Transcript Slide 1

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.
I. Ensemble point of view
Imagine that a quantum system is in the state  j
probability pj .
with
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 outcome k  tr  Pk 
where    pj  j  j is the density matrix.
j
 completely determines all measurement statistics.
Qubit examples
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 


Qubit example
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

 
 p.
Similarly, Pr 1  1  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

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
 '  U U .
†


†
p
U


U
j j j j .
Single-qubit examples
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
0
.

p
Suppose   0 and   1 with equal probabilities 21 .
Then  
I
.
“Completely mixed state”
2
Suppose any unitary gate U is applied.
I
I
Then  '  U U † = .
2
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
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 post-
Pk 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 .
A simple example of quantum noise
X
With probability p the not gate is applied.
With probability 1-p the not gate fails, and nothing happens.
   j pj  j  j

  j pj pX  j  j X  pj 1  p   j  j
 pX X  1  p  
If we were to work with state vectors instead of density
matrices, doing a series of noisy quantum gates would
quickly result in an incredibly complex ensemble of states.

How good a not gate is this?
X
How "good" a not gate is this,
for a particular input  ?
We compare the ideal output,
  E     pX X  1  p   X  , to the actual output.
A quantum operation
The usual way two states a and b are compared is to
compute the fidelity, or overlap:
F (a , b )  a b .
The fidelity measures how similar
the states are, ranging from 0 (totally
dissimilar), up to 1 (the same).
To compare a with    j pj  j
fidelity,
F (a ,  )  a  a .
 j we compute the
Fidelity measures for
two mixed states are a
surprisingly complex topic!
How good a not gate is this?
How "good" a not gate is this,
for a particular input  ?
X
We compare the ideal output,
  E     pX X  1  p   X  , to the actual output.
The fidelity of the gate is thus
F (X  , E    )   XE     X 
 p      1  p   X   X 
 p  1  p   X 
The fidelity ranges between
for   0  1 / 2 .


2
p , for   0 , and 1,
II. Subsystem point of view
Bob
Alice

Pj
  kl kl k l

Pr( j )  tr Pj  I  


*


tr Pj  I  k l  m n
mn
klmn kl


*
 klmn kl mn
l n Pj  I k m

 klm kl ml tr Pj k l
*

 tr Pj A



 klm kl ml* l Pj k
*
where A   klm kl  ml
k l is
known as the reduced density
matrix of system A.
II. Subsystem point of view
Bob
Alice
Pj
  kl kl k l


Pr( j )  tr Pj A , where
A  trB      klm kl  ml* k l
A is the reduced density matrix for system A.
All the statistics for measurements on system A can
be recovered from A .
How to calculate: a method, and an example
An alternative, more convenient definition for the partial
trace is to define:
trB  a1 a2  b1 b2   a1 a2  tr  b1 b2 
 b2 b1 a1 a2
Then extend the definition linearly to arbitrary matrices.
Exercise: Show that this new definition agrees with
*
the old, that is, trB       klm kl  ml
k l when
   kl kl k l .
Example: If the system is in the state a b then
A  trB  a a  b b   b b a a  a a
The example of a Bell state
00  11
Example: Suppose  = 2 . Then the reduced density
matrix for the first system is given by:
A  trB    
trB  00 00   trB  00 11   trB  11 00   trB  11 11 

2
0 0+1 1

2
From Alice's point of view, it's
I

just like having the state 0
2
with probability 21 , and the
state 1 with probability 21 .
Under dynamics and measurement, the density
matrix behaves just as it does in the ensemble
point of view.
III. The density matrix as fundamental object
Postulate 1: A quantum system is described by a positive
matrix (the density matrix), with unit trace, acting on
a complex inner product space known as state space.
A system in state  j with probability pj has density matrix
 j pj  j .
Postulate 2: The dynamics of a closed quantum system are
described by    '  U U † .
Postulate 3: A measurement described by projectors Pk
gives result k with probability tr  Pk   , and the post-
Pk Pk
measurement density matrix is k 
.
tr  Pk Pk 
Postulate 4: We take the tensor product to find the state
space of a composite system. The state of one component
is found by taking the partial trace over the remainder of
the system.
'
Why teleportation doesn’t allow FTL communication
Alice
Bob
Why teleportation doesn’t allow FTL communication
Alice
01
Bob
01
Why teleportation doesn’t allow FTL communication
Alice
Bob
 00  11 
The initial state for the protocol is  

2


Bob's initial reduced density matrix is just the reduced
density matrix for a Bell state, B  I2 .
Why teleportation doesn’t allow FTL communication
Alice
Bob
B1   B2 Z   B3 X   B4 XZ 
01
2
B1  with probability
1
4
;
B2 Z  with probability
1
4
;
B3 X  with probability
1
4
; and
B4 XZ  with probability
1
4
.
Why teleportation doesn’t allow FTL communication
Alice
Bob
Bob's final reduced density matrix is thus
'
B
 
trA  B1 B1     ...
4
   Z   Z  X   X  XZ   ZX

4
2
2
  2  *    2
 *   
 *   




2
2
2
*
*
*
  
    *
      

4
I

2
 *  

2
 
Why teleportation doesn’t allow FTL communication
Alice
Bob
Bob’s reduced density matrix after Alice’s measurement
is the same as it was before, so the statistics of any
measurement Bob can do on his system will be the
same after Alice’s measurement as before!
Fidelity measures for quantum gates
Research problem: Find a measure quantifying
how well a noisy quantum gate works that has
the following properties:
It should have a simple, clear, unambiguous operational
interpretation.
It should have a clear meaning in an experimental
context, and be relatively easy to measure in a stable
fashion.
It should have “nice” mathematical properties that
facilitate understanding processes like quantum
error-correction.
Candidates abound, but nobody has clearly obtained a
synthesis of all these properties. It’d be good to do so!