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

Classical system:
Density function in phase-scape
W(x,p) non negativ, normalized
W ( x , p )

0 
W
Marginal distribution:
( x , p ) dxdp

1 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)

QM system:
Entire system: wavefunction
|
 i

H |
   i a i
|
 i
  
|
  
|
Multiple subsystems:
|
   a i i , j
Entangled: j
|
 i
 
|

|
Not entangled: |

 j


|


1
H
1

|

|

1

|


2
H

2

2


12

|

 i
 i


H i
H i
 
|
Subsystem: density matrix

1

Tr
2
(

12
)
    i
( 1 )
|
  
|
 i
( 1 ) i
Entangled entire system: mixed state

There is not one, but more wavefunction:
|
 i
 i

1 ..
N with p i probability
Not entangled entire system: pure state
Exist a wavefunction:

1

|

1
 
1
|

  

Hermitian
Positive semidefinit real, ≥0 eigenvalues spectral representation:
   i p i
|
 i
  i
|
Normalized:
Tr

2 
1
Tr
 
1



1
 
Pure state
|
 
1
In two level syetem (CSB):

Expectation values:
  
| |
O


:
1
H
 1
Mixed state

H
1

If ρ
1
  pure:
Tr
12
 i
1

If mixed:
( i |

12
|
 j
 j |
 i
)

 p i
Tr
1
(
 
|
 i
|| i
Tr
2

12
)

1
 
  i
| j

|


Tr
1
(
|

 i p i


1
)
 
| i
|

01
|
2 
|
 i


00

11

Full system:
Wavefunction: | S

1
2
(|
 
|

)
Density matrix:

S


0

 0


0
0
0
1 / 2

1 / 2
0
0

1 / 2
1 / 2
0
0
0
0
0





It cannot be written as a product -> Entangled state
Subsystem: first spin:

1

1
2
(|

|
Tr

1
2

|

|)

1 / 2




1 /
0
2
1
Up with ½, down with ½ probability
Not the same as: |
 
1
2
(|
 
|

)
 



1
1 /
/ 2
2
1 /
0
2


1
1 /
/ 2
2



 
2

There’s no operator such as < ψ |E nt
| ψ >= the degree of entanglement
But exists some quanitity, that can tell if the Qstate is entangled or not:
General 2-qubit wavefunction:
|
  c
1
| 00
  c
2
| 01
  c
3
| 10
  c
4
| 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: c
1
 ac c
2
 ad c
3
 bc c
4
 bd so c
1 c
4
 c
2 c
3

0
Statemanet: if c
1 c
4
-c
2 c
3
≠0, then the Qstate is entangled
It can be generalized for bigger systems

As other entropies, it measures the lack of our knowlendge of the
Qstate
S
 
Tr (
 log
2

)
   i p i log
2 p i
(
   i p i
| i
 i | )
Where p i
s are the eigenvalues of ρ
Pure state: S=0, because: ρ=| ψ ><ψ| (p
1
=1, p i≠1
=0)
Subsystem:
S
1
   i p i
( 1 ) log
2 p i
( 1 )
Maximal entropy: p i
(1) =1/M, S
1
=log
2
M diagonal reduced density matrix -> maximally entangled state

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”

Let’s take N qubit, coupled to the same bath
Each qubit gets a phase from the bath:
R z
(

)


1


0
0 e i




| 0

| 0

, | 1
|
 j
 a | 0 j
  b | 1 j

 e i

| 1

The phase has a Gaussian distribution so it’s needed to average out to the phase:
 j
 

 
R z
(

) |
 j
So the density matrix:
  j
| R z

(

) p (

) d

 j


 a
| a
 be
|
2
  ab
 e
  

| b |
2 where p (

)

( 4

)

1
2
 2 e

4


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

|
  
|

1
( t )
A subsystem:

Tr
2
(

12
( t ))

Tr
2
( U ( t )

12
U

( t )) where U ( t )
 exp(
 i

Consider a time evolution for the subsystem (not unitary):
Ht )
 t

1
( t )
  i
[ H ,

1
( t )]

L [

1
( t )]

Where L[ρ
1
(t)] is the so called
Lindblad decoherence term
Mostly L[ρ
1
(t)]=γρ ij
, so it describes an exponential relaxation
 t

11
( t )
  i

[ H ,

( t )]
11

1
T
1

11
 t

00
( t )
  i

[ H ,

( t )]
00

1
T
1

11
T

1
2 T
2
 t

01
( t )
  i

[ H ,

( t )]
01

1
T
2

01
 t

10
( t )
  i

[ H ,

( t )]
10

1
T
2

10

 

00

10
Where r
0 r x r y r z

01


11


00


1
2


01
 i (

01

10
 
10
)
 
00
 k


11

11
0 ,
 y x , r k
, z

1
 k
, and r k
| r
| r

0

|


[
|


1
1 , 1 ] and
( r x
,

 r y
, r z


 
)
1
1
| 1

|

| is the Bloch-vector r |
2 
1

4
Pure state
Mixed state

10

01

Spin measurement: r k

Tr (
 k
) because Tr (
 i
 j
)

2
 ij
4

11

00
Projective operator measurement: p

Tr (

| 1

1 |)

1
2
(

0
  z
)

1
2
( 1
 r z
)
But the output of the measurement can be 1 or 0
-> need to measure multiple times p

Tr [ W

W

(| 1

1 |) ]

Tr [

W

(| 1
Other coefficients:
W

(| 1 l

1 |) l
W

[
1

Recipe: prepare the same state, measure σ x
(3 type of measurement) > calculate ρ ij
, σ y
, σ z
2 many times
> you have ρ

1
W
2
1 |)
  l
W lz
W
]
]

 
1
2
N l
1
...
l
N


0 , x , r l 1 ...
lN
 y , z l 1
  l 2

...
  lN
Tr
  r
00 ...
0

1
4 N -1 real parameter
Tr (

(
 j
1
  j
2

...
  j
N
))
 r l
1
...
l
N
 j
1 l
1
...
 j
N l
N
N Qubit measurement:
 j
1
  j
2

...
  j
N
M qubit measurement: some σ ji
=1
Notation:
If N-qubit measurements are possible -> one qubit operations are enough systems
Multiqubit measurement is not (hardly) realizable is solid-state
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.

H
 l
N  

1
  x , y , z
 l

 l

 l l
N  
,
 m m

1

,
  x , y ,
J z
 lm
 l

  m

H

 
 x , y , z




Without loss of generality: ε lα
, J lm
αβ are positive real numbers
Optimal case: every parameter is switchable
σ z
:
σ y
:
σ x
: p

Tr (

| 1

1 |)

1
(

0
  z
)

1
( 1
 r z
)
2 2
W
W


X
Y (

2
(

2
)
)

 exp(
 exp(
 i

 i
 y
 x

 x t
1
) y t
2
)

 exp(
 exp(
 i
4
 x
) i
4
 y
) where where t
2 t
1




4


 x
4
 y
Notation: In charge qubit system ε ly
-s are always zero.
In most real systems ε lz
-s are not switchable
By setting special J lm
αβ we can get Heisenberg, XXZ, XY etc. Models
Charge qubit: Fully controllable parameters

H
 r z
:


E ch
1
2

E ch
( n g p

)
( n g

)
 z
4 E
C
Tr (

|

( 1
1
1
2
E
J

(

2 n g

1 x
|)
)
)

 x
1
2
E
J
( 1
E
C


(
 x r z
)
)
2 ( C g
 e
2

2 E
J
0
2 C
J
0 cos(
)



0 x )
E
C
 n g

E
J
C g
V g
2 e r y
: Set Ф x
=0 rotation around x axis
POM r x
: Set Ф x
=Ф
0
/2,n g
=0
R z
(t=ħπ/8E
C
)=R z
(π/2)
Set Ф x
=0, n g
=1/2
R x
(t=ħπ/2E
J
(0))=R x
(π/2)
Set Ф x
=Ф
0
/2,n g
=0
R z
(t=3ħπ/8E
C
)=R z
(3π/2) p

Tr ( R x where p

Tr ( t x
( t x

)

R x

2 E
J
R z , x , z


( t x
( 0 )
)

R z

, x , z

|
|
1
1



4 E
J
0
1

|)
1

|)
1
2

( 1
1
2
 r y
( 1
)

POM r x
)

Basic two-qubit operation: time evolution:
Assumption: N=2, ε
1α
=ε
2α
=ε
α
, J lm
αβ
=J lm
α δ
αβ
U
12
( t )
 exp(
 iH
12 t /  )

1
2
( e i
 cos
  e
 i
 cos

) I
 i
( 1
 a
2
) c e
 i

2 sin

(

1 z
 
2 z
)

1
2
( e
 i
 cos
  e i
 cos

)

1 z
 
2 z
 i
2
( e i
 sin
 
2 ace
 i
 sin

)

1 x
 
2 x
 i
( e i
 sin
 
2 ace
 i
 sin

)

1 y
 
2 y
2
Eg.: r zx
 in XY model (J mn x =J
 
Y
1
(

2
) U
12
(

)

U

12
(
 mn y , J
) Y
1

(

2 mn z =0)
) where p

Tr ( | 1

1 |
1
)

( 2
 r x 0
 


8 J x
12
 r zx
) / 2 2
Charge qubit: The interaction is switchable by the flux Ф i
H
 
1
2
E int
(

1 x
,

2 x
) l
2 

1
(

E ch
( n l , g
)
 lz

E
J
(

1 x
) E
J
E
L
(

1 x
)

E
J
(
 lx
)
 lx
)

E int
(

1 x
,

2 x
)

1 y where E
L

(
C
0
J
C qb
)
2
(


2
0
2
L
) ,

1
C qb
 
2 y

( 2 C
J
0
)

1

C g

1

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.: r zzx in the XY model:

U
23
(

)

U
12
(

) Y
1

(

2
)

1 z
Y
1
(

2
) U
12
(

) U
23
(

)
 
2
1
2

1 x

1
4

1 z
 
2 y

1
4

1 z
 
2 z
 
3 x p

2 2

2 r x 00

4 2
2 r zy 0

2 r zzx
Not necessary to do exactly these measurements, its enough to do at least
4 N -1 linearly independent measurment, so you have at least 4 N -1 equation for
4 N -1 variable.
If there are more equation, than variable, solve by RMS method.

I

C dV dt

V
R

I c sin(

) V


2

0 d
 dt
C

2

0 d
2
 dt
2


2

0
1 d

R dt
 d d

[ I c cos(

)

I

]

0
U (

)
 

2

0 [ I c cos(

)

I

]
Washboard potential

Anharmonic potential: different level spacings f
10
=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:
H int

S
(| 01

10 |

| 10

01 |) where
2
At resonance: oscillation with S/h=10MHZ freq between |01> and i|10>
S

C x
C


10
2-qubit operation
Avioded crossing

|00> -> |01>
Not eigenstate
|

( t )

St cos(
2 
) | 01
  i sin(
St
2 
) | 10
 T
1
=130 ns
T
2
*=80 ns t free
=25 ns: entangeled state:
|

1

1
2
(| 01
  i | 10

)
But: pulselength: 10, 4 ns -> not negligible: -> t free
=16 ns
90 z rotation:
|
 
1
2
(| 01
 
| 10

) eigenstate
No oscillation (destruction of coherence?) -> 180 z
|
 
1
2
(| 01
  i pulse
| 10

)

Single qubit fidelities:
F
0
=0.95, F
1
=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

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)