Transcript Quantum State Tomography

Quantum entanglement and
Quantum state Tomography
Zoltán Scherübl
Nanophysics Seminar – Lecture
26.04.2012.
BUTE
What is Quantum Tomography?
Measuring a QM system:
1 measurement (eg. x): 1 physical parameter -> random
1 type of measurement on many copy of same system: |ψ (x)|2
density function
To reconstruct the ψ(x) wavefunction: more type of measurements
needed
Quorum: Complet set of measurable quantities, (operator basis in
the Hilbert-space)
Continuus variables: Wigner function (eg. light polarization)
Discrete variables: density matrix (eg. qubit)
Fidelity: probability of correctly identifying the states
F  Tr
|  0   0 | |  0   0 |
Quantum Process Tomography:
Input
Output
QM Black Box
Using different inputs (complete basis) -> QST on the output
Wigner function
Classical system:
Density function in phase-scape
W(x,p) non negativ, normalized
W ( x, p)  0
 W ( x, p)dxdp  1
Marginal distribution:
pr ( x)   W ( x, p )dp
QM system:
Heisenberg’s uncertainty principle: x and p cannot be measured at
the same time -> neither the phase-space probability density
But: X or P can be measured, so the marginal distributions too
And always exists a quasi-probability density function (Wignerfunction), which:
Normalized
It’s marginal distributions are exists (as above)
But indefinit (not necessarily non negativ)
Entanglement, mixed state, density matrix
QM system:
Entire system: wavefunction
| i  H
|   ai | i 
 |  |
i
Multiple subsystems:
|   ai j | i  |  j   H1  H 2
| | 1 | 2 
Entangled:
Not entangled: | | 1 | 2 
i, j
12 |  |
i  Hi
i  Hi
Subsystem: density matrix
1  Tr2 ( 12 )    i(1) |  | i(1) 
i
Entangled entire system: mixed state
There is not one, but more wavefunction:
| i  i  1..N with pi probability
Not entangled entire system: pure state
Exist a wavefunction: 1 | 1  1 |
Properties of density matrix
  
Hermitian
Positive semidefinit
real, ≥0 eigenvalues
spectral representation:    pi | i  i |
i
Normalized:
Tr  1    |  1
Tr 2  1   1
 1
ˆ : H1
Expectation values: O
Pure state
In two level syetem (CSB):
Mixed state
 | 01 |2  00 11
 H1
 Oˆ  | Oˆ | 
 Oˆ  Tr12 (Oˆ 12 )  Tr1 (Oˆ Tr2 12 )  Tr1 (Oˆ 1 )
If ρ1 pure:
  i | Oˆ |  || i   | Oˆ | 
  j | Oˆ  p |  | j    p 
1
1
i
If mixed:
i
j
i
i
i
i
i
i
| Oˆ | i 
An example: Spin singlet pair
Full system:
1
|
S

(|  |)
Wavefunction:
2
0
0
0
0 1 / 2  1 / 2
Density matrix:  S  
0  1 / 2 1 / 2

0
0
0
0
0
0

0
It cannot be written as a product -> Entangled state
1
2
1 / 2 0 

0
1
/
2


Subsystem: first spin: 1  (||  ||)  
Tr12  1 / 2  1
Up with ½, down with ½ probability
1
(|  |)
Not the same as: |  
2
1 / 2 1 / 2
2




1
/
2
1
/
2


Entanglement measures
There’s no operator such as < ψ |Ent| ψ >= the degree of entanglement
But exists some quanitity, that can tell if the Qstate is entangled or not:
General 2-qubit wavefunction:
|  c1 | 00  c2 | 01  c3 | 10  c4 | 11  ci  C
Not entangled state can be writen as a product:
|  (a | 0 1 b | 1 1 )(c | 0 2 d | 1 2 )
Then: c1  ac c2  ad
c3  bc c4  bd so c1c4  c2c3  0
Statemanet: if c1c4-c2c3≠0, then the Qstate is entangled
It can be generalized for bigger systems
Entanglement measures II - Von Neumann entropy
As other entropies, it measures the lack of our knowlendge of the
Qstate
S  Tr (  log2  )   pi log2 pi
(    pi | i  i |)
i
i
Where pi-s are the eigenvalues of ρ
Pure state: S=0, because: ρ=| ψ ><ψ| (p1=1, pi≠1=0)
(1)
(1)
S


p
log
p
Subsystem: 1
 i 2 i
i
Maximal entropy: pi(1)=1/M, S1=log2M
diagonal reduced density matrix -> maximally entangled state
Decoherence I
Two interacting subsystem (system and environment):
Together a closed system, well defined energy and phase
The energy and phase of subsystem are timedependent/undefined do
to the interaction
Relaxiation: with energy transfer
In Q system always followed by decoherence
Decoherence/dephasing: without energy transfer
Fluctuation of an external parameter (eg flux, magnetic field)
(assumption: Guassian distribution) -> the phase of the system
fluctuates in time -> time average -> decay in coherence -> loss of
phase information
The time average can be seen as ensamble average (eg. Slightly
different N qubit, or spatial fluctuation of the parameter)
Losing ability to interfere
In density matrix picture: rapid vanishing of the off-diagonal elements ->
just the classical occupation probabilities remain.
The off diagonal elements are also called „coherence”
Decoherence II
Let’s take N qubit, coupled to the same bath | j  a | 0 j  b | 1 j 
Each qubit gets a phase from the bath:
1 0 
Rz ( )  
i 
0
e


| 0 | 0 , | 1  ei | 1 
The phase has a Gaussian distribution
so it’s needed to average out to the phase:


1
2
 j   Rz ( ) | j   j | Rz ( ) p( )d where p( )  (4 ) e

So the density matrix:
 | a |2
 j    
a be
ab e  
2 
|b| 

2
4
Decoherence III
Time evolution:
Unitary: a closed system always have unitary time evolution
The state is always pure, so Tr 2  1
it  H
and it 12  [ H , 12 ] where 12 |  |
A subsystem:
i

1 (t )  Tr2 ( 12 (t ))  Tr2 (U (t ) 12U  (t )) where U (t )  exp(  Ht )
Consider a time evolution for the subsystem (not unitary):
i
i
1
 t 1 (t )   [ H , 1 (t )]  L[ 1 (t )]


(
t
)


[
H
,

(
t
)]

11
t 11
11


T1
Where L[ρ1(t)] is the so called
i
1
Lindblad decoherence term
 t  00 (t )   [ H ,  (t )]00  11

T1
Mostly L[ρ1(t)]=-γρij, so it
describes an exponential relaxation
i
1
 t  01 (t )   [ H ,  (t )]01   01

T2
T1  2T2
i
1
 t 10 (t )   [ H ,  (t )]10  10

T2
00

10
Basic idea of QST (1 QUBIT)
01 1

rk k

11 2 k 0, x, y , z
| 0 | and | 1 |
Where r0   00  11  1 , and r  (rx , ry , rz ) is the Bloch-vector
rx   01  10
rk [1,1]   | r |2  1  410 01  41100
ry  i (  01  10 )
| r | 1   1 Pure state
rz   00  11
  1 Mixed state
Spin measurement: rk
 Tr( k )
because Tr( i j )  2 ij
Projective operator measurement: p  Tr (  | 1  1 |) 
1
1
( 0   z )  (1 r z )
2
2
But the output of the measurement can be 1 or 0
-> need to measure multiple times
Other coefficients:
p  Tr[WW  (|1  1 |)l ]  Tr[W  (|1  1 |)l W ]
1 1 
W (| 1  1 |)l W  [  W  lzW ]
2 2

Recipe: prepare the same state, measure σx, σy, σz many times
(3 type of measurement) -> calculate ρij -> you have ρ
QST is multiqubit system

1
2N
r
 l1   l 2  ...  lN
l1...lN
l1 ...l N 0, x , y , z
Tr  r00...0  1
4N-1 real parameter
Tr (  ( j1  j2 ... jN ))  rl1...lN  j1l1 ... jN lN
N Qubit measurement:  j1  j2  ... jN
M qubit measurement: some σji=1
Notation:
If N-qubit measurements are possible -> one qubit operations are enough
Multiqubit measurement is not (hardly) realizable is solid-state
systems
If only single qubit measurements are possible -> one two qubit operation
is required
Theorem: Every M-qubit operation can be decomposed to the
product of single qubit operations and one two operation.
One-qubit measurement
N
H 
l 1


 x, y , z
 l  l 
N
   J    
lm
l , m 1 ,  x , y , z
l m
l
m
H
 


 x, y , z
Without loss of generality: εlα, Jlmαβ are positive real numbers
Optimal case: every parameter is switchable
1
1
p

Tr
(

|
1

1
|)

(



)

(1 r z )
0
z
σz:
2
2

i
i

σy: W  X ( 2 )  exp(   x xt1 )  exp( 4  x ) where t1  4
x

i
i

W

Y
(
)

exp(



t
)

exp(


)
where
t

σx:
y y 2
y
2
2

4
4 y
Notation: In charge qubit system εly-s are always zero.
In most real systems εlz-s are not switchable
By setting special Jlmαβ we can get Heisenberg, XXZ, XY etc. Models
Charge qubit: Fully controllable parameters
One qubit measurement – charge qubit
e2
1
1
H   Ech (ng ) z  E J ( x ) x
EC  EJ
EC 
0
2
2
2(Cg  2CJ )
C gV g
x
0
Ech (ng )  4EC (1  2ng )
EJ ( x )  2 EJ cos(
) ng 
2e
0
1
rz:
p  Tr (  | 1  1 |)  (1  rz )
2
ry: Set Фx=0
rotation around x axis
POM
1
p  Tr ( Rx (t x ) R (t x ) | 1  1 |)  (1  ry )
2


where t x 

2 E J (0) 4 E J0

x
rx: Set Фx=Ф0/2,ng=0
Rz(t=ħπ/8EC)=Rz(-π/2)
Set Фx=0, ng=1/2
1
Rx(t=ħπ/2EJ(0))=Rx(-π/2) p  Tr ( Rz , x , z Rz, x , z | 1  1 | )  (1  rx )
2
Set Фx=Ф0/2,ng=0
Rz(t=3ħπ/8EC)=Rz(-3π/2)
POM
Two qubit measurement
Basic two-qubit operation: time evolution:
Assumption: N=2, ε1α=ε2α=εα, Jlmαβ=Jlmαδαβ
1 i
(1  a 2 )c i
i
U12 (t )  exp(iH12t / )  (e cos  e cos  ) I  i
e sin  ( 1z  2 z) 
2
2
1  i
i
(e cos   ei cos ) 1z   2 z  (ei sin   2acei sin  ) 1x   2 x 
2
2
i i
(e sin   2acei sin  ) 1 y   2 y
2
Eg.: rzx in XY model (Jmnx=Jmny, Jmnz=0)



  ~  Y1 ( )U12 ( ) U12 ( )Y1 ( ) where   x
2
2
8 J12
p  Tr(~ | 1  1 |1 )  ( 2  rx0  rzx ) / 2 2
Charge qubit: The interaction is switchable by the flux Фi
1 2
H    (Ech (nl , g ) lz  EJ (lx ) lx )  Eint (1x , 2 x ) 1 y   2 y
2 l 1
E ( ) E ( )
Eint (1x ,  2 x )  J 1x J 1x
EL
CJ0 2 02
where EL  (
) ( 2 ) , Cqb1  (2CJ0 ) 1  Cg1
Cqb  L
Multi-qubit measurement
Theorem: with one two qubit and all single qubit operation, every m-qubit
operation can be performed
For an m-qubit measurement at least m-1 2-qubit operation needed
Eg.: rzzx in the XY model:


1
1
1
U 23 ( )U12 ( )Y1 ( ) 1zY1 ( )U12 ( )U 23 ( )  
 1x   1z   2 y   1z   2 z   3 x
2
2
4
4
2 2
p
2 2  2rx 00  2rzy 0  2rzzx
4 2
Not necessary to do exactly these measurements, its enough to do at least
4N-1 linearly independent measurment, so you have at least 4N-1 equation for
4N-1 variable.
If there are more equation, than variable, solve by RMS method.
Rehearsal: Josephson junction, phase qubits
I C
dV V
  I c sin( )
dt R
V
 0 d
2 dt
 0 d 2  0 1 d d
C


[ I c cos( )  I ]  0
2
2 dt
2 R dt d
U ( )  
0
[ I c cos(  )  I ]
2
Washboard potential
Measurement of entangled phase qubits I
Anharmonic potential: different level spacings
f10=5.1 GHz, ~30% tunability with bias current
1-qubit operations:
rotation around z: current pulse on bias line
rotation around x/y: microwave pulse
the phase of the pulse defines the rotation axis
the duration defines the rotation angle
Measurement: strong current pulse: |1> tunnels out
Two coupled qubit:
C
S
H int  (| 01  10 |  | 10  01 |) where S  x 10
2
C
At resonance: oscillation with S/h=10MHZ
freq between |01> and i|10>
2-qubit operation
Avioded crossing
Measurement of entangled phase qubits II
St
St
|00> -> |01>
|  (t )  cos( ) | 01  i sin( ) | 10 
2
2
Not eigenstate
T1=130 ns
T2*=80 ns
tfree=25 ns: entangeled state:
|  1 
1
(| 01  i | 10 )
2
But: pulselength: 10, 4 ns ->
not negligible: -> tfree=16 ns
90z rotation:
1
|  (| 01   | 10 )
2
eigenstate
No oscillation (destruction of
coherence?) -> 180z pulse
1
| 
(| 01  i | 10 )
2
Measurement of entangled phase qubits III
Single qubit fidelities:
F0=0.95, F1=0.85
Fidelity for |ψ1> F=0.75
After correction with single qubit
fidelities: F=0.87
Estimated maximal fidelity:
F=0.89
Cause of fidelity loss:
• single qubit decoherence
References
Y. V. Nazarov: Quantum Transport: Introduction to Nanoscience,
Cambridge University Press, 2009
http://qis.ucalgary.ca/quantech
Yu-xi Liu et al. Europhys. Lett. 67 (6), pp. 874-880 (2004)
Yu-xi Liu et al. PRB, 72, 014547 (2005)
M. Steffen et al. Science, 313, 1423 (2006)