Download INTRODUCTION TO NOISE AND DENSITY MATRICES. Slides in PPT.

Quantum Noise
Michael A. Nielsen
University of Queensland
Goals:
1. To introduce a tool – the density matrix – that is used
to describe noise in quantum systems, and to give some
examples.
Density matrices
Generalization of the quantum state used to describe
noisy quantum systems.
Terminology: “Density matrix” = “Density operator”
Ensemble
pj ,  j
Quantum
subsystem
Fundamental point
of view
What we’re going to do in this
lecture, and why we’re doing it
Most of the lecture will be spent understanding the
density matrix.
Unfortunately, that means we’ve got to master a
rather complex formalism.
It might seem a little strange, since the density matrix
is never essential for calculations – it’s a mathematical
tool, introduced for convenience.
Why bother with it?
The density matrix seems to be a very deep abstraction
– once you’ve mastered the formalism, it becomes far
easier to understand many other things, including
quantum noise, quantum error-correction, quantum
entanglement, and quantum communication.
Outer product notation
Let  and  be vectors.
Define a linear operation (matrix)   by
      
Example: 1 0  0   1   1    1
Connection to matrices:
If a   j a j j , and b   j bj j then a b k  bk* a .
 a1 
But a2  b1* b2*
 
 
 1  bk* a .
 
 
 a1 
Thus a b  a2  b1* b2* b3*  .
 
Outer product notation
1 
1
Example: 0 0   1 0  
0 
0
0 
0
Example: 1 1   0 1  
1 
0
0
0 
0
1 
1 0   0 0  1 1
Example: Z  

0

1


1 
0
Example: 0 1    0 1  
0 
0
0 
0
Example: 1 0    1 0  
1 
1
0 1 
Example: X  
 0 1 1

 1 0
1
0 
0
0 
0
Exercise: Find an outer product representation for Y .
Outer product notation
One of the advantages of the outer product notation
is that it provides a convenient tool with which to
describe projectors, and thus quantum measurements.
Recall: The projector P onto sp  e1 , e2  acts as
P  e1   e2   e3    e1   e2
This gives us a simple explicit formula for P , since
 e1 e1  e2 e2   e1   e2   e3    e1   e2
More generally, the projector onto a subspace spanned by
orthonormal vectors e1 ,..., em is given by
P   j ej ej .
Exercise: Suppose e1 ,..., ed
space. Prove that I   j ej
†
is an orthonormal basis for state
ej .
Exercise: Prove that a b = b a .
The trace operation
tr A    j Ajj
0 1 
Examples: X  
tr  X   0;

 1 0
 1 0
I 

0
1


tr I   2.
Cyclicity property: tr AB  =tr BA  .
tr AB    j AB  jj   jk Ajk Bkj   jk Bkj Ajk  k BA kk  tr BA 
Exercise: Prove that tr(|aihb|) = hb|ai.
I. Ensemble point of view
Imagine that a quantum system is in the state  j with
Probability of outcome k being in state j
probability pj .
We do a measurement described by projectors Pk .


Probability of outcome k   Pr k | state  j pj
k
   j Pk  j pj
k

  pj tr  j  j Pk
k

Probability
of being in
state j
Probability of outcome k  tr  Pk 
where    pj  j  j is the density matrix.
j
 completely determines all measurement statistics.
Qubit example: calculate the density matrix
Suppose   0 with probability 1.
1 
 1 0
Then   0 0    1 0  
.

0 
0 0 
Suppose   1 with probability 1.
0 
0 0 
Then   1 1    0 1  
.

1 
0 1 
0 i 1
Suppose  
with probability 1.
2
 0  i 1  0  i 1  1 1
1 1 i 
Then   

  i  1 i   i 1  .
2
2 
2  2 


conjugate
where    pj  j  j is the density matrix.
j
Density matrix is a
generalization of state
Density
matrix
Qubit example: a measurement using density
matrix
Suppose   0 with probability p, and   1 with
probability 1  p .
Then   p 0 0  1  p  1 1
0 
 1 0
0 0   p
 p
 1  p  

.



0 0 
0 1   0 1  p 
Measurement in the 0 , 1 basis yields
0  1 
p
Pr  0   tr   0 0   1 0 
 0 
0
1

p

 
Similarly, Pr 1  1  p .
 p.
Why work with density matrices?
Answer: Simplicity!
The quantum state is:
0 with probability 0.1
1 with probability 0.1
0 1
2
0 1
2
0 i 1
2
0 i 1
2
with probability 0.15
with probability 0.15
with probability 0.25
with probability 0.25
 21 0 

1
0
2

We know the
probabilities of
states and we want
to find or check
the density matrix
Dynamics and the density matrix
Suppose we have a quantum system in the state  j with
probability pj .
The quantum system undergoes a dynamics described
by the unitary matrix U .
The quantum system is now in the state U  j
probability pj .
with
The initial density matrix is    j pj  j  j .
The final density matrix is  '   j pjU  j  j U † .
U


†
p
U


U
j j j j .
 '  U U .
†
Initial density matrix
Dynamics and the density matrix
 '  U U .
†
This way, we can calculate a new density matrix from
old density matrix and unitary evolution matrix U
This is analogous to calculate a new state from old
state and unitary evolution matrix U.
The new formalism is more powerful since it refers also
to mixed states.
Single-qubit example: calculating new density matrix by
operating with an inverter on old density matrix
Suppose   0 with probability p, and   1 with
probability 1  p .
0 
p
Then   
.

0 1  p 
1  p
Suppose an X gate is applied. Then  '  X X  
 0
Suppose   0 and   1 with equal probabilities
I
“Completely mixed state”
Then   .
2
Suppose any unitary gate U is applied.
I
I
Then  '  U U † = .
2
2
0
.

p
1
2
.
How the density matrix changes
during a measurement
Worked Exercise : Suppose a measurement described by
projectors Pk is performed on an ensemble giving rise to
the density matrix  . If the measurement gives result
k show that the corresponding post-measurement density
matrix is
Pk Pk
'
k 
.
tr  Pk Pk 
Characterizing the density matrix
What class of matrices correspond to possible density matrices?
Suppose    j pj  j  j is a density matrix.

Then tr( )   j pj tr  j  j

  j pj  1
For any vector a ,
a  a   j pj a  j  j a
  j pj a  j
2
0
Trace
of a
density
matrix
is one
Summary : tr    =1 and  is a positive matrix.
Exercise : Given that tr    =1 and  is a positive matrix,
show that there is some set of states  j and probabilities
pj such that  = j pj  j  j .
Summary of the ensemble point of view
Definition: The density matrix for a system in state  j
with probability pj is    pj  j  j .
j
Dynamics:    '  U U † .
Measurement: A measurement described by projectors Pk
gives result k with probability tr  Pk   , and the postPk Pk
'
measurement density matrix is k 
.
tr  Pk Pk 
Characterization: tr    =1, and  is a positive matrix.
Conversely, given any matrix satisfying these properties,
there exists a set of states  j and probabilities pj
such that  = j pj  j  j .