Density matrix and its application
Density matrix
• An alternative of state-vector (ket) representation for a certain
set of state-vectors appearing with certain probabilities.
 
*
*

    ,     
 
  0   1
   pi  i  i
i
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Ensembles – pure and mixed states
• Pure sate
 
  *
       
 
2
*








 *    *

*
2
*
       
*
*
• Mixed state: set of pure quantum states with given
probabilities
   pi  i  i
i
• Mixing: weighting with classical probabilities
• Superposition: weighting with quantum probability amplitudes
• E.g. a pure sate can be a superposition
1
0 1
2
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
3
Are density matrices unique?
• Density matrices are not unique. This is the price for being
able to decompose entangled systems
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The Trace
Tr  A  a11  a22 
• where
aii
 ann   aii
i 1
are the elements of the main diagonal
• Eigenvalues and eigenvectors
n
Tr  A   i
i 1
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n
Av   v
Tr  Ak    ik
n
i 1
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Properties of the Trace
Tr  A  B  Tr  A  Tr  B
Tr  sA  sTr  A
Tr  AB  Tr  BA
Tr  A   Tr  AT 
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Trace- properties of density matrices
• The trace of any density matrix is equal to one
Tr     1
• for a pure state
Tr   2   1
• for a mixed state
Tr  2   1
• for a pure entangled system
2
Tr   EPR
 1
• for any mixed subsystem of an EPR pair
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Tr  EPR
 1
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2nd Postulate (evolution)
• The evolution of any closed physical system in time can be
characterized by means of unitary transforms
 n
 †
†
p
U


U

U
p


U

U

U
.





i
i
i
i 
 i i
i 1
 i 1

n
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†
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3rd Postulate (measurement)
• Any quantum measurement can be described bymeans of a set of
measurement operators {Mm}, where m stands for the possible
results of the measurement. The probability of measuring m if the
system is in state v can be calculated as
Pr  j    Tr  Pj  Pj†   Tr  Pj† Pj    Tr  Pj  
• and the system after measuring m gets in state
 n

Pj   pi  i  i  Pj
Pj  Pj
Pj  Pj
i 1


j 


n
 
  Tr  Pj  Pj  Tr  Pj  
Tr  Pj   pi  i  i  Pj 
 
  i 1
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Illustration
  0   1
  2  * 
,
   
2
*
   
• Measurement basis:
0
,1
M  0 0 , 1 1
1
M      M j  M †j
j 0
 0 0 0 0 1 11 1
 0 0   0 0 1 1  1 1
 0 0  0  0  1 1 1 1
 0

2
2
0 0  1
2
1 1
0 0   1 1  p 0 0 1 p 1 1 .
2
  2  * 
 2
M



   
2
*
   
 0
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0 
.
2
 
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Decomposing a system - Partial trace
A  TrB  AB 
• In general
B  TrA  AB 
A i k
B j l
Tr2  i k  j l   i k  Tr  j l

 i k  l j
 l j i k ,
TrB    
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 I
b B
A
 b   I A  b

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Decomposing a system - Partial trace
TrB  A  B  ATr  B
• For product state systems
AB   A  A   B  B
 A  TrB   AB   TrB   A  A   B  B

  A  A Tr   B  B    A  A  B  B .
Tr   1  2    2  1
B B 1
TrB  AB    B  B  A  A   A  A  A
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Decomposing a system - Partial trace
• For entangled systems
pure!
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TrB    
TrB   00  00    I  0
I  0  
00
 I
b B

00
A
 b   I A  b
 00  I  0    I  1   00  00  I  1

 I  0 
1
1
 00  11 
I  1   00   I  1 

 00  11 
2
2
 1 
 1 
 2
 2
 
 
1 0 0 0   0  1 1  1
0 1 0 0  0  1 0 1



0 ,

1,





  
 
2 0
2
0 0 1 0  0 
2 1 
2
0 0 0 1   0 
 1 
 1 
 
 
 2
 2
TrB  00 00
1
1
1
1
1
1
1
0
1
1  0 0  1 1  I.
 0
2
2
2
2
2
2
2
Max mixed!
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
Contains no
information
14
Geometrical interpretation of density
matrices
• Bloch sphere
0 1 
X  

1 0 
0 i 
Y  

i
0


  cos
1

2
0  ei sin

2
1.
0
Z  

0 1
1  rX  X  rY  Y  rZ  Z

2
r   rX , rY , rZ   sin  cos,sin  sin ,cos 
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†
†
1

U
r

U
1

U
r
U

     U U † 

,
2
2
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Pure and mixed states
• The density matrix is not unique!
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