Introduction to Quantum Teleportation & BBCJPW Protocol

Quantum TeleportationTerm Paper - I

Arunabha SahaRoll No: 91/CSE/111033

Supervisors :

Dr. Guruprasad KarPhysics and Applied Mathematics Unit

Indian Statistical Institute, Kolkata

Dr. Pritha BanerjeeDept. of Computer Science & Engineering

University of Calcutta

February 7, 2014Arunabha Saha (CU) Quantum Teleportation February 7, 2014 1 / 31

Outline

1 What is Quantum Teleportation

2 Entanglement

3 Quantum Measurement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

Arunabha Saha (CU) Quantum Teleportation February 7, 2014 2 / 31

What is Quantum Teleportation

2 Entanglement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

Fictions & Facts

In the movie Star Trek a machine introduced, named Transporter.It converts an object or human into energy pattern then send it to atarget site, and at the target site, reconverted into matter.

Human teleportation not possible in practise, but recently its beenclaimed that it possible to teleport information, 143 Kms away.1

1Xiao-song Ma, et al.,2012Arunabha Saha (CU) Quantum Teleportation February 7, 2014 4 / 31

Fictions & Facts

In the movie Star Trek a machine introduced, named Transporter.It converts an object or human into energy pattern then send it to atarget site, and at the target site, reconverted into matter.

Human teleportation not possible in practise, but recently its beenclaimed that it possible to teleport information, 143 Kms away.1

1Xiao-song Ma, et al.,2012Arunabha Saha (CU) Quantum Teleportation February 7, 2014 4 / 31

Fictions & Facts

This is not the idea of Teleportation, actually..!!

Key Idea

Key Ideas

The general idea seems to be that the original object is scanned insuch a way as to extract all the information from it, then thisinformation is transmitted to the receiving location and used toconstruct the replica.2

About to teleport the state(information), not the system itself.

It illustrates another intrinsic feature of quantum information: it canbe swapped but cannot to be cloned.[2]

Superluminal communication not allowed.

Quantum teleportation relies on the phenomenon of entanglementor EPR correlation. [3]

2IBM ResearchArunabha Saha (CU) Quantum Teleportation February 7, 2014 7 / 31

Entanglement

2 Entanglement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

Entanglement

Bipartite System

Suppose we have two parties, Alice and Bob, each having a qubit.Alice‘s qubit: |ψ1〉 = α0 |0〉+ α1 |1〉Bob‘s qubit: |ψ2〉 = β0 |0〉+ β1 |1〉What will be the state of composite system of Alice and Bob ?

given as

|ψ12〉 = (α0 |0〉+ α1 |1〉)⊗ (β0 |0〉+ β1 |1〉)= α0β0 |00〉3 +α0β1 |01〉+ α1β0 |10〉+ α1β1 |11〉

The converse is also true. If we have the bipartite system(|ψ12〉) thenwe can factorise it to get the corresponding qubits involved.

3|0〉 ⊗ |0〉 ≡ |0〉 |0〉 ≡ |00〉Arunabha Saha (CU) Quantum Teleportation February 7, 2014 9 / 31

Entanglement

Bipartite System

given as

|ψ12〉 = (α0 |0〉+ α1 |1〉)⊗ (β0 |0〉+ β1 |1〉)= α0β0 |00〉3 +α0β1 |01〉+ α1β0 |10〉+ α1β1 |11〉

Entanglement

Bipartite System

given as

|ψ12〉 = (α0 |0〉+ α1 |1〉)⊗ (β0 |0〉+ β1 |1〉)= α0β0 |00〉3 +α0β1 |01〉+ α1β0 |10〉+ α1β1 |11〉

Entanglement

Bipartite System

given as

|ψ12〉 = (α0 |0〉+ α1 |1〉)⊗ (β0 |0〉+ β1 |1〉)= α0β0 |00〉3 +α0β1 |01〉+ α1β0 |10〉+ α1β1 |11〉

Entanglement

Entangled State

A pure state of two systems is entangled if it cannot be written as aproduct of two states -

|ψAB〉 6= |ψA〉 ⊗ |ψB〉

We have four such states entangled states called Bell States[4] or EPRStates.

|φ+〉 = 1√2(|00〉+ |11〉)

|φ−〉 = 1√2(|00〉 − |11〉)

|ψ+〉 = 1√2(|01〉+ |10〉)

|ψ−〉 = 1√2(|01〉 − |10〉)

These 4 states are called Maximally Entangled States.Any state of the form a |00〉 ± b |11〉 or a a |01〉 ± b |10〉, where a 6= b, iscalled Pure Entangled State.

Quantum Measurement

2 Entanglement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

Quantum Measurement

Measurement Postulate

Postulate 3

Quantum measurements are described by a collection {Mm} ofmeasurement operators. The index m refers to the measurement outcomesthat may occur in the experiment. If the state of the quantum system is|ψ〉 before experiment, then the probability that result m occurs is given by,

p(m) = 〈ψ|M †m.Mm |ψ〉

And the state of the system after measurement is

Mm|ψ〉√〈ψ|M†

m.Mm|ψ〉

BBCJPW Protocol

2 Entanglement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

BBCJPW Protocol

Quantum Teleportation

Quantum teleportation is a process by which quantum information can betransmitted from one location to another, with the help of classicalcommunication and previously shared quantum entanglement between thesending and receiving location.

The idea was proposed by C. H. Bennett, G. Brassard, C. Crepeau, R.Jozsa, A. Peres and W. K. Wootters in 1993.[1]

BBCJPW Protocol

Quantum Teleportation

It doesn‘t allow superluminal communication.

It doesn‘t violate No Cloning Theorem.[2, 1]

It doesn‘t defy the laws of physics.

The information destroyed at sender site and recreated at receivingsite i.e. the information is not copied.

BBCJPW Protocol

We have two party, Alice(A) and Bob(B). Alice wants to send a qubit|ψ〉 = α |0〉+ β |1〉 to Bob.

But Alice know neither the state of |ψ〉 nor the Bob‘s location. Butstill she can send..!!

An EPR pair is shared among A and B.

Alice interacts her |ψ〉 with the half of the EPR pair and thenmeasures the two qubits. Got one of four classical result 00,01,10,11.Send the result to Bob.

Depending upon the classical message Bob performs one of fouroperations on his half of EPR pair.

He can recover |ψ〉..!

BBCJPW Protocol

The ancilla |ψ〉 = α |0〉+ α |1〉 is to be teleported, α, β are unknown.State prepared:

|ψ0〉 = |ψ〉 |φ+〉 =1√2[α |0〉 (|00〉+ |11〉) + β |1〉 (|00〉+ |11〉)] (1)

The first 2 qubits belongs to Alice and the 3rd qubit to Bob. Alice‘s2nd and Bob‘s qubit are in EPR. Alice sends her qubit through CNOTgate,

|ψ1〉 =1√2[α |0〉 (|00〉+ |11〉) + β |1〉 (|10〉+ |01〉)] (2)

BBCJPW Protocol

The ancilla |ψ〉 = α |0〉+ α |1〉 is to be teleported, α, β are unknown.State prepared:

|ψ0〉 = |ψ〉 |φ+〉 =1√2[α |0〉 (|00〉+ |11〉) + β |1〉 (|00〉+ |11〉)] (1)

The first 2 qubits belongs to Alice and the 3rd qubit to Bob. Alice‘s2nd and Bob‘s qubit are in EPR. Alice sends her qubit through CNOTgate,

|ψ1〉 =1√2[α |0〉 (|00〉+ |11〉) + β |1〉 (|10〉+ |01〉)] (2)

BBCJPW Protocol

She sends her 1st qubit through Hadamard gate

|ψ2〉 =1√2[α(|0〉+ |1〉)(|00〉+ |11〉) + β(|0〉 − |1〉)(|10〉+ |01〉)] (3)

rewriting this, |ψ2〉 = 1√2[|00〉 (α |0〉+ β |1〉) + |01〉 (α |1〉+ β |0〉)

+ |10〉 (α |0〉 − β |1〉) + |11〉 (α |1〉 − β |0〉)] (4)

regardless of ancilla, each of the outcome is equally likely.

BBCJPW Protocol

She sends her 1st qubit through Hadamard gate

|ψ2〉 =1√2[α(|0〉+ |1〉)(|00〉+ |11〉) + β(|0〉 − |1〉)(|10〉+ |01〉)] (3)

rewriting this, |ψ2〉 = 1√2[|00〉 (α |0〉+ β |1〉) + |01〉 (α |1〉+ β |0〉)

+ |10〉 (α |0〉 − β |1〉) + |11〉 (α |1〉 − β |0〉)] (4)

regardless of ancilla, each of the outcome is equally likely.

BBCJPW Protocol

1st term has Alice‘s qubit in |00〉 and Bob‘s qubit in α |0〉+ β |1〉which is the original state |ψ〉similarly,

00 7→ |ψ3(00)〉 ≡ [α |0〉+ β |1〉]−−− > Got |ψ〉 (5)

01 7→ |ψ3(01)〉 ≡ [α |1〉+ β |0〉]−−− > X (6)

10 7→ |ψ3(10)〉 ≡ [α |0〉 − β |1〉]−−− > Z (7)

11 7→ |ψ3(01)〉 ≡ [α |1〉 − β |0〉]−−− > X,Z (8)

BBCJPW Protocol

1st term has Alice‘s qubit in |00〉 and Bob‘s qubit in α |0〉+ β |1〉which is the original state |ψ〉similarly,

00 7→ |ψ3(00)〉 ≡ [α |0〉+ β |1〉]−−− > Got |ψ〉 (5)

01 7→ |ψ3(01)〉 ≡ [α |1〉+ β |0〉]−−− > X (6)

10 7→ |ψ3(10)〉 ≡ [α |0〉 − β |1〉]−−− > Z (7)

11 7→ |ψ3(01)〉 ≡ [α |1〉 − β |0〉]−−− > X,Z (8)

BBCJPW Protocol

BBCJPW Circuit

Figure: Quantum Teleportation Circuit[5]

For N particle system, we need 2log2N classical bits.

Applications

2 Entanglement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

Applications

The most useful and exiting applications of entanglement andteleportation are in quantum cryptography and quantum computing.

1. Quantum Cryptography:It uses this transfer of quantum information to allow securecommunication. The conventional cryptography depends upon thedifficult mathematical problems but if we can se solve them efficiently,that no longer be secure. But due to the uncertain nature or quantumsystems it becomes inherently secure.

2. Quantum Computing:Solves difficult problems more quickly than conventional computers.Used to encrypt internet communications e.g. In last FIFA world Cupquantum encryption was used for data security.

References

2 Entanglement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

References

[1] Teleporting an Unknown Quantum State via Dual Classical andEinstein-Podolsky-Rosen ChannelsC. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. WoottersPhys. Rev. Lett. vol. 70, pp 1895-1899 (1993)

[2] A single quantum cannot be clonedW. K. Wootters & W. H. ZurekNature(London) 299,802(1982)

[3]Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?Einstein, A. and Podolsky, B. and Rosen, N.Phys. Rev. 47, 777-780 (1935)

[4] Quantum Computation and Quantum Information, p-25Nielson, Michael A., and Chuang, Issac L.

Image Courtesy

2 Entanglement

4 BBCJPW Protocol

5 Applications

6 References

7 Image Courtesy

8 Appendix

Image Courtesy

1. http://goo.gl/aeXmyo

2. http://goo.gl/Nv454

3. http://imgs.xkcd.com/comics/quantum_teleportation.png

4. http://goo.gl/MhQRnR

5. http://goo.gl/odHKB5

Image Courtesy

Thank You

Appendix

Appendix A: Tenson Product

Tensor product is a way of putting vector spaces together to form largervector spaces. Denoted by |ψ〉 ⊗ |φ〉.If |ψ〉 is of m dimensional and |φ〉 is of n dimensional, then |ψ〉 ⊗ |φ〉 is ofmn dimensional.

]⊗[23

1 × 21 × 32 × 22 × 3

Appendix

Appendix B: Quantum Gates

Pauli matrices are named after the physicist Wolfgang Pauli.

These are a set of 2 x 2 complex matrices which are Hermitian andunitary.

They look like

σ0 ≡ I ≡[1 00 1

], σ1 ≡ σx ≡ X ≡

[0 11 0

]σ2 ≡ σy ≡ Y ≡

[0 − ii 0

], σ3 ≡ σz ≡ Z ≡

[1 00 − 1

Appendix

They look like

σ0 ≡ I ≡[1 00 1

], σ1 ≡ σx ≡ X ≡

[0 11 0

]σ2 ≡ σy ≡ Y ≡

[0 − ii 0

], σ3 ≡ σz ≡ Z ≡

[1 00 − 1

Appendix

They look like

σ0 ≡ I ≡[1 00 1

], σ1 ≡ σx ≡ X ≡

[0 11 0

]σ2 ≡ σy ≡ Y ≡

[0 − ii 0

], σ3 ≡ σz ≡ Z ≡

[1 00 − 1

Appendix

Appendix C: CNOT Gate

The CNOT gate flips the second qubit (the target qubit) if and only if thefirst qubit (the control qubit) is 1.

Appendix

Appendix D: Hadamard Gate

The Hadamard gate acts on a single qubit.

It maps the basis state |0〉 to |0〉+|1〉√2

and |1〉 to |0〉−|1〉√2

Hadamard

Matrix, H = 1√2

[1 11 − 1

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