random walks Andre Kochanke MaxPlanckInstitute of Quantum Optics 7272011 Motivation 2 Motivation 3 Overview Density matrix formalism Randomness in quantum ID: 136170
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Slide1
Quantum random walks
Andre Kochanke
Max-Planck-Institute of Quantum Optics
7/27/2011Slide2
Motivation
2Slide3
Motivation
3
?
?
?
?Slide4
Overview
Density matrix formalismRandomness
in quantum mechanics
Transition from classical to quantum
walksExperimental realisation
4Slide5
Density matrix approach
Two state system
5
-1
1
0Slide6
Density matrix approach
Two state systemDensity operator
6
0
-1
1Slide7
Density matrix approach
Density operator
7
Pure
state
Mixed
state
0
-1
1Slide8
Galton box
8
Binomial distributionSlide9
Galton box
Statistical mixtureFirst four
steps9Slide10
Quantum analogy
Used Hilbert space
Specify subspaces10
0
-1
-2
-3
1
2
3Slide11
Quantum analogy
Evolution with
shift and coin operators 11
0
-1
-2
-3
1
2
3Slide12
Quantum analogy
Evolution with
shift and
coin operators 12
0
-1
-2
-3
1
2
3Slide13
Quantum analogy
Evolution with
shift and
coin operators 13
0
-1
-2
-3
1
2
3Slide14
Quantum analogy
State transformation
Density matrix transformation
14Slide15
Quantum analogy
15Slide16
Quantum analogy
16
Positionpc
pq
Variances
pcpq
p
c
p
q
Position
Position
100
stepsSlide17
Phase
shift
Transformed density matrix
Average
Decoherence
effect
Decoherence
17Slide18
Different
realisations
C. A. Ryan et al
.
,
“Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”,
PRA 72, 062317 (2005)
M.
Karski
et al
.
,
“Quantum Walk in Position Space with Single Optically Trapped Atoms”,
Science 325, 174 (2009)
A. Schreiber et al
.
,
“Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”,
PRL 104, 050502 (2010)
F.
Zähringer
et al
.
,
“Realization
of a
Quantum Walk
with One and Two Trapped
Ions”,
PRL 104, 100503 (2010)
18Slide19
Setup
19
CCD
Microwave
Dipole
trap laser
Objective
Fluorescence picture
Cs
Microwave
M.
Karski
et al
.
,
Science 325, 174 (2009
)Slide20
Setup
20
Polarizations
and
Slide21
Setup
21
Polarizations
and
Slide22
Results
22
M.
Karski
et al
.
,
Science 325, 174 (2009
)Slide23
Results
23
M. Karski
et al.,
Science 325, 174 (2009)
Theoretical expectation
6
stepsSlide24
Results
24
Theoretical expectation
M.
Karski
et al
.
,
Science 325, 174 (2009
)
6
stepsSlide25
Results
25
Theoretical expectationSlide26
Results
26
Theoretical expectation
M.
Karski
et al
.
,
Science 325, 174 (2009
)Slide27
Results
27
Gaussian fit
M.
Karski
et al
.
,
Science 325, 174 (2009
)Slide28
Conclusion
The density matrix formalism allows you to describe cassical and quantum behavior
Karski et al. showed how to prepare a quantum walk with delocalized atomsThe quantum random walk is not random at all
28
M.
Karski et al
.
,
Science 325, 174 (2009
)Slide29
29Slide30
References
C. A. Ryan et al., “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor”,
PRA 72, 062317 (2005)M. Karski et al.
, “Quantum Walk in Position Space with Single Optically Trapped Atoms”, Science 325, 174 (2009)
SOM for “Quantum Walk in Position Space with Single Optically Trapped Atoms”,
Science 325, 174 (2009)
A. Schreiber et al.
,
“Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations”,
PRL 104, 050502 (2010
)
F.
Zähringer
et al.
,
“Realization of a
QuantumWalk with One and Two Trapped Ions”, PRL 104, 100503 (2010)
M. Karksi, „State-selective transport of
single neutral atoms”, Dissertation, Bonn (2010)C. C. Gerry
and P. L. Knight, „Introductory Quantum
Optics“, Cambridge University Press, Cambridge (2005)M. A. Nielsen
and
I. A.
Chuang
,
„Quantum
Computation
and
Quantum Information“
, Cambridge University Press, Cambridge (2000)
30