About the use of fidelity the access quantum resources Antonio Mandarino Supervisor: Matteo G. A. Paris Phd School in Physics, Astrophysics and Applied Physics End Year Seminar
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About the use of fidelity the access quantum resources
Antonio MandarinoSupervisor : Matteo G. A. Paris
Phd School in Physics, Astrophysics and Applied Physics
End Year Seminar
Quantum Teleportation
⇢ 6=X
pk⇢Ak ⌦ ⇢Bk
| i = 1p2[|0iA|1iB � |1iA|0iB ]
Uhlmann Fidelity
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
If the input state is pure
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
If the input state is pure
F (|�i, ⇢2) = |h�|⇢2|�i|2
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
If the input state is pure
F (|�i, ⇢2) = |h�|⇢2|�i|2
F = 0
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
If the input state is pure
F (|�i, ⇢2) = |h�|⇢2|�i|2
F = 0 The states do not overlap
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
If the input state is pure
F (|�i, ⇢2) = |h�|⇢2|�i|2
F = 1
F = 0 The states do not overlap
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
If the input state is pure
F (|�i, ⇢2) = |h�|⇢2|�i|2
F = 1 The states belong to the same ray F = 0 The states do not overlap
F (⇢1, ⇢2) =
✓Tr
qp⇢1⇢2
p⇢1
◆2
Uhlmann Fidelity
If the input state is pure
F (|�i, ⇢2) = |h�|⇢2|�i|2
F = 1 The states belong to the same ray F = 0 The states do not overlap
In cauda venenum
A Brain-Teaser about fidelity
A Brain-Teaser about fidelity
Does a high value of fidelity imply that the two states are very close in the Hilbert space?
A Brain-Teaser about fidelity
Does a high value of fidelity imply that the two states are very close in the Hilbert space?
1�pF (⇢1, ⇢2)
1
2k⇢1 � ⇢2k1
p1� F (⇢1, ⇢2)
A Brain-Teaser about fidelity
Does a high value of fidelity imply that the two states are very close in the Hilbert space?
1�pF (⇢1, ⇢2)
1
2k⇢1 � ⇢2k1
p1� F (⇢1, ⇢2)
DB =p
2[1� F (⇢1, ⇢2)]
If two states have a fidelity close to unit should we conclude that they share very similar physical properties ?
A Brain-Teaser about fidelity
Does a high value of fidelity imply that the two states are very close in the Hilbert space?
1�pF (⇢1, ⇢2)
1
2k⇢1 � ⇢2k1
p1� F (⇢1, ⇢2)
DB =p
2[1� F (⇢1, ⇢2)]
If two states have a fidelity close to unit should we conclude that they share very similar physical properties ?
We found examples where this is not quite true if “fidelity close to unit” is intended to mean the usual values 0.9, 0.99 or so.
A Brain-Teaser about fidelity
Does a high value of fidelity imply that the two states are very close in the Hilbert space?
1�pF (⇢1, ⇢2)
1
2k⇢1 � ⇢2k1
p1� F (⇢1, ⇢2)
DB =p
2[1� F (⇢1, ⇢2)]
0.6 0.7 0.8 0.9 1.0 m
0.5
1.5
2.0
s
m = 0.9s = 1
m = 0.7s = 1.6
m = 0.7s = 0.6
F > 0.99
⇢sµ = S(r)⌫(N)S†(r)
S(r) = exp {r(a†2 � a2)}
s = e2r
µ =1
2N + 1
⌫(N) =Na†a
(1 +N)a†a
Single-mode Gaussian State
Region of classicality regionSingular Glauber P-function
0.6 0.7 0.8 0.9 1.0 m
0.5
1.5
2.0
s
m = 0.9s = 1
m = 0.7s = 1.6
m = 0.7s = 0.6
F > 0.99
0.6 0.7 0.8 0.9 1.0 m
0.5
1.5
2.0
s
m = 0.9s = 1
m = 0.7s = 1.6
m = 0.7s = 0.6
F > 0.99
Is a cure achievable?
mean photon number differing at most 10%
&photon number fluctuations differing at most 10%
mean photon number differing at most 10%
0.6
0.8
1.
m0.81.21.62
s
0.5
1.
1.5
2.
a
R =h�n2i
n< 1
⇢G = D(x)⇢sµD†(x)
D(x) = exp{↵a† � ↵̄a}
µ = 0.9 s = 1.4 ↵ = 0.5
µ = 0.7 s = 1.2 ↵ = 1.5
Variation on a theme
superPoissonian Target State
subPoissonian Target State
Fano Factor
F > 0.97
1.0 1.5 2.0 2.5s
0.8
1.0
1.2
1.4
1.6
1.8
2.0a
s = 1.5a = 1.5
s = 1.0a = 0.8
F > 0.97m = 0.8
mean photon number differing at most 10%
&photon number fluctuations differing at most 10%
mean photon number differing at most 10%
0.0
0.1
0.2
0.3
b
0.00.5
1.0g
0 1 23N
N = 2.5 � = 0.2 � = 0.5
N = 1 � = 0.13 � = 0.5
Two-mode Gaussian State
Entangled Target StateSeparable Target State
⇢N�� = S2(r)⌫1(N1)⌦ ⌫2(N2)S†2(r)
� =N1
N1 +N2
N = N1 +N2 + 2(1 +N1 +N2)Ns
� = Ns/N
S2(r) = exp {r(a†b† � ab)}
F > 0.99
Separability Region (PPT Criterion)
N = 2 � = 0.2 � = 0.5 N = 2 � = 1
Fidelity And Discord
F > 0.95 0.95 < F < 0.99
Conclusion And Outlooks
Conclusion And Outlooks
Two states with a fidelity “close” to unit may not share the same physical properties
Conclusion And Outlooks
Two states with a fidelity “close” to unit may not share the same physical properties
Fidelity is a quantity that should be employed with more caution to assess quantum resources
Conclusion And Outlooks
Two states with a fidelity “close” to unit may not share the same physical properties
Fidelity is a quantity that should be employed with more caution to assess quantum resources
Which is the minimal set of quantities to be specified in order to certify quantum resources?
Thanks for your attention
If you would like to know more of it, please ask questions in the next 5 mins or feel free to come upstair (5th floor) or take
a look at arXiv:quant-ph/1309.5325
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