Transcript Chaos and Entanglement - Strona domowa CFT PAN

Chaos and Entanglement
friends or enemies?
Rafał Demkowicz-Dobrzański (CFT PAN)
Classical Chaos
Deterministic laws
Exponential sensitivity
to initial perturbation
p
(t)
 (t )  et (0)
 – positive Lyapunov exponent
(0)
x
Quantum Chaos
• Do quantum systems which in the classical limit are chaotic
differ form the ones which are classically regular?
Quantum world:
Classical world:
Regular systems
Chaotic systems
• What are the quantum signatures of chaos?
– Distribution of energy levels
– Stability of quantum motion under perturbed Hamiltonian
– Entanglement ???
Entanglement
• Bipartite quantum system:
H1
1 , 2 ,, d1  H1
H2
Η  H1  H 2
1 , 2 ,, d2  H2
• Arbitrary pure state:
 
d1 , d 2
c
i 1, j 1
• Separable state (product state):
• Entangled state:
ij
i  j
  
 
(more than one term in the Schmidt decomposition)
min(d1 , d 2 )
 
i 1
i
i
 i
Measure of entanglement
d
   i  i  i
i 1
• Reduced density matrix of the first subsystem:
1   i 2  i  i
1  Tr2  
i
• Mixedness of the reduced density matrix (linear entropy):
E()  1  Tr(  )
2
1
d
E ( )  1   i
4
i 1
0  E ( )  1 
product state
  
1
d
maximally entangled state
1
 
d
d

i 1
i
 i
Entangling properties of quantum
evolutions
• How an initially product state is being entangled?
U – unitary operator acting in H  H1  H 2
U
   
E () - entanglement produced
• Entangling power of U
EU   d ( )d ( ) E U    
Su(d) – Haar measure
• Does chaos influence entangling properties?

Two weakly coupled chaotic
systems
weak coupling
k
k
chaoticity parameter in the
Hamiltonian of the system
• How entanglement production changes when varying k,
under fixed coupling?
- initial entanglement growth rate
- long time behaviour of entanglement
• Does it depend on the type of initial product state chosen?
Kicked Top
j – total spin
 j ,, j  H - basis
d  dim H  2 j  1
• Hamiltonian:
H(t)  pJ y  2kj J z
2

  (n  t )
n  
• One period evolution:
U e
i 2kj J z 2 ipJ y
e
Kicked Top
U e
i 2kj J z 2 i  J y
2
e
• Heisenberg picture:
 i kj ( J x  12 )
~

1
J x  U J xU  2 ( J z  iJ y )e
 h.c.
 i kj ( J x  12 )
~

i
J y  U J yU  2 ( J z  iJ y )e
 h.c.
~
J z  U  J zU  J x
• Direction operators:
1
[ X ,Y , Z ]  [J x , J y , J z ]
j
For large j they commute
Classical limit for the kicked top
• j
~
X  Z cos(kX )  Y sin(kX )
~
Y  Z sin(kX )  Y cos(kX )
~
Z  X
X 2 Y 2  Z 2 1
discreet dynamics on a sphere
Classical limit for the kicked top
• j
~
X  Z cos(kX )  Y sin(kX )
~
Y  Z sin(kX )  Y cos(kX )
~
Z  X
X 2 Y 2  Z 2 1
discreet dynamics on a sphere
Classical limit for the kicked top
• j
~
X  Z cos(kX )  Y sin(kX )
~
Y  Z sin(kX )  Y cos(kX )
~
Z  X
X 2 Y 2  Z 2 1
discreet dynamics on a sphere
Classical limit for the kicked top
• j
~
X  Z cos(kX )  Y sin(kX )
~
Y  Z sin(kX )  Y cos(kX )
~
Z  X
X 2 Y 2  Z 2 1
discreet dynamics on a sphere
Most classical quantum states
• Spin-coherent states:
 ,  R( ,) j
R( , )  exp i J x sin   J y cos 
- minimal uncertainty with respect
to angular momentum components
- overcomplete set
J  J x  J y  J z
• Phase space picture of quantum states
Hussimi function:
2 j 1
Q( ,  ) 
 ,   ,
4
Hussimi function of
a spin-coherent state (j=20):
2
2
2
2
Evolution of a spin-coherent state
• j=20
  0.89,  3.77
k=1
k=6
Coupled kicked tops
H  H1  H2  Hint
• Hamiltonian:
H i (t)  2 J yi  2kj J zi
2

  (n  t )
n  
Hint (t)  j J z1 J z2

  (n  t )
n  
• One period evolution operator:
U  U1 U 2 Uint
• Entanglement evolution
  0.89,  0.63
j=20
0.01
Chaos enhances initial production rate!?
Short time behaviour of
entanglement
• Perturbative formula (Tanaka et al. 2002)
U  U1 U 2 Uint
U0

Uint  exp  i j J z1 J z2
Interaction picture:
 
 t ˆ
t
ˆ
A(t )  U 0 AU 0 
 
~

 (t )  U 0

t
 (t )
~
~
 (t )  U int (t )  (t  1)
• Entanglement of (t ) to the second order in :
t
t
E (t )  1  Tr ( 1 )  2 2 j 2  C1 (t1 , t2 )C2 (t1 , t2 )
2

t1 1 t 2 1
1
C1 (t1 , t2 )  2 J z1 (t1 ) J z1 (t2 )  J z1 (t1 ) J z1 (t2 )
j
time correlation function

Chaos and time correlation
function
t
t
E (t )  2 2 j 2  C 2 (t1 , t2 )
t1 1 t 2 1
• t1  t2

1
2
C (t , t )  2 J z (t )  J z (t )
j
2

very low for coherent state:
 1j
for random state:
1
Chaos increases C(t,t) !
• t1  t2
but kills corelations for t1  t2
Chaotic motion:

C(t, t  t ) t

 0
Regular motion:
t 
C(t , t  t ) 
C
Chaos induces initial linear
entanglement increase
t
t
E (t )  2 2 j 2  C 2 (t1 , t2 )
• Chaotic regime:
E (t )  2 j
2
t
2
c
linear increase
• Regular regime:
E (t )  2 j
t2
2 2
r2
quadratic increase
t1 1 t 2 1
Initial entangling power for the
coupled kicked tops
• Initial entanglement growth rate, averaged over either random or
coherent states:
Chaos always diminishes initial entangling power!
Long time entangling properties
• j=20 0.01
  0.89,  0.63
Chaos helps achieving high asymptotic entanglement!
Averaged asymptotic behaviour
and eigenvectors entanglement
• Asymptotic entanglement, averaged over either random or
coherent states:
• Averaged asymptotic entanglement and eigenvectors entanglement
Easymp  2Eeigen 1
Conclusions
Chaos and entanglement are....
• Friends:
a) Chaos drives low-uncertainty states into highly smeared states and
thus increases initial entanglement growth rate
b) Chaos assures high asymptotic entanglement
c) In different approaches, where chaos is not ,,localized’’ in the subsystems
and the coupling is strong, chaos helps entanglement growth.
• Enemies:
a) For certain choices of parameters (j, , regular dynamics, thanks to
non-vanishing time correlations, outperforms chaotic dynamics in
terms of initial entanglement production.
b) For weakly coupled systems initial entangling power is always worst
in chaotic case
c) In the case of coupled kicked tops, very regular dynamics has equally high
(even a little bit higher) asymptotic entanglement than chaotic cases.
Chaos and Entanglement
friends or enemies?
Rafał Demkowicz-Dobrzański (CFT PAN)