Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Quantum Teleportation :

Theory and Experiment

Chithrabhanu P

chithrabhanu@prl.res.in

THEPH, PRL

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Quantum bits

Bit :- Fundamental unit of classical information {0,1}

Qubit :-Quantum analog to bit.

|ψ〉 = α|0〉+ β|1〉 (1)

The state of the qubit is a vector in an two-dimensional

complex vector space. Qutrit, qudit :- 3 and higher

dimensions respectively.

|0〉 , |1〉 :- Computational basis states forming orthonormal

basis of the vector space. |α|2 :- Probability that system is

in |0〉; |β|2 :- Probability that system is in |1〉

Example of qubit states:- Two polarization states { |H〉,|V 〉}, spin states { | ↑〉,| ↓〉} etc.

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Entanglement

Non local quantum correlation between particles.

A two particle entangled state cannot be written as

product of two single particle states.

Ψ12 6= φ1 ⊗ ξ2 (2)

Bell states :- Maximally entangled state of two qubits.

|Ψ±〉 =1√2

(|0〉|1〉 ± |1〉|0〉) (3)

|Φ±〉 =1√2

(|0〉|0〉 ± |1〉|1〉) (4)

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Quantum gates

Basic unit of a quantum circuit.

NOT gate { X }

X (α|0〉+ β|1〉)→ α|1〉+ β|0〉 (5)

Z gate

Z (α|0〉+ β|1〉)→ α|0〉 − β|1〉 (6)

Hadamard gate {H}

H (α|0〉+ β|1〉) = α|0〉+ |1〉√

2+ β|0〉 − |1〉√

2(7)

CNOT gate :- Two qubit state. Flips the second qubit

(target) if the first qubit (control) is 1. Similar to XOR

|A,B〉 → |A,B ⊕ A〉

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Quantum gates cont..

Hadamard and CNOT operation to prepare Bell states.

x, y are |0〉 or |1〉 logic. βxy - Bell states.

In case of polarization; a half wave plate (HWP), can

perform many single qubit operations by keeping its fast

axis at different angle with respect to the incident

polarization. { 0→ Z , π4 → X , π8 → H }

Polarization CNOT :- not trivial. Requires interaction of

two qubits (Zhao et al., PRL 2005 ; Bao et al., PRL 2007).

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Quantum Teleportation

VOLUME 70 29 MARCH l993 NUMBER 13

Teleporting an Unknown Quantum State via Dual Classical andEinstein-Podolsky-Rosen Channels

Charles H. Bennett, ~ ) Gilles Brassard, ( ) Claude Crepeau, ( ) ( )

Richard Jozsa, ( ) Asher Peres, ~4) and William K. Wootters( )' IBM Research Division, T.J. watson Research Center, Yorktomn Heights, ¹mYork 10598

( lDepartement IIto, Universite de Montreal, C.P OI28, Su. ccursale "A", Montreal, Quebec, Canada HBC 817( lLaboratoire d'Informatique de 1'Ecole Normale Superieure, g5 rue d'Ulm, 7M80 Paris CEDEX 05, France~ i

l lDepartment of Physics, Technion Israel In—stitute of Technology, MOOO Haifa, Israell lDepartment of Physics, Williams College, Williamstoivn, Massachusetts OIP67

(Received 2 December 1992)

An unknown quantum state ]P) can be disassembled into, then later reconstructed from, purelyclassical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To doso the sender, "Alice," and the receiver, "Bob," must prearrange the sharing of an EPR-correlatedpair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantumsystem, and sends Bob the classical result of this measurement. Knowing this, Bob can convert thestate of his EPR particle into an exact replica of the unknown state ]P) which Alice destroyed.

PACS numbers: 03.65.Bz, 42.50.Dv, 89.70.+c

The existence of long range correlations betweenEinstein-Podolsky-Rosen (EPR) [1] pairs of particlesraises the question of their use for information transfer.Einstein himself used the word "telepathically" in thiscontempt [2]. It is known that instantaneous informationtransfer is definitely impossible [3]. Here, we show thatEPR correlations can nevertheless assist in the "telepor-tation" of an intact quantum state from one place toanother, by a sender who knows neither the state to beteleported nor the location of the intended receiver.

Suppose one observer, whom we shall call "Alice, " hasbeen given a quantum system such as a photon or spin-&particle, prepared in a state ]P) unknown to her, and shewishes to communicate to another observer, "Bob," suf-ficient information about the quantum system for him tomake an accurate copy of it. Knowing the state vector]P) itself would be sufficient information, but in generalthere is no way to learn it. Only if Alice knows before-hand that ~qb) belongs to a given orthonormal set can shemake a measurement whose result will allow her to makean accurate copy of [P). Conversely, if the possibilitiesfor ~P) include two or more nonorthogonal states, then nomeasurement will yield sufhcient information to prepare

a perfectly accurate copy.A trivial way for Alice to provide Bob with all the in-

formation in [P) would be to send the particle itself. If shewants to avoid transferring the original particle, she canmake it. interact unitarily with another system, or "an-cilla, " initially in a known state ~ap), in such a way thatafter the interaction the original particle is left in a stan-dard state ~Pp) and the ancilla is in an unknown state]a) containing complete information about ~P). If Al-ice now sends Bob the ancilla (perhaps technically easierthan sending the original particle), Bob can reverse heractions to prepare a replica of her original state ~P). This"spin-exchange measurement" [4] illustrates an essentialfeature of quantum information: it can be swapped fromone system to another, but it cannot be duplicated or"cloned" [5]. In this regard it is quite unlike classicalinformation, which can be duplicated at will. The mosttangible manifestation of the nonclassicality of quantuminformation is the violation of Bell s inequalities [6) ob-served [7] in experiments on EPR states. Other rnanifes-tations include the possibility of quantum cryptography[8), quantum parallel computation [9], and the superior-ity of interactive measurements for extracting informa-

1993 The American Physical Society 1895

A non classical transfer of an unknown quantum state

using entanglement.

Sender (Alice) knows neither the state to be teleported

nor the location of the receiver (Bob )

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Teleportation protocol

Alice and Bob initially share a pair of entangled particles

(say 2 & 3).

Alice receives the particle with unknown state (say 1) .

Alice does a joint Bell operator measurement on the

unknown state particle and her entangled particle.

Projective measurement. 1 & 2 gets destroyed due to the

measurement.

Alice sends the outcome of her measurement to Bob

through a classical channel.

Bob does a unitary transformation on his particle (particle

3) with respect to Alice’s measurement results.

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

How teleportation works?

Initially, the unknown state and entangled pair are given by

|φ1〉 = α|0〉+ β|1〉 ; |Ψ−23〉 =1√2

(|01〉 − |10〉) (8)

Total wave function

|Ψ123〉 = 1√2

(α|0〉+ β|1〉)⊗ (|01〉 − |10〉) (9)

It can be written as

|Ψ123〉 = 1√2

(α|00〉12|1〉3 − α|01〉12|0〉3 +

β|10〉12|1〉3 + β|11〉12|0〉3) (10)

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

How teleportation works?

From the Bell states (Eq.3 & Eq.4), we can have

|00〉 = |Φ+〉+|Φ−〉√2

; |11〉 = |Φ+〉−|Φ−〉√2

(11)

|01〉 = |Ψ+〉+|Ψ−〉√2

; |10〉 = |Ψ+〉−|Ψ−〉√2

(12)

Substituting in Eq.10 and rearranging the terms

|Ψ123〉 =1

2{ |Ψ−12〉(−α|0〉3 − β|1〉3) +

|Ψ+12〉(−α|0〉3 + β|1〉3) +

|Φ−12〉(α|1〉3 + β|0〉3) +

|Φ+12〉(α|1〉3 − β|0〉3)

} (13)

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

How teleportation works?

Outcome Unitary operator

Ψ− σ0

Ψ+ σ3

Φ− σ1

Φ+ σ3 σ1

In polarization case

σ0 −→ Free space propagation

σ3 −→ HWP in 00

σ1 −→ HWP in π4

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Quantum circuit for teleportation

Single/double lines :- classical/quantum channels.

HCNOT :- Bell state preparation; CNOT H :- Bell state

projection/detection

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Experimental teleportation

Bouwmeester et al.(Nature 1997) demonstrated quantum

teleportation using photons.

Figure: Experimental teleportation- Bouwmeester et al.(1997)

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Experimental teleportation

Entangled pair :- parametric down converted photons

Bell projection :- beam splitter and detectors

Figure: Experimental teleportation- Bouwmeester et al.(1997)

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Experimental teleportation

Only particles with anti symmetric wave function ( |Ψ−〉)will emerge from both ends of beam splitter (Loudon, R.

Coherence and Quantum Optics VI ).

Coincidence in detectors f1&f2 only when state is |Ψ−12〉.

Unitary operation :- free space propagation.

Initial state is prepared in +45 (-45) polarization states .

ie 1√2

(|H〉 ± |V 〉)

PBS differentiate +45 & -45 polarization. Detector on

each port (d1&d2)

A delay is given in photon 2 path.

Delay 6= 0 - no mixing - f1f2 coincidence 50% - f1f2d1 &

f1f2d2 coincidence 25%

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation

Teleportation results

initial state +45.

Delay 0 - f1f2 coincidence 25% - f1f2d1 coincidence 25% -

f1f2d2 coincidence 0%

Figure: Bouwmeester et al.(1997)The absence of coincidence corresponding to zero delay

confirms the teleportation.

Quantum

Teleportation

:- Theory and

experiment

Chithrabhanu

P

Introduction

Quantum

Teleportation THANK YOU