Download Quantum Noise and Quantum Operations

More Quantum Noise and Distance
Measures for Quantum Information
(Some of Ch8 and Ch 9)
Patrick Cassleman
EECS 598
11/29/01
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Outline
• Types of Quantum Noise
–
–
–
–
–
–
Bit Flip
Phase Flip
Bit-phase Flip
Depolarizing Channel
Amplitude Damping
Phase Damping
• Distance measures for Probability Distributions
• Distance measures for Quantum States
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Background – The Bloch Sphere
• Remember :
  cos
q
2
ij
0  e sin
q
2
1
• The numbers q and j define a point on the unit
three-dimensional sphere 0
q

j
1
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise – Bit Flip
• A bit flip channel flips the state of a qubit from |0> to |1> with
probability 1-p
• Operation Elements:
1 0
E0  p I  p 

0
1


0 1 
E1  1  p X  1  p 

1 0
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise – Bit Flip
• Bloch sphere representation:
– Before y
-After
z
x
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise - Phase Flip
• Corresponds to a measurement in the |0>, |1> basis, with the
result of the measurement unknown
• Operation Elements:
1 0
E0  p I  p 

0
1


1 0 
E1  1  p Z  1  p 

0  1
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantun Noise – Phase Flip
• Bloch vector is projected along the z axis, and the x and y
components of the Bloch vector are lost
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise – Bit-phase Flip
• A combination of bit flip and phase flip
• Operation Elements:
1 0
E0  p I  p 

0
1


0  i 
E1  1  pY  1  p 

i 0 
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise – Bit-phase Flip
• Bloch vector is projected along y-axis, x and z components of the
Bloch vector are lost
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise – Depolarizing
Channel
• Qubit is replaced with a completely mixed state I/2 with
probability p, it is left untouched with probability 1-p
• The state of the quantum system after the noise is:
pI
E(  )   (1 p) 
2
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise –
Depolarizing Channel
• The Bloch sphere contracts uniformly
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise –
Depolarizing Channel
• Quantum Circuit Representation

I/2
(1-p)|0><0|+p|1><1|
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise –
Amplitude Damping
• Noise introduced by energy dissipation from the
quantum system
– Emitting a photon
• The quantum operation:
E AD (  ) E0 E0 E1 E1
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise –
Amplitude Damping
• Operation Elements:
1
E0  
0

1   
0
E1  
0

0

0 
2
•   sin q can be thought of as the probability of
losing a photon
• E1 changes |1> into |0> - i.e. losing energy
• E0 leaves |0> alone, but changes amplitude of |1>
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise –
Amplitude Damping
• Quantum Circuit Representation:
 in
0
 out
R y (q )
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise –
Amplitude Damping
• Bloch sphere Representation:
• The entire sphere shrinks toward the north pole, |0>
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise – Phase Damping
• Describes the loss of quantum information without the loss of
energy
• Electronic states perturbed by interacting with different charges
• Relative phase between energy eigenstates is lost
• Random “phase kick”, which causes non diagonal elements to
exponentially decay to 0
• Operation elements:
1
E0  
0
•
0 
1  l 
0
E1  
0
0 

l
l = probability that photon scattered without losing energy
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Examples of Quantum Noise – Phase Damping
• Quantum Circuit Representation:
 in
0
 out
R y (q )
• Just like Amplitude Damping without the CNOT gate
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Box 8.4 – Why Shrodinger’s Cat Doesn’t Work
• How come we don’t see superpositions in the world we observe?
• The book blames: the extreme sensitivity of macroscopic
superposition to decoherence
• i.e it is impossible in practice to isolate the cat and the atom in
their box
– Unintentional measurements are made
• Heat leaks from the box
• The cat bumps into the wall
• The cat meows
• Phase damping rapidly decoheres the state into either alive or dead
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Distance measures for Probability Distributions
•
•
•
•
We need to compare the similarity of two probability distributions
Two measures are widely used: trace distance and fidelity
Trace distance also called L1 distance or Kalmogorov distance
Trace Distance of two probability distributions px and qx:
1
D( p x , q x )   | p x  q x |
2 x
• The probability of an error in a channel is equal to the trace
distance of the probability distribution before it enters the channel
and the probability distribution after it leaves the channel
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Distance measures for Probability Distributions
• Fidelity of two probability distributions:
1
F ( px , qx )   px qx
2 x
• Fidelity is not a metric, when the distributions are equal, the
fidelity is 1
• Fidelity does not have a clear interpretation in the real world
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Distance measures for Quantum States
• How close are two quantum states?
• The trace distance of two quantum states  and s:
1
D(  , s )  tr |   s |
2
• If  and s commute, then the quantum trace distance between
 and s is equal to the classical trace distance between their
eigenvalues
• The trace distance between two single qubit states is half the
ordinary Euclidian distance between them on the Bloch sphere
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Trace Preserving Quantum Operations
are Contractive
• Suppose E is a trace preserving quantum operation. Let  and
s be density operators. Then
D( E (  ), E (s )) D(  , s )
• No physical process ever increases the distance between two
quantum states
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Fidelity of Two Quantum States
F (  , s )  tr  s
1/ 2
1/ 2
• When  and s commute (diagonal in the same basis),
degenerates into the classical fidelity, F(ri, si) of their eigenvalue
distributions
• The fidelity of a pure state  and an arbitrary state  :
F(  , )    
• That is, the square root of the overlap
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Uhlmann’s Theorem
• Given  and s are states of a quantum system Q, introduce a
second quantum system R which is a copy of Q Then:
F (  , s )  max |   |
 ,
• Where the maximizaion is over all purifications |> of  and |j>
of s into RQ
• Proof in the book
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Turning Fidelity into a Metric
• The angle between states  and s is:
A(  , s )  arccos F (  , s )
• The triangle inequality:
A(  , )  A(  , s ) A(s , )
• Fidelity is like an upside down version of trace distance
– Decreases as states become more distinguishable
– Increases as states become less distinguishable
– Instead of contractivity, we have monotonicity
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman
Monotonicity of Fidelity
• Suppose E is a trace preserving quantum operation, let  and s
be density operators, then:
F ( E (  ), E (s ))  F (  , s )
• Trace distance and Fidelity are qualitatively equivalent measures
of closeness for quantum states
– Results about one may be used to deduce equivalent results about the
other
– Example:
1  F (  , s )  D(  , s )  1  F (  , s )
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2
Quantum Noise and Distance
Patrick Cassleman
Conclusions
• Quantum Noise is modeled as an operator on a state and the
environment
• Quantum Noise can be seen as a manipulation of the Bloch
sphere
• Fidelity and Trace distance measure the relative distance
between two quantum states
• Quantum noise and distance will be important in the
understanding of quantum error correction next week
Advanced Computer Architecture Lab
University of Michigan
Quantum Noise and Distance
Patrick Cassleman