bound states and recurrence properties of quantum walks
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Bound States and Recurrence Properties of Quantum Walks
Albert H. Werner
Joint work with:
Andre Ahlbrecht, Christopher Cedzich,
Volkher B. Scholz (now ETH), Reinhard F. Werner (Hannover)
Andrea Alberti & Dieter Meschede (Bonn)
Alberto F. Grünbaum (Berkeley)
Luis Velázquez (Zaragoza)
Autrans
18.07.2013
What are Quantum Walks?
a single particle
with internal degree
of freedom
A. H. Werner
Dynamics of
01
What are Quantum Walks?
a single particle
with internal degree
of freedom
on a lattice
A. H. Werner
Dynamics of
01
What are Quantum Walks?
a single particle
with internal degree
of freedom
on a lattice
in discrete timesteps
A. H. Werner
Dynamics of
01
What are Quantum Walks?
a single particle
with internal degree
of freedom
on a lattice
in discrete timesteps
strictly local
A. H. Werner
Dynamics of
01
Why?
Step towards quantum-simulators
Simulation of lattice systems in discrete time steps
Simulation of one particle-effects
Quantum Biology
“quantization” of random walks
Searching in graphs
Quantum computer
02
source: ucm.es
source: wikipedia.org
A. H. Werner
phase space of
trapped ions
Experimental Realisations
wave guide arrays
optical fibres
atom in
optical lattice
03
source: iap.uni-bonn.de/
source: Peruzzo et al. (2010) source: Matjeschk et al. (2012)
source: Schreiber et al. (2011)
A. H. Werner
Outline
Propagation properties
Bound states in interacting quantum walks
Recurrence properties
04 A. H. Werner
Outline
Propagation properties
Bound states in interacting quantum walks
Recurrence properties
04 A. H. Werner
1D Example: Coined Quantum Walk
Basis:
Walk operator:
Time evolution:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
Basis:
Walk operator:
Time evolution:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
Basis:
Walk operator:
Time evolution:
Coin operator:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
U
Basis:
Walk operator:
Time evolution:
Coin operator:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
U
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
U
S
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
U
S
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
U
S
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 05 A. H. Werner
1D Example: Coined Quantum Walk
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 05 A. H. Werner
Ballistic scaling Diffusive scaling
Asymptotic position distribution
Position observable
Characteristic function of
Find minimal for the existence of
07 A. H. Werner
Propagation properties
translation invariance
co
here
nce
Ballistic transport
Anderson localisation
Diffusive transport
Diffusive transport
09 A. H. Werner
Outline
Propagation properties
Bound states in interacting quantum walks
Recurrence properties
10 A. H. Werner
Outline
Propagation properties
Bound states in interacting quantum walks
Recurrence properties
10 A. H. Werner
1D Example: Coined Quantum Walk
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 11 A. H. Werner
1D Example: Coined Quantum Walk
Basis:
Walk operator:
Time evolution:
Coin operator:
Shift operator:
Hilbert space: 11 A. H. Werner
Interacting Quantum Walks
Two particles on the line
Free evolution:
Projection on collision space:
Interaction on collision:
Coin on collision:
12 A. H. Werner
Example: Interacting Hadamard Walk
Two particles on the line
Free evolution:
Projector on collision space:
Interaction on collision:
Interaction phase:
Initial state:
Walk preserves symmetric/antisymmetric subspaces
12 A. H. Werner
Fourier Description II
Write Walk operator in terms of and
Conserved by
translation invariance.
External parameter
15 A. H. Werner
Fourier Description II
Write Walk operator in terms of and
Conserved by
translation invariance.
External parameter
Walk in this variable is perturbed
by on subspace of -
constant functions
15
Family of 1D QWs with perturbation
at the origin indexed by
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Interacting Hadamard Walk
Compare with
free band structure
depth=p1-p2
Jointly diagonalize and
on a ring of length L
L=8 L=32
A. H. Werner
Quantum Walk with Point Perturbation
projection onto the subspace
Finite rank perturbation essential spectrum unchanged
Look for eigenvalues
independent of
Look for eigenvalues in band gap of
Consistency condition:
16 A. H. Werner
Interacting Quantum Walks
17 A. H. Werner
Result: For all values in the band gap, there is an
interaction such that is an eigenvalue of .
The corresponding eigenvectors satisfy
Example: Interacting Hadamard Walk
Two particles on the line
Free evolution:
Projection on collision space:
Interaction on collision:
Interaction phase:
Initial state:
Walk preserves symmetric/antisymmetric subspaces
17 A. H. Werner
Interacting Hadamard Walk
Result:
Explicit formula for quasi-energy of the bound state.
A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. Werner New J. Phys. 14 (2012)
Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012)
A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák ,V.Potoček, C.Hamilton ,I.Jex, C.Silberhorn Science (2012)
A. H. Werner
18
Interacting Hadamard Walk
Result:
Explicit formula for quasi-energy of the bound state.
Effective theory of molecule as QW
A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. WernerNew J. Phys. 14 (2012)
Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012)
A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák ,V.Potoček, C.Hamilton ,I.Jex, C.Silberhorn Science (2012)
A. H. Werner
18
Interacting Hadamard Walk
Result:
Explicit formula for quasi-energy of the bound state.
Effective theory of molecule as QW
A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. WernerNew J. Phys. 14 (2012)
Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012)
A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák ,V.Potoček, C.Hamilton ,I.Jex, C.Silberhorn Science (2012)
A. H. Werner
18
Interacting Hadamard Walk
Result:
Explicit formula for quasi-energy of the bound state.
Effective theory of molecule as QW
Molecule exponentially localized
A.Ahlbrecht, A. Alberti, D.Meschede, V.B.Scholz, AHW, R.F. WernerNew J. Phys. 14 (2012)
Y.Lahini, M.Verbin, S.D.Huber, Y.Bromberg, R.Pugatch, Y.Silberberg; Phys. Rev. A 86, (2012)
A.Schreiber, A.Gábris, P.Rohde, K.Laiho, M.Štefaňák ,V.Potoček, C.Hamilton ,I.Jex, C.Silberhorn Science (2012)
A. H. Werner
18
Outline
Propagation properties
Bound states in interacting quantum walks
Recurrence properties
A. H. Werner
19
Outline
Propagation properties
Bound states in interacting quantum walks
Recurrence properties
A. H. Werner
19
Recurrence in Random Walks
„Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend
die Irrfahrt im Straßennetz“
George Pólya Does the walker return with certainty?
recurrent transient
20
Georg Pólya; Mathematische Annalen 84(2), (1921)
A. H. Werner
Markov Process
21
Georg Pólya; Mathematische Annalen 84(2), (1921)
Countable state space
Transition matrix
Probability to move from to
Trajectory
Fix initial state
Probability to return to in exactly steps
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Return Probabilities
Return in exactly steps:
First return after exactly
steps (conditioned)
Generating functions
Recurrence:
22 A. H. Werner
Recurrence in Time Discrete Quantum Systems
Scenario: separable Hilbert space
unitary operator
evolution: ,
Question: Given , does the system return
with certainty to this initial state?
24 A. H. Werner
Return Amplitudes
Return after exactly steps
Generating function
Conceptional problem: First return probabilities .
25
Idea: Use renewal equation
A. H. Werner
Return Amplitudes
Return after exactly steps
Generating function
Conceptional problem: First return probabilities .
25
Idea: Use renewal equation
A. H. Werner
Simple Counter Example
26
time step
Way out:
Directly use classical Polya criterium for
M. Stefanak, I. Jex, T. Kiss Phys. Rev. Lett. 100, 020501 (2008)
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Operational Approach
Test for return in each time step
projective measurement:
Modified dynamics:
1. Unitary time step
2. Measurement: System in state ?
Yes No
System returned.
End of experiment.
System in state
27 A. H. Werner
First Return Amplitudes
First return after steps
Generating function
Total return probability
Definition: recurrent iff .
28 A. H. Werner
Renewal Equation
Generating function
and differ by rank perturbation: Krein Formula
Identify scalar product on RHS with
29
Renewal-equation
A. H. Werner
Random Walk vs. Quantum Case
Random Walk:
probabilities
Quantum Case:
amplitudes
Return
First return
Return probability
Renewal equation
30
F.A. Grünbaum, L. Velázquez, A. H. Werner, R. F. Werner; Com. Math. Phys. 320(2) (2013)
A. H. Werner
Recurrence Criteria
Characterization in terms of spectral measure
For matrices
pure point singular continuous absolutely continuous
32 A. H. Werner
Recurrence Criteria
Characterization in terms of spectral measure
Theorem: is recurrent iff has no absolutely
continuous component.
F.A. Grünbaum, L. Velázquez, A. H. Werner, R. F. Werner; Com. Math. Phys. 320(2) (2013)
32
pure point singular continuous absolutely continuous
A. H. Werner
RAGE Theorem
sequence of compact operators strongly convergent to
the identity, unitary operator
33
pure point singular continuous absolutely continuous
A. H. Werner
Comparison
is recurrent iff
contains no
absolutely continuous
component
Distinguishes between
singular and non-
singular spectrum
is localized iff the
spectral measure is
pure point
Distinguishes between
continuous and pure
point spectrum
Recurrence RAGE theorem
A. H. Werner
34
Proof idea I: Measures on the unit circle
Given a probability measure on , the unit circle,
define for with two analytic functions
Stieltjes function:
Schur function :
Boundary behaviour of and for characterizes .
Theorem: The absolutely continuous part of is
supported on the subset of , where .
A. H. Werner
35
Proof idea II Identify RHS of renewal equation with Schur function
For to be recurrent we need
Using this implies for the Schur function
Since bounded by we need
for almost all , which is equivalent to having no
absolutely continuous part.
A. H. Werner
36
Expected Return Time
Given first return amplitudes
Consider expected return time
Proof idea: Identify with winding number of the phase of the
Schur function on the unit circle
Result: If the pair is recurrent, the expected
return time is infinite or an integer!
counts point masses in .
A. H. Werner
37
Summary translation invariance
co
here
nce
Pure Point
spectrum
Singular continuous
spectrum
absolutely continuous
spectrum
A. H. Werner
Summary translation invariance
co
here
nce
Pure Point
spectrum
Singular continuous
spectrum
absolutely continuous
spectrum
A. H. Werner
Thank you for your
attention!
References I
A. Ahlbrecht, V.B. Scholz, A. H. Werner; J. Math. Phys. 52, 102201 (2011)
A. Ahlbrecht, H. Vogts, AHW, and R. F. Werner J. Math. Phys. 52, 042201 (2011)
G. Grimmett, S. Janson, P.F. Scudo; Phys. Rev. E, 69, 026119 (2004).
F.A. Grünbaum, L. Velázquez, A. H. Werner, R. F. Werner; Com. Math. Phys.
320(2) (2013)
A. Joye; CMP 307(1) (2011)
A. Joye, M. Merkli; J. Stat. Phys. 140(6) (2010)
M. Karski, L. Förster, JM. Choi, A. Steffen, W. Alt, D. Meschede, A. Widera;
Science 325 (2009)
N. Konno, J. Math. Soc. Japan Volume 57, Number 4 (2005)
R. Matjeschk, A. Ahlbrecht, M. Enderlein, Ch. Cedzich, A. H. Werner, M. Keyl, T.
Schaetz, R. F. Werner; Phys. Rev. Lett. 109, 240503 (2012)
A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.
Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G.
Thompson, J. L. O'Brien; Science, 329(5998) (2010)
G. Pólya; Mathematische Annalen 84(2), (1921)
A. H. Werner
References II
A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, C. Silberhorn Phys.
Rev. Lett. 106, 180403 (2011)
H. Schmitz, R. Matjeschk, Ch. Schneider, J. Glueckert, M. Enderlein, T. Huber,
and T. Schaetz; Phys. Rev. Lett. 103, 090504 (2009)
ucm.es: http://pendientedemigracion.ucm.es/info/giccucm/index.php/Quantum_Computation.html
Zugriff: 03.07.13
wikipedia.org: http://en.wikipedia.org/wiki/D-Wave_Systems Zugriff. 03.07.13
iap.uni-bonn.de: http://quantum-technologies.iap.uni-bonn.de/ Zugriff. 03.07.13
F. Zähringer, G. Kirchmair,, R. Gerritsma, E. Solano, R. Blatt and C. F. Roos;
Phys. Rev. Lett. 104, 100503 (2010)
A. H. Werner
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