Swapping between Two Nonorthogonal Entangled Coherent States (and Branching of Measurement Results)
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This document describes entanglement swapping between two pairs of nonorthogonal entangled coherent states. It outlines how entanglement swapping works by having the first pair interact with the second pair via a series of quantum measurements. This transfers the entanglement between the original pairs to new pairs. The results of the measurements can branch into different outcomes depending on the photon densities of the initial states. Specifically, if the densities are different, the measurement outcomes will split into branches with unique fidelities rather than remaining identical. This demonstrates how entanglement swapping between nonorthogonal states can lead to branching of the measurement results.
![|φ 12 = N 12[|α,α12 - z|-α,-α12]
|χ 34 = N34[|2α,α34 - |-2α,-α34]
|Ψ 1234 = |φ 12 |χ 34
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nonorthogonal entangled coherent states
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![|φ 12 = N 12[|α,α12 - z|-α,-α12]
|χ 34 = N34[|2α,α34 - |-2α,-α34]
|Ψ 1234 = |φ 12 |χ 34
initialglobalstateAlice
Bob
nonorthogonal entangled coherent states
21](https://image.slidesharecdn.com/5-140310052014-phpapp01/85/Swapping-between-Two-Nonorthogonal-Entangled-Coherent-States-and-Branching-of-Measurement-Results-21-320.jpg)
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Now Alice and Bob both possess an entangled pair each.
Alice makes a measurement on states 6 and 8, which effectively
amounts to rewriting them using the expression:](https://image.slidesharecdn.com/5-140310052014-phpapp01/85/Swapping-between-Two-Nonorthogonal-Entangled-Coherent-States-and-Branching-of-Measurement-Results-23-320.jpg)
Swapping between Two Nonorthogonal Entangled Coherent States (and Branching of Measurement Results)
- 1. Swapping between Two Nonorthogonal Entangled Coherent States (and Branching of Measurement Results) Vasudha Pande Advisor: Dr. Shivani A. Kumar Course: M.Sc. AP (Sem-III) 23 August 2013
- 2. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 2
- 3. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 3
- 4. Nonorthogonal Entangled Coherent States qubits correspond to wave functions pure or mixed 4
- 5. |Ψ = α|0 + β|1 basis states complex numbers vector in complex Hilbert Space superposition over basis states 5
- 6. |Ψ = α|0 + β|1 basis states complex numberssuperposition over basis states vector in complex Hilbert Space superposition of eigenstates of an observable measurement leads to collapse probabilistic outcome destructive process 6
- 7. Nonorthogonal Entangled Coherent States 7 can’t be distinguished perfectly with certainty not completely distinguishable (|0 + |1) and (|0 + z|1)
- 8. Nonorthogonal Entangled Coherent States 8 remain coherent with time, but α evolves as α(t) = α0 e-iωt minimum uncertainty states
- 9. 9 0 22 0 ! † 22 n A n een n e A|α = α|α, where α is a complex number eigenstate of annihilation operator energy eigenstates normalisation constant
- 10. Nonorthogonal Entangled Coherent States cause: temporary physical interaction effect: nonlocal quantum correlation 10 measurement of one particle affects state of the other
- 11. Nonorthogonal Entangled Coherent States particles’ wave functions cannot be separated cause: temporary physical interaction effect: nonlocal quantum correlation coupling of quantum systems monogamous representation independent 11 unaffected by spatial separation measurement of one particle affects state of the other
- 12. entangled state 12 separable state inseparable vectors of particles’ Hilbert spaces measurement outcomes correlated measurement outcomes uncorrelated mixture of product of particles’ states
- 13. NE (α|0012 + β|1112) NS (ρ 1 ⊗ ρ2) 13
- 14. NE (α|0012 + β|1112) 1 2 Alice Bob quantumchannel Entanglement is a resource. 14
- 15. NE (α|0012 + β|1112) Applications: • superdense coding • quantum teleportation • entanglement swapping • entanglement diversion • quantum cryptography Entanglement is a resource. 15
- 16. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 16
- 17. The following material is taken from: Shivani A. Kumar and Vasudha Pande, Branching of Measurement Results for Swapping between Two Nonorthogonal Entangled Coherent States. (In press: World Journal of Science & Technology Research, August 2013.) 17
- 18. 18 Transfer of nonlocal correlations between quantum systems.
- 19. Swapping Protocol: Schematic Diagram 19
- 20. |φ 12 = N 12[|α,α12 - z|-α,-α12] |χ 34 = N34[|2α,α34 - |-2α,-α34] |Ψ 1234 = |φ 12 |χ 34 initialglobalstate nonorthogonal entangled coherent states 20
- 21. |φ 12 = N 12[|α,α12 - z|-α,-α12] |χ 34 = N34[|2α,α34 - |-2α,-α34] |Ψ 1234 = |φ 12 |χ 34 initialglobalstateAlice Bob nonorthogonal entangled coherent states 21
- 22. |α3 →|-iα5 |β2 →|β + iγ6 / √2 |γ5 →|γ + iβ7 / √2 |δ7 →|-iδ8 22
- 23. global state 23 68146814 3412 1468 3,,,3,[ 2 NN ],3,3,, 68146814 zz , 2 1 , 2 )1( 0 2 ODD x NZE x x Now Alice and Bob both possess an entangled pair each. Alice makes a measurement on states 6 and 8, which effectively amounts to rewriting them using the expression:
- 24. Final Global State, or Possible Measurement Outcomes 24
- 25. Alice Bob 25
- 26. |T' 14 |T 14 26
- 27. |T' 14 |T 14 27
- 28. |T 14 28
- 30. 30 2224 2 12 2 *)1()1(*)1)(1( 4 zxzx NN F I I 2224 22 12 )1)(1()1)(1()1( 4 zxzxx NN FFF II VIIIVIIII 22424 22 12 )*1)(1()*1)(1( 4 zzzxzzzx NN FFFFFF III XVIIXIIIIXVIVIII 22422 12 )1)(1( zxNNF IVIV 22424 22 12 )*1)(1()*1)(1( 4 zzzxzzzx NN FF X XVX 224 22 12 )1)(1( 4 zx NN FF XI XIVXI 2 2224 22 12 )1)(1()*1)(1()1( 4 zxzzxx NN FF XII XVIXII Fidelity
- 31. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 31
- 32. 32 We obtain vacuum state only when the number of photons in both output modes is found to be zero. No swapping or branching of measurement result is observed in this case. A difference in photon densities of initial quantum states causes the expected measurement results to split into outcomes with unique fidelities. Some of these branches may regroup and share the same fidelities. However, the new distribution is not identical to the one we would expect for initial states with same photon density.
- 34. Questions? 34