Multi-Particle Entanglement & It s Application in Quantum Networks

Transcription

1 Lecture Note 5 Multi-Particle Entanglement & It s Application in Quantum Networks

2 Polarization Entangled Photons ( ) ( ) ± = Ψ ± = Φ ± ± H V V H V V H H [P. G. Kwiat et al., Phys. Rev. Lett. 75, 4337 (995).]

3 Post Selection H + V H + V ( ) ( ) H + iv H + iv ( ) ( ) ' ' ' ' ( ) HH VV + i VH + HV ( ) ( ) ( ) ' ' ' ' ' ' ( HH VV) ' ' ' '

4 Initial State: 3-Photon ' ' ' ) ( ' avb Va H b H avb Va H b ( H Four-fold coincidence H H H V or H VV H T 3 T 3 ) Final State: H T ( H H V + V V 3 H 3 )

5 Bell s Inequality and Violation of Local Realism [J. S. Bell, Physics, 95 (964)] Bell s inequality states that certain statistical correlations predicted by QM for measurements on two-particle ensembles cannotbe understood within a realistic picture based on local properties of each individual particle. LR prediction: MAX An unstatisfactory feature! QM prediction: S = S MAX In the derivation of BI such a local realistic and thus classical picture can explain perfect correlations and is only in conflict with statistical prediction of quantum mechanics.

6 conflict with local realism Consider a three-photon GHZ state written in ψ = + ( H H H V V V ) H =, V = denotes the ± eigenstate of σ z respectively 0 σ z basis Linear polarizatoin basis σ : ' ( ) x H = H + V, V ' = ( H V ). circular polarization basis σ : ( ) y R = H + i V, L = ( H i V ).

7 σ σ σ : = = = ( ' ' ' ' RLH LRH RRV LLV ) y y 3x σ σ σ : ( ' ' ' ' RH L LH R RVR LVL ) y x 3y σ σ σ : ψ ψ ψ conflict with local realism ( ' ' ' ' H RL HLR VRR VLL) x y 3 y therefore state ψ is the eigenstate of operators 3 σ σ σ σ σ σ σ σ σ y y 3x y x 3y x y 3y with value -

8 EPR reality criterion: the individual value of any local operator is predetermined. There exists an element of local reality S ix corresponding to σ i =,, 3. operator ( ) ix conflict with local realism All six of the elements of reality S ix and S iy have to be there, each with the values + and! S S S =, y y 3x S S S y x 3y S S S x y 3y =, =.

9 What outcomes are possible? Consider measurement of 45 linear polarization basis local realism S S S = S ( S ) S ( S ) S ( S ) x x 3x x y x y 3x 3y = ( S S S )( S S S )( S S S ) = Possible outcomes: x y 3y y x 3y y y 3y VVV, HHV, HVH, V H H ' ' ' ' ' ' ' ' ' ' ' '

10 quantum physics What outcomes are possible? ψ = Possible outcomes: ( H ' HH ' ' HVV ' ' ' VHV ' ' ' VVH ' ' ' ) HHH S S S =! x x 3 x, HVV, VHV, VVH ' ' ' ' ' ' ' ' ' ' ' ' Whenever local realism predicts a specific result definitely to occur for a measurement for one of the photons based on the results for the other two, quantum physics definitely predicts the opposite result!

11 Experimental Results J.-W. Pan et al., Nature (London) 403, 55 (000)

12 An improved 3- & 4-photon source + + Φ Φ = 34 H H + V V H H + V V ( H H H H V V V V ) ( H V )( H H H V V V ) [J.-W. Pan et al., Rev. Lett. 86, 4435 (00) ]

13 Advanced Quantum Cryptography Quantum secret sharing A procedure for splitting a message into several parts so that no subset of parts is sufficient to read the message, but the entire set is. [M. Hillery et al., Phys. Rev. A 59, 89 (999).] Third-Man Quantum Cryptography A procedure that the third man, a controller, can control whether the users, say Alice and Bob, can communicate in a secure way while he has no access whatsoever on the content of the communication between Alice and Bob. [M. Żukowski et al., Acta Phys. Pol. 93, 87 (998).]

14 Schemes for QSS and TQC ψ = + ( H H H V V V ) abc a b c a b c x y ± ± = = ( H ± V ), ( H ± i V ) A xxx measurement + Ψ = (( x x + x x ) x abc + a + b a b + c ( x+ x a x b x a + b) x c) + +

15 Quantum Secret Sharing = = = = ( ) ( ) ( x x x x x x x x x x ) ψ + a + b a b + c + a b a + b c ( ) ( ) ( y y y y x y y y y x ) a b a b c a b a b c ( ) ( ) ( y x y x y y x y x y ) a b a b c a b a b c ( ) ( ) ( x y x y y x y x y y ) a b a b c a b a b c xxx, xyy, yxy, yyx xyx, yxx, xxy, xyx

16 Setup A ultra-stable high intensity source: four-fold coincidence per second! 00 times brighter! stable for a few months! [Z. Zhao et al., Phy. Rev. Lett (003).

17 Result for QSS From bits of raw key with a QBER of.9%, after security check and error reduction, Alice and Bob jointly generate bits cured key with Charlie with a QBER of 0.3%.

18 Third-Man s Control If all of them randomly select the base to measure the polarization. Any two of them can create a coding by being told the other one s measurement result. If Charlie rejects to tell them his selection or just does not make any selection then Alice and Bob can get nothing useful for the cryptography.

19 Result for TQC With the permission of Charlie, after security check and error reduction Alice can generate a bits cured key with Bob, with the same QBER. Otherwise, Without knowing Charlie's results, the only thing Alice and Bob can do is to randomly guess Charlie's results and continue the same encoding and error reduction procedure. In our experiment, after performing twice error reductions, the QBER remains %. [Y.-A. Chen et al., Phy. Rev. Lett. 95, 0050 (005) ]

20 conflict with local realism in 4-photon case ψ = H V V H + V H H V ( ) σ xσσσ x x x, σσσσ x y x y, σσσσ x x y y σ xσ yσ yσ x σ yσσσ y y y, σσσσ y x y x, σσσσ y y x x σ yσ xσ xσ y σ xσσσ x x x : σx: H' = H + V V = H V σy: R = H + iv L = H iv ψ = ( H ' H ' H ' H ' H ' H ' V ' V ' H ' V ' H ' V ' + H ' V ' V ' H ' ( ) ' ( ) ( ) ( ) + V ' H ' H ' V ' V ' H ' V ' H ' V ' V ' H ' H ' + V ' V ' V ' V ' )

21 Violation of Local Realism [Z. Zhao et al., Phy. Rev. Lett (003).

22 A new protocol for 3-Photon ( H + V ) ( HH + VV ) 3 3 ( HHH VVV ) [J. G. Rarity and P. R. Tapster, Phys. Rev. A 59, R35 (999).]

23 5-Photon ( H + V ) ( HH + VV ) ( HH + ) VV Five-fold Coincidence: ( HHHHH + ) 345 VVVVV

24 Encoding operation for simple quantum error correction Ψ = Ψ Φ [ + = Φ ( α H 3 H 4 H 5+ β V 3 V 4 V 5 ) + Φ ( α H H H β V V V ) Ψ ( α V V V + β H H H ) Ψ ( α V V V β H H H )] This implies that a joint Bell measurement on photons and would project the state of photons 3, 4 and 5 into one of the four corresponding states, which can be used for either one bit-flip error or phase-shift error correction in quantum communication.

25 Quantum State Sharing & Open-destination Teleportation α H + β V α HHH β VVV 345 If we perform a +45- degree measurement on photons 4 and 5, then photon 3 is left in the state of photon. In a similar manner the initial state of photon can also be teleported either onto photon 4 or photon 5. [A. Karlsson, et al., Phys. Rev. A 58, 4394 (998) ]

26 Further Demonstration In contrast to the original teleportation scheme, after the encoding operation the destination of teleportation is left open until we perform a polarization measurement on two of the remaining three photons. Even though photons 3, 4 and 5 are far away from each other, one can still choose which particle should act as the output where the initial state of photon is transferred to. This is why we have called the encoding-decoding procedure as open-destination teleportation. It is therefore a generalization of standard teleportation, when no prior agreement on the final destination of the teleportation is necessary. It is also a generalization of Quantum State Sharing. No subset of parts is sufficient to decode the state, but the entire set is. It broadens the scope of quantum information networks allowing quantum communication between multiple nodes, while providing security against malicious parties in the network as well as node and channel failures.

27 Setup for Five-photon GHZ Entanglement [Z. Zhao et al., Nature 430, 54 (004). ]

28 CNOT operation for twoindependent photons +/-? H/V? ( α H + β V ) ( HV + VH ) ( γ H + δ V ) 34 5 ( α HH V β VV H ) ( γ H δ V ) PBS + + T( H); R( V) ( α HH β VV )( γ δ)( H V )( H V ) PBS T( H+ V); R( H V) Conditional 3 at H+V Conditional 4 at H ( α HH β VV )( γ δ)( H V )( H V ) α H ( γ V + δ H ) + β V ( γ H + δ V )

29 CNOT Gate A full Bell state Measurement for 00% Teleportation Control H Target H Control H Target V [T. B. Pittman, PRA 64,063(00)] H V V V H V H V V H H V [S. Gasparoni et al., Phys. Rev. Lett. 93, (004); Z. Zhao et al., Phys. Rev. Lett. 94, (005).]

30 Most recently... 6-Photon [Q. Zhang et al.,in preparation for Science ]