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

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|0012 + β|1112) NS (ρ 1 ⊗ ρ2)
13

14. NE (α|0012 + β|1112)
1
2
Alice
Bob
quantumchannel
Entanglement is a resource.
14

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. 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,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. |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. |α3 →|-iα5
|β2 →|β + iγ6 / √2
|γ5 →|γ + iβ7 / √2
|δ7 →|-iδ8
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

29. 29
2
TF Fidelity

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.

33. Thanks for listening.
33

34. Questions?
34