Swapping between
Two Nonorthogonal
Entangled Coherent States
(and Branching of Measurement Results)
Vasudha Pande
Advisor:...
Outline
1. Nonorthogonal Entangled
Coherent States
2. Entanglement Swapping
3. Conclusion
2
Outline
1. Nonorthogonal Entangled
Coherent States
2. Entanglement Swapping
3. Conclusion
3
Nonorthogonal Entangled Coherent States
qubits
correspond to wave functions
pure or mixed
4
|Ψ = α|0 + β|1
basis states
complex numbers
vector in complex Hilbert Space
superposition over
basis states
5
|Ψ = α|0 + β|1
basis states
complex numberssuperposition over
basis states
vector in complex Hilbert Space
superpositio...
Nonorthogonal Entangled Coherent States
7
can’t be distinguished perfectly with certainty
not completely distinguishable
(...
Nonorthogonal Entangled Coherent States
8
remain coherent with time,
but α evolves as α(t) = α0 e-iωt
minimum uncertainty ...
9





0
22
0
!
†
22
n
A
n
een
n
e 



A|α = α|α, where α is a complex number
eigenstate of annihilation ope...
Nonorthogonal Entangled Coherent States
cause: temporary physical interaction
effect: nonlocal quantum correlation
10
meas...
Nonorthogonal Entangled Coherent States
particles’ wave functions cannot be separated
cause: temporary physical interactio...
entangled state
12
separable state
inseparable vectors of
particles’ Hilbert spaces
measurement
outcomes correlated
measur...
NE (α|0012 + β|1112) NS (ρ 1 ⊗ ρ2)
13
NE (α|0012 + β|1112)
1
2
Alice
Bob
quantumchannel
Entanglement is a resource.
14
NE (α|0012 + β|1112)
Applications:
• superdense coding
• quantum teleportation
• entanglement swapping
• entanglement di...
Outline
1. Nonorthogonal Entangled
Coherent States
2. Entanglement Swapping
3. Conclusion
16
The following material is taken from:
Shivani A. Kumar and Vasudha Pande, Branching of Measurement Results for Swapping
be...
18
Transfer of nonlocal correlations between
quantum systems.
Swapping Protocol: Schematic Diagram
19
|φ 12 = N 12[|α,α12 - z|-α,-α12]
|χ 34 = N34[|2α,α34 - |-2α,-α34]
|Ψ 1234 = |φ 12 |χ 34
initialglobalstate
nonort...
|φ 12 = N 12[|α,α12 - z|-α,-α12]
|χ 34 = N34[|2α,α34 - |-2α,-α34]
|Ψ 1234 = |φ 12 |χ 34
initialglobalstateAlice
B...
|α3 →|-iα5
|β2 →|β + iγ6 / √2
|γ5 →|γ + iβ7 / √2
|δ7 →|-iδ8
22
global state
23
68146814
3412
1468
3,,,3,[
2
 
NN
],3,3,, 68146814
  zz
 ,
2
1
,
2
)1(
0
2...
Final Global State, or Possible Measurement Outcomes
24
Alice
Bob
25
|T' 14
|T 14 26
|T' 14
|T 14 27
|T 14 28
29
2
TF Fidelity
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 
2242...
Outline
1. Nonorthogonal Entangled
Coherent States
2. Entanglement Swapping
3. Conclusion
31
32
We obtain vacuum state only when the number of photons
in both output modes is found to be zero. No swapping or
branchi...
Thanks for listening.
33
Questions?
34
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Swapping between Two Nonorthogonal Entangled Coherent States (and Branching of Measurement Results)

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At Amity Institute of Applied Sciences, Amity University, Noida, India. 23 Aug 2013.

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Swapping between Two Nonorthogonal Entangled Coherent States (and Branching of Measurement Results)

  1. 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. 2. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 2
  3. 3. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 3
  4. 4. Nonorthogonal Entangled Coherent States qubits correspond to wave functions pure or mixed 4
  5. 5. |Ψ = α|0 + β|1 basis states complex numbers vector in complex Hilbert Space superposition over basis states 5
  6. 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. 7. Nonorthogonal Entangled Coherent States 7 can’t be distinguished perfectly with certainty not completely distinguishable (|0 + |1) and (|0 + z|1)
  8. 8. Nonorthogonal Entangled Coherent States 8 remain coherent with time, but α evolves as α(t) = α0 e-iωt minimum uncertainty states
  9. 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. 10. Nonorthogonal Entangled Coherent States cause: temporary physical interaction effect: nonlocal quantum correlation 10 measurement of one particle affects state of the other
  11. 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. 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. 13. NE (α|0012 + β|1112) NS (ρ 1 ⊗ ρ2) 13
  14. 14. NE (α|0012 + β|1112) 1 2 Alice Bob quantumchannel Entanglement is a resource. 14
  15. 15. NE (α|0012 + β|1112) Applications: • superdense coding • quantum teleportation • entanglement swapping • entanglement diversion • quantum cryptography Entanglement is a resource. 15
  16. 16. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 16
  17. 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. 18 Transfer of nonlocal correlations between quantum systems.
  19. 19. Swapping Protocol: Schematic Diagram 19
  20. 20. |φ 12 = N 12[|α,α12 - z|-α,-α12] |χ 34 = N34[|2α,α34 - |-2α,-α34] |Ψ 1234 = |φ 12 |χ 34 initialglobalstate nonorthogonal entangled coherent states 20
  21. 21. |φ 12 = N 12[|α,α12 - z|-α,-α12] |χ 34 = N34[|2α,α34 - |-2α,-α34] |Ψ 1234 = |φ 12 |χ 34 initialglobalstateAlice Bob nonorthogonal entangled coherent states 21
  22. 22. |α3 →|-iα5 |β2 →|β + iγ6 / √2 |γ5 →|γ + iβ7 / √2 |δ7 →|-iδ8 22
  23. 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. 24. Final Global State, or Possible Measurement Outcomes 24
  25. 25. Alice Bob 25
  26. 26. |T' 14 |T 14 26
  27. 27. |T' 14 |T 14 27
  28. 28. |T 14 28
  29. 29. 29 2 TF Fidelity
  30. 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. 31. Outline 1. Nonorthogonal Entangled Coherent States 2. Entanglement Swapping 3. Conclusion 31
  32. 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.
  33. 33. Thanks for listening. 33
  34. 34. Questions? 34

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