Topological phase and quantum criptography Topological phase and quantum criptography

with spin-orbit entanglement of the photonwith spin-orbit entanglement of the photonTopological phase and quantum criptography Topological phase and quantum criptography

with spin-orbit entanglement of the photonwith spin-orbit entanglement of the photon

Universidade Federal FluminenseUniversidade Federal Fluminense

Instituto de Física - Niterói – RJ - BrasilInstituto de Física - Niterói – RJ - Brasil

Antonio Zelaquett KhouryAntonio Zelaquett Khoury

Financial Support: Financial Support: CNPq - CAPES – FAPERJCNPq - CAPES – FAPERJ

INSTITUTO DO MILÊNIO DE INSTITUTO DO MILÊNIO DE

INFORMAÇÃO QUÂNTICAINFORMAÇÃO QUÂNTICA

OutlineOutline

• Geom. phase for a spin ½ in a magnetic field

• Geometric quantum computation

• The Pancharatnam phase• Beams carrying OAM• Topological phase for entangled states

• BB84 QKD without a shared reference frame

• Conclusions

Geometric phase of a spin 1/2 in a magnetic Geometric phase of a spin 1/2 in a magnetic fieldfield

Spin 1/2 in a time dependent magnetic field Spin 1/2 in a time dependent magnetic field

)(ˆ)())(( tBStBtBH

)(tB

| (0) | , (0)u

0

( / ) ( ( ')) '( )| ( ) | , ( )ˆ

t

n

n

i E B t dti tt e e u t

BERRY PHASE

0( ) ( )ˆB t B u t

Geometric quantum computationGeometric quantum computation

Geometric conditional phase gateGeometric conditional phase gate

]ˆ[]ˆ[ 000 BSIIBSH BBABAA

)(ˆ)(ˆ20 tBStBSSSJHH BBAABzAz

i

i

i

i

e

e

e

e

2

2

2

2

000

000

000

000

| | | |Conditional phase gate

J.A. Jones, V. Vedral, A. Ekert, G. Castagnoll,

NATURE V.403, 869 (2000)

L.-M. Duan, J.I. Cirac, P.Zoller

SCIENCE V.292, 1695 (2001)

The Pancharatnam phaseThe Pancharatnam phase

Pancharatnam phasePancharatnam phase

2/g

S. Pancharatnam, Proc. Indian Acad. Sci. Sect. A, V.44, 247 (1956)

Collected Works of S. Pancharatnam, Oxford Univ. Press, London (1975).

2/ 2/

Beams carrying orbital angular momentumBeams carrying orbital angular momentum

Gauss-Laguerre beams carrying OAMGauss-Laguerre beams carrying OAM

2 , )2 0t

ψ (r zψ + ikz

(Paraxial Wave Equation)(Paraxial Wave Equation)(Paraxial Wave Equation)(Paraxial Wave Equation)

0V

s oL r p dv L L

Angular momentumAngular momentum

Hermite-Gauss (HG)Hermite-Gauss (HG)

RectangularRectangular

Laguerre-Gauss (LG)Laguerre-Gauss (LG)

CylindricalCylindrical

Poincaré representation for beams carrying OAMPoincaré representation for beams carrying OAM

Poincaré representation of first order Gaussian modes

Cylindrical lenses at 45o

Astigmatic mode converter

2

1

i 2

1

Geometric phase from astigmatic mode conversionGeometric phase from astigmatic mode conversion

2/g

E.J. Galvez, P.R. Crawford, H.I. Sztul, M.J. Pysher, P.J. Haglin, R.E. Williams,

Physical Review Letters V.90, 203901 (2003)

Topological phase for entangled statesTopological phase for entangled states

C. E. Rodrigues de Souza, J. A. O. Huguenin and A. Z. KhouryC. E. Rodrigues de Souza, J. A. O. Huguenin and A. Z. Khoury

IF-UFFIF-UFF

P. Milman P. Milman

LMPQ – Jussieu - FranceLMPQ – Jussieu - France

Geometric representation for two-qubit statesGeometric representation for two-qubit states

TWO QUBITS

| , |1z

| , y | , x

| , | 0z

| , |1z

| , y | , x

| , | 0z

Two Bloch spheres??

Only for product states!!!

Bloch sphere

(or Poincaré sphere)

| , |1z

| , y | , x

| , | 0z

ONE QUBIT

| | 0 |1

Geometric representation for two-qubit PURE statesGeometric representation for two-qubit PURE states

Bloch ball| 0 0 |

|1 1 |

SO(3) sphere

(opposite points identified)

| | 00 | 01 |10 |11

Two-qubit

PURE STATES

21

2

C

C

(Concurrence)

Maximally entangled state 1 0C Bloch ball colapses to a point!!!!

P. Milman and R. Mosseri, Phys. Rev. Lett. 90, 230403 (2003).

P. Milman, Phys. Rev. A 73, 062118 (2006).

Topological phase for maximally entangled statesTopological phase for maximally entangled states

* *| | 00 | 01 |10 |11

Cyclic evolutions preserveing maximal entanglement (“Closed” trajectories)

Two homotopy classes:

0top

top

0-type trajectories

π-type trajectories

| ( ) | (0)T

| ( ) | (0)T

SO(3) sphere

0

Separable polarization-OAM modesSeparable polarization-OAM modes

0( ) ( ) ( ) ˆ ˆH VE r E r r e e

( )r

( )r

ˆHe

Ve

Nonseparable polarization-OAM modesNonseparable polarization-OAM modes

* *( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆH V H VE r r e r e r e r e

Geometric representation on the SO(3) sphere 1

23

4

0 01 2

0 03 4

( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ2 2

( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ2 2

H V V H

V H H V

E EE r r e r e E r r e r e

iE iEE r r e r e E r r e r e

Nonseparable mode preparationNonseparable mode preparation

Holographic preparation of the LG modesHolographic preparation of the LG modesHolographic preparation of the LG modesHolographic preparation of the LG modes

LG LG LG0-1 00 01

(a)

(b)Laser

2/

PBS

1

E

Interferometric measurementInterferometric measurement

1

23

4

4’

1 2 3 4

4 1 (θ = 00) / 4’ 1 (θ = 900)41

/ 2 / 2 / 2

/ 4

/ 4

CCDθ = 45 0 θ = - 45 0θ = 0 0

θ = 0 0

θ = 0 0, 22.5 0, 45 0, 67.5 0, 90 0

Experimental resultsExperimental results

Unseparable mode

Separable mode

θ=00 θ=22.50 θ=450 θ=67.50 θ=900

θ=00 θ=22.50 θ=450 θ=67.50 θ=900

Theoretical expressionsTheoretical expressions

Unseparable mode

2 2( ) ( ) 2 ( ) 1 cos2 cos sin 2 sin 2 siniqyE r e E r r qy qy

2 2( ) ( ) 2 ( ) 1 cos2 cosiqyE r e E r r qy

Separable mode

Calculated imagesCalculated images

Unseparable mode

Separable mode

Partial separability and concurrencePartial separability and concurrence

Partially separable mode

0( ) ( ) 1 ( )ˆ ˆ

H VE r E r e r e

Interference pattern (θ=450)

2( ) 2 ( ) 1 2 (1 ) sin 2 sin

I r r qy

CONCURRENCE

BB84 Quantum key distribution without a BB84 Quantum key distribution without a shared reference frameshared reference frame

C. E. Rodrigues de Souza, C. V. S. Borges, C. E. Rodrigues de Souza, C. V. S. Borges,

J. A. O. Huguenin and A. Z. KhouryJ. A. O. Huguenin and A. Z. Khoury

IF-UFFIF-UFF

L. Aolita and S. P. Walborn L. Aolita and S. P. Walborn

IF-UFRJIF-UFRJ

The BB84 protocolThe BB84 protocol

0

0

1

1

0

0

1

1

ALICE

Bennett and Brassard

1984Polarizers

HV

+/-

HV

+/-

Polarizers

BOB

H - 45o45oV

Photons

010111100Result

HVHVHV+/-HV+/-+/-+/-HVBasis

000111101Result

HV+/-HVHVHV+/-HV+/-+/-Basis

0 1 1 0 0

Alice and Bob check their basis, but not their results !

ALICE

BOB

Spin-orbit entanglementSpin-orbit entanglement

1

2[ ]

L 0L 1L

Logic basis +/-

Logic basis 0/1

1

2[ ]1L

1

2[ ]

1

2[ ]0L 1

2[ ]

Invariant under rotations ! ! ! !

L. Aolita and S. P. Walborn

PRL 98, 100501 (2007)

BB84 without frame alignmentBB84 without frame alignment

BASIS BASIS

{ 0L 1L },

{ L L },

{ 0L 1L },

{ L L },Photons

0L 1L, L L,,

Robust against alignment noise ! ! ! !

ALICE BOB

Procedure sketchProcedure sketch

??

0L 1LL L

0L 0L

0 1

+ -

BOB

CNOTXX

R(φ)

ALICE

R(θ)

Experimental setupExperimental setup

Experimental resultsExperimental results

Bob’s detector 1

State sent by Alice

Bob’s detector 0

Rotation of Alice’s setup

Bob’s detector 1Alice sends 1

Bob’s detector 0

{ 0L 1L },

Bob`s detection basis:

ConclusionsConclusions

ConclusionsConclusions

• Spin-orbit entanglement

• Topological phase for spin-orbit transformations

• Potential applications to conditional gates

• Quantum criptography without frame alignment