investigation of high-precision phase estimation in the ... · 1 motivation 2 the quantum...
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Investigation of high-precision phase estimationwith trapped ions in the presence of noise
Sanah Altenburg, S. Wolk and O. Guhne
Department Physik, Universitat Siegen, Siegen, Germany
Bilbao, 23. April 2015
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 1
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Siegen
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 2
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Motivation: Why Quantum Metrology?
Classical setup:
N particles in a classical state,The Standard Quantum Limit(SQL):
(∆ϕ)2 ∝ 1/N
Quantum setup:
N particles in a quantumstate.The Heisenberg Limit (HL):
(∆ϕ)2 ∝ 1/N2
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Motivation: Why Quantum Metrology?
The Cramer-Rao bound
(∆ϕ)2 ≥ 1/FQ [%,Λϕ]
With the tool;Quantum Fisher Information (QFI) FQ [%,Λϕ]
But noise destroys the advantage of using quantum states.
The Question:
For a given set-up, with a given noise model;Which quantum state should be used?
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1 Motivation
2 The Quantum Fisher-Information
3 Quantum Metrology with Ions
4 Metrology with different states
5 Conclusions and next steps
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 5
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1 Motivation
2 The Quantum Fisher-Information
3 Quantum Metrology with Ions
4 Metrology with different states
5 Conclusions and next steps
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 6
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The Quantum Fisher-Information
The QFI is propor-tional to the statisticalvelocity of the evolu-tion of a given state
(∂tDH)2 ∝ F
with the Hellinger Dis-tance DH (distance inprobability space).
Picture from G. Toth
and I. Apellaniz, J. Phys.
A 42, 424006 (2004)
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The Quantum Fisher-Information
Initial state in its eigendecomposition % =∑i
λi |i〉〈i |
Unitary time evolution U = exp [−iθM]
Definition of the Quantum Fisher-Information
F (%,M) = 2∑α,β
(λα − λβ)2
λα + λβ|〈α|M |β〉|2
Example:
The fully mixed state % = 1/d is insensitive to any kind of unitarytime evolution, so that F (%,M) = 0.
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1 Motivation
2 The Quantum Fisher-Information
3 Quantum Metrology with Ions
4 Metrology with different states
5 Conclusions and next steps
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 9
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Quantum Metrology with IonsSetup:
N trapped ions in achain.
Magnetic field~B0 = B0~ez
Dynamics:
Ideal dynamicsH = γB0t︸ ︷︷ ︸
ϕ
Sz
Noise: Magnetic field fluctuations B = B0 + ∆B(t)
η = ϕ+ γ
∫ t
0dτ∆B(τ)︸ ︷︷ ︸δϕ(t)
Estimate ϕ for a given time t = T
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The noise model1
Dynamics
U = exp[−iϕSz ]︸ ︷︷ ︸Uϕ
exp[−iγ∫ T
0dτ∆B(τ)Sz ]︸ ︷︷ ︸
Unoise
The state %0 evolves in a noisy state %T = Unoise%0U†noise
Time correlation function for the magnetic field fluctuations ∆B(τ)
〈∆B(τ)∆B(0)〉 = exp[− ττc
] 〈∆B2〉
Assume Gaussian phase fluctuations with 〈δϕ(T )〉 = 0
Average 〈%T 〉 and calculate the QFI FQ [〈%T 〉 ,Sz ]
For frequency estimation with ϕ = ωT :
FωQ = T 2FϕQ
1T. Monz et al.: 14-Qubit Entanglement: Creation and Coherence, PRL 106 (2011)
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1 Motivation
2 The Quantum Fisher-Information
3 Quantum Metrology with Ions
4 Metrology with different states
5 Conclusions and next steps
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 12
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The product-state
The classical state....
...is a Product-state:
|PN〉 =
(1√2
(|0〉+ |1〉))⊗N
Observations:
For T = 0 we find FQ = N, thisis the SQL.
For T > 0 the QFI decreases.
The more ions the faster theQFI decreases. 0 10 20 30 40 50
T in a.u.
0.0
0.2
0.4
0.6
0.8
1.0
FQ
[%,Sz]/N
N=2N=4N=6
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The GHZ-state
The GHZ-state...
...is defined as
|Ψ〉GHZ =1√2
(|000...0〉+ |111...1〉).
The QFI is
FQ = N2e−N2C(T ).
Solving the master equationleads to
FQ = N2e−N2C(T ).
[F.Frowis et al., New Journal of Phys.8 (2014)]
0 10 20 30 40 50T in a.u.
0
1
2
3
4
5
6
FQ
[%,Sz]/N
N=2N=4N=6
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The sym. Dicke-state
Definition: Dicke-state
|DNsym.〉 ∝
∑j
Pj|0〉⊗N/2 ⊗ |1〉⊗N/2
Rotation around the X -axis: UNx (α) |Dsym.
N 〉
α = 0 α = π/4 α = π/2
x
y
|0〉⊗N
|1〉⊗N
x
y
|0〉⊗N
|1〉⊗N
x
y
|0〉⊗N
|1〉⊗N
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The sym. Dicke-state
For N = 4:
For α = 0, FQ = 0
For α = π/2 and T = 0,FQ = N(N + 2)/2
For α = π/4, somewherein between
For a given T and N, we areable to tell which state givesthe best precision for phaseand frequency estimation.
|0〉⊗N
|1〉⊗N
|0〉⊗N
|1〉⊗N
|0〉⊗N
|1〉⊗N0 5 10 15 20 25 30
T in a.u.
0
1
2
3
4
FQ
[%,Sz]/N
GHZ stateProduct state
9.5 13.5
Details
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Dicke state under any rotation angle α
N = 4
Rotation around the X -axis: UNx (α) |Dsym.
N 〉
xy
|0〉⊗N
|1〉⊗N
xy
|0〉⊗N
|1〉⊗N
xy
|0〉⊗N
|1〉⊗N
0 π4
π2
3π4
α
0.0
0.5
1.0
1.5
2.0
2.5
3.0FQ
[%,Sz]/N
T=0
T=4
T=8
T=12
T=16
T=20
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1 Motivation
2 The Quantum Fisher-Information
3 Quantum Metrology with Ions
4 Metrology with different states
5 Conclusions and next steps
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 18
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Conclusions
Trapped ions in the presence ofnoise
Sym. Dicke state under noise
0 π4
π2
3π4
α
0.0
0.5
1.0
1.5
2.0
2.5
3.0
FQ
[%,Sz]/N
T=0
T=4
T=8
T=12
T=16
T=20
Optimization over states
|0〉⊗N
|1〉⊗N
|0〉⊗N
|1〉⊗N
|0〉⊗N
|1〉⊗N0 5 10 15 20 25 30
T in a.u.
0
1
2
3
4
FQ
[%,Sz]/N
GHZ stateProduct state
9.5 13.5
Details
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 19
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Next steps
Differential interferometry withtrapped Ions
Trapped ions withHamiltonian
H = γ
∫ t
0dτ∆B(τ)(Sz ⊗ Sz)
+γφ(1⊗ Sz).
For photons and atoms seeM. Landini et al., New J. ofPhys. 11, 113074 (2014)
Other states
Investigation of the usefulness ofother Dicke-states, e.g. theW-state.
Collaboration with Ch.Wunderlich (Siegen)
I. Baumgart et al., arXiv:1411.7893(2014)
Trapped ions with Hamiltonian
H = γ
∫ t
0
dτ∆B(τ)Sz
+γ(Ω + ε)tSx .
Optimization for the estimationof ε.
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Acknowledgements
This work was founded by the Friedrich-Ebert-Foundation.
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Thank you for your Attention! Questions?
Trapped ions in the presence ofnoise
Sym. Dicke state under noise
0 π4
π2
3π4
α
0.0
0.5
1.0
1.5
2.0
2.5
3.0
FQ
[%,Sz]/N
T=0
T=4
T=8
T=12
T=16
T=20
Optimization over states
|0〉⊗N
|1〉⊗N
|0〉⊗N
|1〉⊗N
|0〉⊗N
|1〉⊗N0 5 10 15 20 25 30
T in a.u.
0
1
2
3
4
FQ
[%,Sz]/N
GHZ stateProduct state
9.5 13.5
Details
Sanah Altenburg Investigation of high-precision phase estimation in the presence of noise 22