Quantum Dense coding and Quantum Teleportation

Quantum Dense coding and Quantum Teleportation

Lecture Note 3Quantum Dense coding andQuantum TeleportationJian-Wei Pan2006.5.17

| Ψ+〉12| Φ+〉Bell states – maximally entangled states:121=2=12( | H〉| H〉+ | V 〉 | V 〉 )12Dense Coding+( | H〉1| V 〉2+ | V 〉1| H〉2) = ˆ σ1x| Φ 〉1212| Φ| Ψ−−〉〉12121=21=2( | H 〉 | H 〉 − | V 〉 | V 〉 )1211z+( | H〉1| V 〉2− | V 〉1| H 〉2) = −iˆ σ1y| Φ 〉122= ˆ σ| Φ+〉12Theory:[C. H. Bennett & S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992)]

Dense Coding1. Alice and Bob share an entangled photon pair in the+state of Φ .2. Bob chooses one of the four unitary transformation{ I,σ }z, σ x, σ y on his photon. The information of whichchoice is 2 bit.3. Bob sends his photon to Alice.4. Alice does a joint Bell-state measurement on thephoton from Bob and her photon.5. With the measurement result, she can tell Bob’sunitary transformation and achieve the 2 bitinformation.

Teleportation Classical PhysicsScanning and Reconstructing Quantum PhysicsHeisenberg Uncertainty Principle Forbidden Extracting All the Informationfrom An Unknown Quantum State

Quantum Teleportation--- “Six Author Scheme”ψ±±φ==1( 021( 0210±±110 )1 )Bell states – maximally entangled statesΦ= α 0 + β 1A AΨ ABC ==+++A2( α + β = 1)Φ A ⊗ Φ +BCΦ + ⊗ ( α 0 C + β 1 )ABCΦ −AB⊗ ( α 0 C − β 1 C )Ψ + ⊗ ( α 1 C + β 0 C )ABΨ −AB⊗ ( α 1 C − β 0 C )2[C.H. Bennett et al., Phys. Rev. Lett. 73, 3801 (1993)]

Teleportation of entanglement----Entanglement Swapping| Ψ〉1234==+| Φ 〉+| Φ 〉+| Ψ 〉141212+⊗|Φ 〉+⊗|Φ 〉+⊗|Ψ 〉233434−+ | Φ 〉−+ | Ψ 〉1212⊗|Φ 〉−34−34⊗|Ψ 〉+[M. Zukowski et al., Phys. Rev. Lett. 71, 4287 (1993)]

SPDC—time bin entanglementΨ=12(short short +ablongalongb)[J. Brendel et al., Phys. Rev. Lett. 82, 2594 (1999)]

SPDC—momentum entanglementΨ=12(u +d d u1 2 12)[Z. B. Chen et al., Phys. Rev. Lett. 90, 160408 (2003)]

SPDC—polarization entanglement| Φ| Ψ±±〉〉1212==1212( | H〉| H〉± | V 〉 | V 〉 )1( | H〉| V 〉 ± | V 〉 | H〉)1221122[P. G. Kwiat et al., Phys. Rev. Lett. 75, 4337 (1995)]

Bell state analyzer with Linear OpticsControlled-NOT gateHH → HV , HV → HHVH →VH,VV →VV( H + V ) H → HV + VH| Φ| Ψ| Ψ±+−〉〉〉121212===121212[H. Weinfurter, Eruophys. Lett. 25, 559 (1995)][J.-W. Pan et al., Phys. Rev. A (1998)]( | H〉| H〉± | V 〉 | V 〉 )1( | H〉| V 〉 + | V 〉 | H 〉 )1( | H 〉 | V 〉 − | V 〉 | H〉)1222111222

Bell state analyzer with Linear Optics• To realize a Bell state measurement,one has to make the two inputSPDC photons indistinguishable on the BS. The method is to makethe two photons spatially and temporally overlapped on the BSperfectly.• However, the timing overlap is difficult since the SPDC photon has aultrasmall coherent time about 100fs, definitely shorter than the timeresolution of state-of-the-art single photon detector.• What we can do is to scan the interference fringes to make it surethat the two photons arrive at the same time. But a cw laser will…No interference fringes

Bell state analyzer with Linear Optics• A solution is to use Pulse laserThe pulse will bring some time jitter to the SPDC photon, we insert anarrow band filter can extend the coherent time of the SPDC photon

Experimental Dense Coding0 ≡1 ≡2 ≡ΦΨΨ−+−= ˆ= ˆ •= ˆ ◦[K. Mattle et al., Phys. Rev. Lett. 76, 4656 (1996)]

Experimental Realizations of quantumteleportationD. Bouwmeester, et al., Nature 390, 575-579 (1997) (photons)D. Boschi, et al., Phys. Rev. Lett. 80, 1121-1125 (1998) (photons).J-W. Pan, et al., Phys. Rev. Lett. 80, 3891–3894 (1998) (mixed state ofphotons).A. Frusawa, et al., Science 282, 706 (1999) (continuous-variable)M. Riebe, et al., Nature 429, 734-737 (2004) (trapped calcium ions).M.D. Barret, et al., Nature 429, 737-739 (2004) (trapped beryllium ions)I. Marcikic, et al., Nature 421, 509-513 (2003) (long distance)R. Ursin, et al., Nature 430, 849 (2004) (long distance)Z. Zhao, et al., Nature 430, 54 (2004) (open destination teleportation)

Experimental Quantum TeleportationThe setup[D. Bouwmeester et al., Nature 390, 575 (1997)]

Experimental Quantum TeleportationThe result[D. Bouwmeester et al., Nature 390, 575 (1997)]

Experimental Entanglement SwappingThe setupThe result[J.-W. Pan et al., Phys. Rev. Lett. 80, 3891 (1998)]

A Two-Particle Quantum teleportationexperiment1. Sharing EPR Pairψ = α H + β V1 111Ψ + = ( H V + V H )1 2 1 221→ ( a1a2+ b1b2) H22. Initial State Preparation1V2[D. Boschi et al., Phys. Rev. Lett. 80, 1121 (1998)]3. BSMΦ1= ( a1a2+ b1b2) ψ V1 221= [ a1V + b1H )( β a11 2+ α b2+ ( a V − b H )( β a −αb++aa111HV111+−bb111VH111)( α a)( α a222+ β b− β b2222) V) V) V) V222]2

Applications of Entanglement SwappingQuantum telephone exchangeSpeed up the distribution of entanglement[S. Bose et al., Phys. Rev. A 57, 822 (1998)]

Applications of Entanglement SwappingBSME( N)⊗ E(3)⎯⎯→E(N + 1) ⊗ E(2)[S. Bose et al., Phys. Rev. A 57, 822 (1998)]

Open-Destination TeleportationSharing a secret quantum stateof single particles| Ψ〉V 〉| Ψ〉1= α | H〉1+β |12345==1| Ψ〉1| Φ〉23451 +[| Φ 〉12( α | H〉2−+ | Φ 〉 ( α | H〉+ | Ψ+ | Ψ+−〉〉121212( α | V 〉( α | V 〉3333||H〉H〉| V 〉| V 〉4444||H〉| V 〉| V 〉H〉5555+ β | V 〉− β | V 〉+ β |− β |H〉H〉3333||| V 〉| V 〉H〉H〉4444| V 〉||| V 〉H〉H〉[A. Karlsson et al., Phys. Rev. A 58, 4394 (1998)][R. Cleve et al., Phys. Rev. Lett. 83, 648 (1999)]5555))))]

Experimental Setup

Experimental Results[Z. Zhao et al., Nature 430, 54 (2004)]

Teleportation of a composite system----Setup

StateTeleportation of a composite system----ResultFidelityFidelity withnoise reductionFidelity:HVHV-VH0.86±0.030.60±0.030.97±0.030.71±0.03Pure stateF=iniψψTel2(H+V)(H-iV)0.75±0.020.83±0.02Mixed stateF = Tr( ρ ψ ψ )iniAverage0.74±0.030.84±0.03Well beyond theclone limit 0.40[A. Hayashi et al., Phys. Rev. A. 72, 032325 (2005).]Teleported State of HV-VH is Still Entangled!Tr(ρW)1= Tr[ρ(HH2+ − − − − −HHRL= −0.23± 0.04 < 0+RLVV−LRVV[M. Barbieri, et al. Phys. Rev. Lett. 91, 227901 (2003).]+LR+ +)]+ +

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