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