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RECURRENCE:
RANDOM WALKS vs
QUANTUM WALKS
MARTES CUÁNTICO 05/05/2015
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
particles independently change container at rate N ∆t
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
particles independently change container at rate N ∆t
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
N − k k
particles independently change container at rate N ∆t
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Even !!![N, 0]Any state has return probability R = 1
N − k k
particles independently change container at rate N ∆t
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Even !!![N, 0]Any state has return probability R = 1
N − k k
particles independently change container at rate N ∆t
Differences come from the expected return time
τ [N−k,k] =2N
(
N
k
) ∆t
Example: the Ehrenfest model
Proposed by Tatiana and Paul Ehrenfest to reconcile reversibility of classical mechanics and irreversibility of thermodynamics
It is a simple model for the exchange of gas molecules between two isolated bodies
RECURRENCE
RECURRENCE RETURN PROPERTIES≡
Even !!![N, 0]Any state has return probability R = 1
N − k k
particles independently change container at rate N ∆t
Differences come from the expected return time
τ [N−k,k] =2N
(
N
k
) ∆t
millions of years ≈ 3×age of Universeτ[N,0] ≈ 40 000
secτ[N/2,N/2] ≈ 1.25× 10−11
For instance, if and sec N = 100 ∆t = 10−12
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
chaos non-equilibriummicro-thermodynamics
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
chaos non-equilibriummicro-thermodynamics
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
The Ehrenfest example shows that simple models (random, discrete) may help to uncover central aspects of recurrence avoiding unnecessary complications
chaos non-equilibriummicro-thermodynamics
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
The Ehrenfest example shows that simple models (random, discrete) may help to uncover central aspects of recurrence avoiding unnecessary complications
RECURRENCEOLD RANDOM WALKS (RW)
chaos non-equilibriummicro-thermodynamics
KEY RETURN PROPERTIES
Return probability
Expected return time
RECURRENCE
CLASSIFICATION OF STATES
Expected Return Time
< .
ReturnProbability
< 1 Transient
1 RecurrentPositive
Recurrent
∞∞
The Ehrenfest example shows that simple models (random, discrete) may help to uncover central aspects of recurrence avoiding unnecessary complications
RECURRENCEOLD RANDOM WALKS (RW)
NEW! QUANTUM WALKS (QW)
chaos non-equilibriummicro-thermodynamics
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
R = 1 R = 1 R ≈ 0.34
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
R = 1 R = 1 R ≈ 0.34CRITICAL
DIMENSIOND= 3
⇒UNBIASED
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
RANDOM WALKS (RW)
RANDOM WALKS formalization of a path consisting in successive random steps, widely used in physics, chemistry, biology, psychology, economy and computer science
≡
George Pólya
George Pólya was puzzled by the fact that he recurrently run into a couple while strolling in a park
He reduced the problem to the recurrence of a single walker and studied the return probability
1D 2D 3D
R = 1 R = 1 R ≈ 0.34
τ = ∞ τ = ∞
CRITICAL DIMENSION
D= 3
⇒UNBIASED
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
RW RECURRENCE
STATES: elements of a countable set Ωi ∈ Ω
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
Prob(in steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1j j
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1j jj
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
ji
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
i
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
Example: the Ehrenfest modelN − k k
≡ 0, 1, . . . , NΩ = [N, 0], [N − 1, 1], . . . , [0, N ]
pk,k+1 =
N − k
Npk,k−1 =
k
N= 1−
k
N
10 2 3 N
1− 1/N 1− 2/N 1− 3/N1
1/N 2/N 3/N 4/N 1
1/N
i
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
OVERCOUNTING!!!
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
FIRST TIMEProb(i → i) =
X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Ri = Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)RETURN
PROBABILITY
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
i
i
SIMPLE LOOP
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Ri = Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
EXPECTED RETURN TIME
RETURNPROBABILITY
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
q(n)i∆t( )τi =
X
n≥1
n
i
i
SIMPLE LOOP
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
PROBABILITY of RETURNING to
for the first time in the -th STEPn
i
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)iFIRST TIME
Ri = Prob(i → i) =X
n≥1
Prob(in steps−−−−−−→ i)
EXPECTED RETURN TIME
RETURNPROBABILITY
CONVENTION
∆t = 1[ [
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
q(n)i
τi =
X
n≥1
n
i
i
SIMPLE LOOP
RANDOM EVOLUTION: given by a stochastic matrix P = (pij)
iii
RW RECURRENCE
pij = Prob(i1 step−−−−→ j) ≥ 0
= Pn
iProb(i
n steps−−−−−→ ) =
X
ik
pii1pi1i2 · · · pin−1RETURN PROB.
in STEPSn
FIRST RETURN PROB.
in STEPSnFIRST TIME
Prob(in steps−−−−−−→ i) =
X
ik 6=i
pii1pii2 · · · pin−1i
p(n)i
=
q(n)i
=
=
X
n≥1
q(n)i
Ri =
EXPECTED RETURN TIME
RETURNPROBABILITY
CONVENTION
∆t = 1[ [
ik 6=i
STATES: elements of a countable set Ωi ∈ Ω
SIZE of
SIZE of
P
=
Ω
X
j
pij = Prob(i1 step−−−−→ Ω) = 1
q(n)i
τi =
X
n≥1
n
i
i
SIMPLE LOOP
RW RECURRENCE: GENERALITIES
RW RECURRENCE: GENERALITIES
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
RW RECURRENCE: GENERALITIES
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
p(n)i
q(n)i≥
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
p(n)i
q(n)i≥
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
p(n)i
q(n)i≥
T R
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
FINITE systems may have TRANSIENT states
p(n)i
q(n)i≥
T R
Ω
SUBSET RECURRENCE
i
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
FINITE systems may have TRANSIENT states
p(n)i
q(n)i≥
T R
Ω
SUBSET RECURRENCE
Si
11
RW RECURRENCE: GENERALITIES
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RETURNPROBABILITY
Ri =
X
n≥1
q(n)i
is RECURRENT ifi ∈ Ω Ri = 1
POSITIVE RECURRENT if τi < ∞
RETURN PROB.
in STEPSnp(n)i
= Pn
iiP = (pij)
q(n)i
FIRST RETURN PROB.
in STEPSn
?
τi can be ANY real number in [1,∞]
FINITE systems may have TRANSIENT states
p(n)i
q(n)i≥
Prob(i → S) Prob(i → i)≥ = Ri
T R
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities?
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
= bqi(1)
=
dbqidz
∣∣∣∣z=1
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
= bqi(1)
=
dbqidz
∣∣∣∣z=1
= 1−1
bpi(1)
= limz→1
1
(1− z)bpi(z)
RECURRENTi ∈ Ω Ri = 1⇔
POSITIVE RECURRENT ⇔ τi < ∞
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
= bqi(1)
=
dbqidz
∣∣∣∣z=1
= 1−1
bpi(1)
= limz→1
1
(1− z)bpi(z)
RECURRENTi ∈ Ω Ri = 1⇔
POSITIVE RECURRENT ⇔ τi < ∞
RETURNPROBABILITY
=
X
n≥1
q(n)i
Ri
EXPECTED RETURN TIME
τi =
X
n≥1
nq(n)i
RECURRENCE & GENERATING FUNCTIONS
How to calculate these quantities? P p(n)i
= Pn
ii
Pn
q(n)i
?
bpi(z) =X
n≥0
p(n)i
zn bqi(z) =
X
n≥1
q(n)i
zn
RETURN g.f. FIRST RETURN g.f.
bqi(z) = 1−1
bpi(z)RENEWAL EQUATION
⇔ bpi(1) = ∞
⇔ limz→1
(1− z)bpi(z) > 0
= bqi(1)
=
dbqidz
∣∣∣∣z=1
= 1−1
bpi(1)
= limz→1
1
(1− z)bpi(z)
RECURRENCE & SPECTRUM
RECURRENCE & SPECTRUM
=
X
n≥0
Pn
iizn
P bpi(z)
RECURRENCE & SPECTRUM
=
X
n≥0
Pn
iizn
P bpi(z)
= (1− zP )−1
ii
RECURRENCE & SPECTRUM
P bpi(z) = (1− zP )−1
ii
RECURRENCE & SPECTRUM
P bpi(z)
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?τi = lim
z→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?τi = lim
z→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
RECURRENT⇔⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN IN
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
RECURRENT⇔⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN IN
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN IN
P bpi(z)
SPECTRAL SHORTCUT?
P FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Under quite general conditions P becomes self-adjoint with kPk 1
τi = limz→1
1
(1− z)bpi(z)
RENEWAL Eq.Ri = 1−
1
bpi(1)= (1− zP )−1
ii ONLY
mattersz → 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN INP FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Under quite general conditions P becomes self-adjoint with kPk 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN INP FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Recurrence ONLY depends on the spectral decomposition around λ = 1
Under quite general conditions P becomes self-adjoint with kPk 1
If = (0, . . . , 0, 1, 0, . . . )v
i)
τi =1
kvλ=1k2then
vλ=1
veig. λ=1
FINITEΩ
RECURRENCE & SPECTRUM
IN INP FINITE matrix with spectrum in [−1, 1]
Any vector has a spectral decomposition
v
X
k
vλkv =
eigenvector with EIGENVALUE λk
contribution from CONTINUOUS SPEC.
dv(λ)+
Z
RECURRENT⇔
⇔ vλ=1 6= 0i ∈ ΩPOSITIVE
RECURRENT or dkv(λ)k2 = 1Z
1
1− λ
Recurrence ONLY depends on the spectral decomposition around λ = 1
FINITE systems: RECURRENT POSITIVE RECURRENT⇒
Under quite general conditions P becomes self-adjoint with kPk 1
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
source: http://physik.uni-paderborn.de/?id=178571
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
RW
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
source: http://physik.uni-paderborn.de/?id=178571
RecurrenceQUANTUM WALKS (QW)
QUANTUM WALKS models for a quantum particle in discrete space-time≡
1993 Aharonov et al:
quantum version of RW spreads out much faster
Aharonov
RW QW
1940s Feynman: checkerboard model for path integral of free Dirac eq.Later related to D Ising model1
Feynman
1+1
source: http://physik.uni-paderborn.de/?id=178571
RecurrenceQUANTUM WALKS (QW)
Why?
RecurrenceQUANTUM WALKS (QW)
Why?
Simple models for quantum dynamics
RecurrenceQUANTUM WALKS (QW)
Why?
Simple models for quantum dynamics
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
source: http://physik.uni-paderborn.de/?id=178571
RecurrenceQUANTUM WALKS (QW)
Why?
Step towards information processing machines using quantum mechanical laws (Feynman ’82 proposal) quantum simulators, quantum computers
Simple models for quantum dynamics
Experimental realizations
Atoms in optical lattices
Trapped ions
Wave guide arrays
Optical fibres
Quantum biology quantum coherence in photosynthetic energy transfer
The widespread use of RW in randomized algorithms motivated the use of QW in quantum algorithms exponential speedup over any classical algorithm!!!
source: http://physik.uni-paderborn.de/?id=178571
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ UψEVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
Example: D coined QW1
Generalization of checkerboard model: particle with spin on a lattice x ∈ Zs ∈ ↑, ↓
spanned by states H = Hspace ⊗ Hspin |xi |si
U = S(1space ⊗ C) unitary step
C =
a b
c d
∈ U(2) spin rotation (‘coin flip’)
conditional shiftS =
X
x∈Z
|x− 1ihx| |#ih#|+ |x+ 1ihx| |"ih"|
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
FIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
FIRST RETURN PROB.
in STEPSnFIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
PROBLEM: the quantum measurement to check the return after every step collapses the state altering irreversibly the “natural” evolution
FIRST RETURN PROB.
in STEPSnFIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
STATES: (up to phases) unit vectors of a Hilbert spaceψ (H , h · | · i)
QW RECURRENCE
ψ1 step−−−−→ Uψ
PROBABILITYAMPLITUDE
MEASUREMENT: Probability of measuring the state when the system is in state
φ
ψ
Probψ(φ) = |hφ|ψi|2
EVOLUTION: given at every step by a unitary operator U
PROBLEM: the quantum measurement to check the return after every step collapses the state altering irreversibly the “natural” evolution
We will take the collapse as an intrinsic ingredient of monitored quantum recurrence.
This is in QM spirit, which gives to measurements a role absent in classical physics
FIRST RETURN PROB.
in STEPSnFIRST TIME
= Probψ(Unψ) = |hψ|Unψi|2Prob(ψ
n steps−−−−−→ ψ) RETURN PROB.
in STEPSn
Prob(ψ → ψ) =X
n≥1
Prob(ψn steps−−−−−→ ψ)
RETURNPROBABILITY
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ|
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ|
Qφψ
PROBABILITY OF FINDING φ
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Qφψ
PROBABILITY OF FINDING φ
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Two possible results when measuring at state φ ψ
Qφψ
PROBABILITY OF FINDING φ
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Two possible results when measuring at state φ ψ
φ is found:Pφψ
k · k= φψ
COLLAPSE−−−−−−−−→
Qφψ
PROBABILITY OF FINDING φ
Qφψ
kQφψk
MEASUREMENT & COLLAPSE
ψ
φ⊥
φ
Pφψ
PROBABILITY OF NOT FINDING φ
ORTHOGONAL PROJECTION onto φ
Qφ = I − Pφ ORTHOGONAL PROJECTION onto φ⊥
Pφ = |φihφ| 1 = kψk2 = kPφψk2+ kQφψk
2
=
Probψ(φ)
=
Probψ(φ⊥)
Two possible results when measuring at state φ ψ
φ is found:Pφψ
k · k= φψ
COLLAPSE−−−−−−−−→
φ is NOT found:Qφψ
k · kψ
COLLAPSE−−−−−−−−→
Qφψ
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
Dynamics perturbed by measurements≡
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
Dynamics perturbed by measurements≡
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
Dynamics perturbed by measurements≡
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
NO ψ NO ψ
step 3−−−−−→ · · ·
step n−1−−−−−−−→ eUn−1ψ
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
step n
−−−−−→ U eUn−1ψNO ψ NO ψ
step 3−−−−−→ · · ·
step n−1−−−−−−−→ eUn−1ψ
MEASUREMENT
Return to ?ψ
UNITARY STEP
U
FIRST RETURN
PROBABILITY
in STEPSn
= |hψ|U eUn−1ψi|2FIRST TIME
Prob(ψn steps−−−−−→ ψ)
MONITORED RECURRENCE How to calculate first return probabilities?
QW RECURRENCE
YES END
=eU
projection onto ψ⊥
NO UQψ
Dynamics perturbed by measurements≡
NO return to prior to -th step meansnψ
NO ψψ
step 1−−−−−→ eUψ
NO ψ
step 2−−−−−→ eU2ψ
step n
−−−−−→ U eUn−1ψNO ψ NO ψ
step 3−−−−−→ · · ·
step n−1−−−−−−−→ eUn−1ψ
QW RECURRENCE
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
projection onto ψ⊥
eU = QψU
FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
FIRST TIMEProb(ψ
n steps−−−−−→ ψ)= |a
(n)ψ |2
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
projection onto ψ⊥
eU = QψU
FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
FIRST TIMEProb(ψ
n steps−−−−−→ ψ)= |a
(n)ψ |2
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
RETURNPROBABILITY
Rψ = Prob(ψ → ψ) =X
n≥1
|a(n)ψ |2
EXPECTED RETURN TIME
τψ =
X
n≥1
n |a(n)ψ |2
projection onto ψ⊥
eU = QψU
FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
FIRST TIMEProb(ψ
n steps−−−−−→ ψ)= |a
(n)ψ |2
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi = |µ
(n)ψ |2Prob(ψ
n steps−−−−−→ ψ)
QW RECURRENCE
is RECURRENT if ψ Rψ = 1
POSITIVE RECURRENT if τψ < ∞
RETURNPROBABILITY
Rψ = Prob(ψ → ψ) =X
n≥1
|a(n)ψ |2
EXPECTED RETURN TIME
τψ =
X
n≥1
n |a(n)ψ |2
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
|a|2 + |c|2 = 1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
|a|2 + |c|2 = 1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
UNBIASED RW
CRITICAL DIMENSION
D= 3
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
|a|2 + |c|2 = 1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Example: D coined QW1
-1 10
-1 10
a
b
c
d
b
a
c
d
C =
a b
c d
∈ U(2)
COIN FLIPSITE
UNBIASED COINED QW
|a|2 = |b|2 = |c|2 = |d|2 =1
2
D ≥ 3
D ≤ 2 RECURRENT
TRANSIENT
UNBIASED RW
CRITICAL DIMENSION
D= 3
Every state is TRANSIENT
already for !!! D= 1
Example: cyclic shift
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
Example: cyclic shift
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
Example: cyclic shift
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
Prob(ψ2 steps−−−−−→ ψ) =
9
16FIRST TIMEa(2)ψ = hψ|U eUψi = −
3
4
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
Example: cyclic shift
<
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
Prob(ψ2 steps−−−−−→ ψ) =
9
16FIRST TIMEa(2)ψ = hψ|U eUψi = −
3
4
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
Example: cyclic shift
QUANTUM PARADOX
FIRST return probabilities can be greater than return probabilities!!!
<
Consider a system with Hilbert state space and unitary step H = span|0i, |1i, |2i
Prob(ψ2 steps−−−−−→ ψ) =
1
4
Suppose the system initially in the state ψ =1p2(|1i − |2i)
µ(2)ψ = hψ|U2ψi = −
1
2
Prob(ψ2 steps−−−−−→ ψ) =
9
16FIRST TIMEa(2)ψ = hψ|U eUψi = −
3
4
2 1
0
1 1
1
U = |1ih0|+ |2ih1|+ |0ih2|
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
For amplitudes
instead of
probabilities!!!
=
X
n≥1
|a(n)ψ |2Rψ
=
X
n≥1
n|a(n)ψ |2
RETURNPROBABILITY
EXPECTED RETURN TIME τψ
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
For amplitudes
instead of
probabilities!!!
=
X
n≥1
|a(n)ψ |2Rψ
=
X
n≥1
n|a(n)ψ |2
RETURNPROBABILITY
EXPECTED RETURN TIME τψ
RECURRENCE & GENERATING FUNCTIONS
RETURN
AMPLITUDESµ(n)ψ = hψ|Unψi FIRST RETURN
AMPLITUDESa(n)ψ = hψ|U eUn−1ψi
bµψ(z) =X
n≥0
µ(n)ψ z
nRETURN g.f. baψ(z) =
X
n≥1
a(n)ψ z
nFIRST RETURN g.f.
QUANTUM RENEWAL EQUATION
baψ(z) = 1−1
bµψ(z)
=
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πi
For amplitudes
instead of
probabilities!!!
RECURRENCE & SPECTRUM
RECURRENCE & SPECTRUM
U bµψ(z) =X
n≥0
hψ|Unψi zn
RECURRENCE & SPECTRUM
U bµψ(z) =X
n≥0
hψ|Unψi zn
= hψ|(1− zU)−1ψi
RECURRENCE & SPECTRUM
U bµψ(z) = hψ|(1− zU)−1ψi
RECURRENCE & SPECTRUM
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
U unitary ⇒spectrum in
unit circle
eiθ
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+
Zdψ(eiθ)
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
SPECTRAL SHORTCUT?
U bµψ(z) = hψ|(1− zU)−1ψiRENEWAL Eq.
QUANTUMRψ =
Z2π
0
|baψ(eiθ)|2dθ
2π
=
Z2π
0
baψ(eiθ) ∂θbaψ(eiθ)dθ
2πiτψ
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
dimH = ∞
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS
dimH = ∞
ABSOLUTELY CONTINUOUS SINGULAR
RECURRENCE & SPECTRUM
EVERY point matterseiθ
U unitary ⇒spectrum in
unit circle
eiθ
Any vector has a spectral decompositionψ
⇒
ψ =
X
k
ψλk+ +
Zdψsc(e
iθ)
Zw(θ) dθ
contribution from CONTINUOUS SPEC.
eigenvector with EIGENVALUE λk
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS
τψ = number of EIGENVECTORS
dimH = ∞
RECURRENCE & SPECTRUM
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
RECURRENCE & SPECTRUM
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
Recurrence depends an ALL the spectral decomposition
RECURRENCE & SPECTRUM
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
RECURRENCE & SPECTRUM
QUANTIZATION of EXPECTED RETURN TIME!!!
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
RECURRENCE & SPECTRUM
QUANTIZATION of EXPECTED RETURN TIME!!!
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
θα
eiθ e
iα
baψ(eiθ)
=∆α
2πτψ
WINDING NUMBER of baψ(eiθ)
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
EXPECTED RETURN TIME: Topological meaning INTEGER
RECURRENCE & SPECTRUM
QUANTIZATION of EXPECTED RETURN TIME!!!
ψ ONLY SINGULAR part RECURRENT ⇔
⇔POSITIVE RECURRENT ONLY finite EIGENVECTORS τψ = number of EIGENVECTORS
θα
eiθ e
iα
baψ(eiθ)
=∆α
2πτψ
WINDING NUMBER of baψ(eiθ)
Recurrence depends an ALL the spectral decomposition
FINITE DIMENSIONAL systems only have POSITIVE RECURRENT states
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
-1 10
-1 10
Example: site recurrence in D coined QW1
SUBESPACE RECURRENCE
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
-1 10
-1 10
Example: site recurrence in D coined QW1
|0i |"iψ =
−→
Prob???
span|0i |"i, |0i |#iV =
SUBESPACE RECURRENCE
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
-1 10
-1 10
Example: site recurrence in D coined QW1
|0i |"iψ =
−→
Prob???
span|0i |"i, |0i |#iV =
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
The generating functions become matrix functions acting on V
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
The generating functions become matrix functions acting on V
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ
The generating functions become matrix functions acting on V
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ
Rψ(V ) = Prob(ψ → V ) -RETURNPROBABILITYV
The generating functions become matrix functions acting on V
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ=
Z2π
0
k bψ(θ)k2 dθ
2πRψ(V ) = Prob(ψ → V ) -RETURN
PROBABILITYV
The generating functions become matrix functions acting on V
Even more interesting than the return properties of a state are the return properties of a subspace : starting at , what is the probability of returning to ? ψ ∈ V VV ⊂ H
SUBESPACE RECURRENCE
orthogonal projection onto=Pψ ψ
orthogonal projection onto=Qψ ψ⊥
orthogonal projection onto= VPV
orthogonal projection onto= V⊥QV
STATE RECURRENCEProb(ψ → ψ) SUBSPACE RECURRENCEProb(ψ → V )
first return g.f. SCALAR
baψ(z) baV (z)first -return g.f.
MATRIX
V
LOOP in V
bψ(θ) := baV (eiθ)ψ=
Z2π
0
k bψ(θ)k2 dθ
2πRψ(V ) = Prob(ψ → V ) -RETURN
PROBABILITYV
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi EXPECTED -RETURN TIMEV
The generating functions become matrix functions acting on V
1/ dimV
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
1/ dimV
V
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
Averaging over we find again “topological integers”ψ ∈ Vτψ(V )
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
Averaging over we find again “topological integers”ψ ∈ Vτψ(V )
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
hτψ(V )iψ∈V =N
dimV
N =
WINDING NUMBER
of detbaV (eiθ)
θ
eiθ
α
eiα
detbaV (eiθ)
BERRY PHASE
of the loop
EXPECTED -RETURN TIME = geometrical phase NOT QUANTIZEDV
bψ(θ) := baV (eiθ)ψ
1/ dimV
V
Averaging over we find again “topological integers”ψ ∈ Vτψ(V )
τψ(V ) =
Z2π
0
h bψ(θ)|∂θ bψ(θ)idθ
2πi
EXPECTED
-RETURN
TIME
V
STATE RECURRENCE Expected return time was a “topological integer”
SUBESPACE CURRENCE
hτψ(V )iψ∈V =N
dimV
N =
WINDING NUMBER
of detbaV (eiθ)
θ
eiθ
α
eiα
detbaV (eiθ)
QUANTIZATION of MEAN EXPECTED -RETURN TIME!!!
Topological meaning INTEGER MULTIPLE of
V
1/ dimV
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
Example: site recurrence in D coined QW1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
|a|2 + |c|2 = 1
R|xi|si(V ) =2
π|c|2
|ac|+ (1− 2|a|2) arcsin |a|
SITE RECURRENCE
Example: site recurrence in D coined QW1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
STATE RECURRENCE
|a|2 + |c|2 = 1
R|xi|si(V ) =2
π|c|2
|ac|+ (1− 2|a|2) arcsin |a|
SITE RECURRENCE
Example: site recurrence in D coined QW1
|c|0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
-1 10
-1 10
a
b
c
d
b
a
c
d
−→
Prob???
span|0i |"i, |0i |#iV =
ψ = |0i |si
R|xi|si =2
π|c|4
(1 + 2|a|2)|ac|+ (1− 4|a|2) arcsin |c|
STATE RECURRENCE
|a|2 + |c|2 = 1
R|xi|si(V ) =2
π|c|2
|ac|+ (1− 2|a|2) arcsin |a|
SITE RECURRENCE
Example: site recurrence in D coined QW1
As intuition suggests, . Is this a general fact?Prob(ψ → V ) ≥ Prob(ψ → ψ)
Example: combined shifts
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
Example: combined shifts
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
ψ = α|0i+ |"ip
2+ β|#iFor and we obtainV = span|0i+ |"i, |#i
Rψ = Prob(ψ → ψ) =1− 1
2|α|2
1 + 1
2|α|2
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
|α|
Rψ(V ) = Prob(ψ → V ) = 3
4− 1
4|α|2
Example: combined shifts
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
ψ = α|0i+ |"ip
2+ β|#iFor and we obtainV = span|0i+ |"i, |#i
Rψ = Prob(ψ → ψ) =1− 1
2|α|2
1 + 1
2|α|2
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
|α|
Rψ(V ) = Prob(ψ → V ) = 3
4− 1
4|α|2
Example: combined shifts
QUANTUM PARADOX
Return probability to a subspace can be smaller than to the state!!!
U =
X
x∈Z
|x+ 1ihx|+ |#ih"|+ |"ih#|
-1 101 1
1
1
Consider a Hilbert space spanned by with unitary step|xix∈Z [ |"i, |#i
ψ = α|0i+ |"ip
2+ β|#iFor and we obtainV = span|0i+ |"i, |#i
Rψ = Prob(ψ → ψ) =1− 1
2|α|2
1 + 1
2|α|2
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
|α|
Rψ(V ) = Prob(ψ → V ) = 3
4− 1
4|α|2
RW vs QW
Random Walks Quantum Walks
Spectral shortcut NOT always applicable
Stochastic Self-adjoint Spectrum on [-1,1]
Spectral shortcut always applicable
Unitary Spectrum on unit circle
ONLY spectrum around 1 matters for recurrence ALL the spectrum matters for recurrence
Recurrence Singular part Recurrence ONLY singular part
eigenvector Positive recurrence
with eigenvalue 1
ONLY finite eigenvectors Positive recurrence
with ANY eigenvalue
Finite system ONLY recurrent states ONLY positive
Finite-dim system recurrent states
Expected return time is NOT quantized Expected return time is quantized
First return probabilities
are NOT greater than return probabilities
First return probabilities
can be greater than return probabilities
Return probability to a subset
is NOT smaller than to the initial state
Return probability to a subspace
can be smaller than to the initial state
⇔;:
⇔⇔
; ⇒
? ⇒ ⇒
∃
RW vs QW
Random Walks Quantum Walks
Spectral shortcut NOT always applicable
Stochastic Self-adjoint Spectrum on [-1,1]
Spectral shortcut always applicable
Unitary Spectrum on unit circle
ONLY spectrum around 1 matters for recurrence ALL the spectrum matters for recurrence
Recurrence Singular part Recurrence ONLY singular part
eigenvector Positive recurrence
with eigenvalue 1
ONLY finite eigenvectors Positive recurrence
with ANY eigenvalue
Finite system ONLY recurrent states ONLY positive
Finite-dim system recurrent states
Expected return time is NOT quantized Expected return time is quantized
First return probabilities
are NOT greater than return probabilities
First return probabilities
can be greater than return probabilities
Return probability to a subset
is NOT smaller than to the initial state
Return probability to a subspace
can be smaller than to the initial state
⇔;:
⇔⇔
; ⇒
? ⇒ ⇒
∃
RECURRENCE COLLABORATORS
Reinhard WernerLeibniz U Hannover
Albert WernerFreie U Berlin
Alberto GrünbaumUC Berkeley
Jon WilkeningUC Berkeley
Jean BourgainIAS Princeton
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