Download Quantum Chaos, Information Gain and Quantum

Quantum Chaos,
Information Gain and
Quantum Tomography
Vaibhav Madhok
UBC
1
The Big Picture
The universe, as we know, is quantum.
Classical mechanics gives an excellent description
of the macroscopic world !
2
A tale of two theories: Quantum and
Classical Physics
Quantum superposition
Newton’s Laws
and
interference
Chaos
Entlanglement
How does classical behavior emerge out of quantum mechanics?
Are there any footprints of classical chaos in the quantum world?
3
What is Classical Chaos?
• Aperiodic long term behavior in a
deterministic system.
• Sensitive dependence to initial conditions.
xn = e
4
n
x0
Quantum Chaos?
• Quantum mechanics preserves the inner
product between state vectors. Unitary
evolution gives us:
(0)|⇥(0)⇥ =
U † |U ⇥⇥ = (t)|⇥(t)⇥
• What are other ways to characterize chaos in
quantum mechanical systems?
• Quantum Chaology :
What are the quantum
signatures of classical chaos?
• We have a new quantum signature of chaos.
Michael 5Berry 1989 Phys. Scr. 40 335
Information Gain: A
Classical Analogue
Ht =
pj log pj
j
Chaotic trajectory acts as an information source
As one follows a single trajectory, one gains information about
the initial conditions of the trajectory
Rate of information gain about the initial
conditions on increasingly finer scales
> 0 for a chaotic system
6
K-S Entropy:
HKS
Continuous Measurement
Quantum Tomography
N copies
r0
Unknown
Quantum
State
Evolution &
Measurement
M(t)
Measured
Signal
Gaussian
Estimator
M (t) = N O(t)⇥ + (t)
r̄
Reconstructed
Quantum
State
We gain information about the initial state by
continuously measuring the system.
Silberfarb et al. PRL 95, 030402 (2005)
G. Smith et al. PRL 97, 180403 (2006)
Riofrío et al. J. Phys. B. 15, 154007 (2011).
7
Reconstruction Algorithm
Mi = T r(Oi 0 ) + ⇥Wi
0
1
= I+
d
Measurement at time
i
d2 1
E
r E
Orthonormal basis of hermitian
matrices
=1
d2 1
Mi =
r T r(Oi E ) + Wi
=1
Least square estimate
where
8
Reconstruction Algorithm
d2 1
Mi =
r T r(Oi E ) + Wi
=1
p (M |r) / exp
(
1
/ exp(
(r
2
Maximum likelihood solution =
2
1 X
2
[Mi
↵
i
T
rM L ) C
1
(r
rM L =
is the Covariance Matrix C
9
6
X
Oi↵ r↵ ]2
rM L ))
)
Tomography and Chaos
Mi = T r(Oi 0 ) + ⇥Wi
†i
Let Oi = U O0 U i -- repeated application of the same map
U the tomography process
affect
?
How does the nature of
Tomography by the kicked top dynamics
H = ↵FX +
2F
1
X
n⌧ )Fz2 . Kicked top floquet map
(t
n= 1
is the chaoticity parameter
1
0.8
0.6
0.4
0.2
Y
Y
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
Z
z
0.2
0.4
0.6
0.8
1
Phase space (mixed) for the kicked top ,
= 2.5
10
= 1.4
The Kicked Top Phase Space
Kicked top floquet map
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
Y
Y
is the chaoticity parameter
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
Z
0.2
0.4
0.6
0.8
−1
−1
1
−0.8
−0.6
−0.4
= 0.5
1
−0.2
0
Z
0.2
0.4
0.6
0.8
1
= 2.5
1
0.8
0.8
0.6
0.4
0.4
0.2
0.2
0
0
Y
Y
0.6
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1
−1
−0.8
−0.6
−0.4
= 3.0
−0.2
0
Z
0.2
0.4
0.6
0.8
1
11
−0.8
−0.6
−0.4
−0.2
0
Z
0.2
= 7.0
0.4
0.6
0.8
1
Tomography Fidelity Vs Chaoticity
0.9
λ = 0.5
λ = 2.5
0.8
λ = 3.0
λ = 7.0
0.7
Fidelity
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
tn
60
70
80
90
100
Tomography and Information
gain
Fisher Information
For a family of probability densities, f (x; ✓)
E[...|✓] is the expectation with respect to f (x; ✓)
given a particular ✓
Fisher information tells us how well we can estimate
a parameter based on a sequence of data
For n iid observations :
In (✓) = nI(✓)
Fisher Information
For example : For a gaussian density
1
f (x; ✓) = N (0, ✓) = p
e
2⇡✓
x2
2✓
= 1
Fisher information =
✓
Fisher Information and parameter estimation
Cramer Rao
Bound
V✓ (T (X1 , ..., Xn ))
1
In (✓)
Multivariate Fisher Information
For a single variable
Multivariable
For a multivariate
gaussian
1
P (r|M ) ⇥ exp(
(r
2
I( ) = C
1
rM L ) T C
1
(r
rM L ))
Multivariate Fisher Information
I( ) = C
For a multivariate gaussian
T rh{(⇢0
2
⇢¯) }i =
✏
X
↵
2
h( r↵ ) i
h
I(✓)
1
1
i
↵↵
= T r(C)
T r(C)
Thus, we get a single number to capture the
information gain in tomography
1/T r(C) is the collective Fisher information.
Fisher Information vs Chaoticity
1.6
λ = 0.5
λ = 2.5
1.5
λ = 3.0
λ = 7.0
Fisher Information
1.4
1.3
1.2
1.1
1
0.9
0
200
400
600
800
1000
tn
1200
1400
1600
1800
2000
Information content in the
Covariance matrix
•
Eigenvectors of the inverse of the covariance matrix give specify
the direction in the operator space that have been measured. They
are the “observables” that we measure.
•
The corresponding eigenvalues give the uncertainty with which we
measure those “observables” given a signal to noise ratio.
•
Entropy of the eigenvalues of the inverse of the covariance matrix
captures the collective information gain in tomography
•
We expect high fidelity reconstruction for higher entropy values in
the presence of noise.
19
Shannon Entropy Vs Chaoticity
4.5
4
3.5
Shanon Entropy
3
2.5
2
λ = 0.5
λ = 2.5
λ = 3.0
λ = 7.0
1.5
1
0.5
0
0
10
20
30
40
50
tn
60
70
80
90
100
Random Matrix theory and Quantum Chaos
How well our analysis agrees with
the Random Matrix theory ?
21
The Random Matrix conjecture of
Quantum Chaos.
When Hamiltonian or Floquet map is
completely chaotic, then its statistical properties
can be described by a Random Matrix picked
from an appropriate ensemble.
Signatures of chaos/integrability: Level statistics, Level repulsion are
described by the Random Matrix theory.
O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett 52, 1 (1984.)
F. Haake, Quantum Signatures of Chaos (Spring-Verlag, Berlin, 1991.)
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RMT predictions for the kicked top
1
0.9
kicked top( full chaos regime)
COE
0.8
0.7
Fidelity
Fidelity
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
1.5
5
1.4
Fisher Information
6
4
E
Shannon
Entropy
1.6
3
COE
upper bound on the entropy
kicked top (full chaos regime)
2
90
100
Fisher
Information
1.3
kicked top (full chaos regime)
COE
1.2
1.1
1
0
80
(a)
tn
7
70
1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0
200
400
600
800
1000
1200
tn
tn
(b)
0.9
23
(c)
1400
1600
1800
2000
RMT predictions for a Floquet map
without time reversal symmetry
U⌧ = e
i
Jx2 i↵1 Jx
1
e
i
2
Jy2 i↵2 Jy
e
i
2
3 Jz
i↵3 Jz
.
1
CUE
0.9
Floquet map without time reversal invariance symmetry
0.8
0.7
Fidelity
0.6
Fidelity
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
tn
60
70
80
90
100
(a)
5
7
4.5
CUE
6
Floquet map without time reversal invariance symmetry
4
5
Fisher Information
3.5
4
E
Shannon
Entropy
Map without time reversal
upper bound on the entropy
CUE
3
Fisher
Information
3
2.5
2
2
1.5
1
1
0
0
200
400
600
800
1000
t
1200
1400
1600
1800
2000
n
0.5
0
200
400
600
800
1000
tn
1200
(c)
(b)
24
1400
1600
1800
2000
Conclusion
• We have a new quantum signature of classical chaos!
• Quantum chaos intimately related to information gain
and consequences are manifested in the fidelities of
reconstruction.
• Information gain in chaotic systems has a remarkable
agreement with the Random Matrix theory.
25
Choice of the appropriate ensemble of
Random Matrix
Appropriate ensemble is given by fundamental symmetries of the
Hamiltonian e.g Time reversal invariance
The kicked top is Time reversal Invariant.
U =e
T U1 T
1
2
Fz
i⇥ 2F
=U
e
1
i Fx
=e
i Fx
2
Fz
i⇥ 2F
e
where, T is an anti-unitary operator.
For the kicked top:
T =e
K
2
and : T = 1
Thus, the appropriate ensemble for the kicked top is the OE.
i Fx
26
The System: Description of
the dynamics
B
F=I+J
J
"
"F
I
H = AI!· J +
U⌧ = e
(t
n⌧ )BJz
⌘e
i↵F 2 /2
n= 1
• Floquet map:
i↵I·J
1
!X
e
i Jz
e
i Jz
Rotation of J about Z axis by an angle
followed by a precession of I and J about F
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↵|F |
by an angle
Evidence of Entanglement as a signature of
chaos: Dynamics
Spin size
I = J = 150
We consider
Fz = 0 subspace.
Initial state: Product state of spin coherent state associated with each spin
"
(a)
"
!
!
"# 0
"# 0
!
!
"#
"#
0
"#
2"
!
0
2"
"#
(b)
(a)
!
(b)
!
!
!
!
(a) The classical phase space
(b )Long time average entanglement as a function of the
initial position of the coherent states
28