Download Density matrices

Density Matrices and Quantum
Noise
Michael A. Nielsen
University of Queensland
Goals:
1. To review a tool – the density matrix – that is used to
describe noise in quantum systems.
2. To give more practical 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 mastering the density
matrix.
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.
Review: 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*  .
 
As we remember,
this is a matrix, we
showed how to
calculate it
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 .
REMINDER:
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 REMINDER: 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 and change
kets to bras
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
Pr(0)
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.
Pr(1)
Why work with density matrices?
Answer: Simplicity!
The quantum (mixed) 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
We know the
probabilities of
states and we
want to find or
check the density
matrix
 21 0 

1
0
2

with probability 0.25
with probability 0.25
Sum of these
probabilities must be
equal one
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.
S1 = U * S0
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 .
tr A    j Ajj
Problems to Solve
0 1 
Examples: X  
tr  X   0;

 1 0
Cyclicity property: tr AB  =tr BA  .
 1 0
I 

0
1


tr I   2.
Illustrate on
matrices
tr AB    j AB  jj   jk Ajk Bkj   jk Bkj Ajk  k BA kk  tr BA 
Exercise: Prove that tr(|aihb|) = hb|ai.
Exercise: Suppose e1 ,..., ed
space. Prove that I   j ej
†
Illustrate on
matrices
is an orthonormal basis for state
ej .
Exercise: Prove that a b = b a .