info abt

DC Capacity networkDC capacity network

maximization over all encodings i.e. over all {p{i}, U{i} }C = Max Iacc = Max Holevo quantity obtained by Bob

DC Capacity NetworkC =

Bruss, DAriano, Lewenstein, Macchiavello, ASD, Sen, PRL 04Bruss, Lewenstein, ASD, Sen, DAriano, Macchiavello, Int. J. Quant. Info. 05DC Capacity NetworkC =

Bruss, DAriano, Lewenstein, Macchiavello, ASD, Sen, PRL 04Bruss, Lewenstein, ASD, Sen, DAriano, Macchiavello, Int. J. Quant. Info. 05Tamoghnas PosterDC Capacity: Known/UnknownSingle Sender Single Receiver

Many Senders Single Receiver

Solved

Dense Coding Network 3

Distributed DC: Two receiversAlice (A1)Alice (A2)Bob (B1)Bob (B2)

Distributed DC: Two receiversAlice (A1)Alice (A2)Bob (B1)Bob (B2)

LOCCi1i2Distributed DC: Two receiversAlice (A1)Alice (A2)Bob (B1)Bob (B2)

Distributed DC: Two receiversAlice (A1)Alice (A2)Bob (B1)Bob (B2)

Alices send her particles to BobsDistributed DC: Two receiversBob (B1)Bob (B2)

Bobs task: gather info abt ik by LOCCDistributed DC: Two receiversBob (B1)Bob (B2)

Bobs task: gather info abt ik by LOCCLOCCC = Max

Distributed DC: Two receivers

C = Max Max

LOCC Holevo boundMaximization over all encodings i.e. over all {pi, Ui }

Distributed DC: Two receivers

C = Max Max

LOCC Holevo boundMaximization over all encodings i.e. over all {pi, Ui }

Badziag, Horodecki, ASD, Sen, PRL03Distributed DC: Two receivers

C = Max Max

LOCC Holevo boundMaximization over all encodings i.e. over all {pi, Ui }

Bruss, DAriano, Lewenstein, Macchiavello, ASD, Sen, PRL 04

Distributed DC: Two receivers

DC Capacity: Known/UnknownSingle Sender Single Receiver

Many Senders Single Receiver

Solved

DC Capacity: Known/UnknownSingle Sender Single Receiver

Many Senders Single Receiver

Solved Many Senders Two Receivers

DC Capacity: Known/UnknownSingle Sender Single Receiver

Many Senders Single Receiver

Solved Many Senders Two Receivers

Partially Solved

DC Capacity: Known/UnknownSingle Sender Single Receiver

Many Senders Single Receiver

Solved Many Senders Two Receivers

Partially Solved Many Senders Many Receivers

Not Solved