### Bayesian Quantum Tomography

```Byron Smith
December 11, 2013
What is Quantum State Tomography?
What is Bayesian Statistics?
1.
2.
1.
2.
3.
Example: Schrodinger’s Cat
3.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Conditional Probabilities
Bayes’ Rule
Frequentist vs. Bayesian
Interpretation
Analysis with a Non-informative Prior
Analysis with an Informative Prior
Sources of Error in Tomography
Error Reduction via Bayesian Analysis
Adaptive Tomography
Conclusion
References
Supplementary Information

Tomography comes from tomos meaning section.

Classically, tomography refers to analyzing a 3dimensional trajectory using 2-dimensional slices.

Quantum State Tomography refers to identifying a
particular wave function using a series of
measurements.
P ( A)  P ( A  B )  P ( A  C )
P(A | B) 
P(A  B)
P(B)
P(A | B)
P(A | C )
P ( B | A) 
P( A | B)P(B)
 ( x | y ) is the Likelihood
P ( A)
 ( y ) is the Prior Probability
 ( y | x) 
 ( x | y ) ( y )
  ( x | y ) ( y ) dy
(or just prior)
 ( y | x ) is the Posterior
Probability (or just
posterior)

Inference on probability
arises from the frequency
that some outcome is
measured and that
measurement is random.

Inference on probability
arises from a prior
probability assumption
weighted by empirical
evidence.

In other words, a
measurement is random
only due to our ignorance.

Measurements are
inherently random.
Frequentist
Bayesian
Frequentist: Given N cats, there
are some that are alive and some
that are dead. The probability
that the cat is alive is associated
with the fact that we sample
randomly and a*N of the cats are
alive.
 cat  Palive alive
 a alive
alive  Pdead dead
alive  (1  a ) dead
dead
dead
Bayesian: The probability a is a
random variable (unknown) and
therefore there is an inherent
probability associated with each
particular cat.
p ( x | a )  a (1  a )
x
N
( x | a) 

a i (1  a )
x
1 x i
1 x
 a (1  a )
Y
i 1
N
 cat  Palive alive
 a alive
alive  Pdead dead
alive  (1  a ) dead
Y 
dead
x
i
i 1
dead

a
 ( x | a )  0  aˆ 
Y
N
N Y
N
( x | a) 
a
xi
(1  a )
1 x i
 a (1  a )
Y
N Y
i 1
 (a ) 
  (a | x) 
 cat  Palive alive
 a alive
alive  Pdead dead
alive  (1  a ) dead
 (   )
 ( )  (  )
a
 1
(1  a )
 (    N )
 (  Y )  (   N  Y )
a
 1
  Y 1
(1  a )
dead
dead
a | x ~ Beta (  Y ,   N  Y )
 aˆ 
 Y
  N
  N  Y 1
True value of a = 0.3
N
Frequentist
Bayesian
3
0.67
0.6
25
0.2
0.22
75
0.24
0.25
250
0.336
0.337
Flourine-18, halflife=1.8295 Hours
Cat has been in for
3 hours


N
Frequentist
Bayesian
3
0.67
0.39
25
0.28
0.29
75
0.267
0.272
250
0.316
0.316

Error in the measurement basis.

Error in the counting statistics.

Error associated with stability.
◦ Detector efficiency
◦ Source intensity

Model Error
aˆ 
Y
aˆ 
N
var( aˆ )  var(
Y
)
Y (N  Y )
N
N
3
var( aˆ ) 
 Y
  N
N (  Y )(   N  Y )
(    N )
N
Var(a)
N
Var(a)
3
0.027
3
0.013
25
0.0074
25
0.0057
75
0.0029
75
0.0026
250
0.00079
250
0.00076
Frequentist
3
Bayesian (α=3.1, β=6.9)


There are several
measures of “lack of fit.”
Likelihood

Infidelity

Shannon Entropy
  N
 ( nˆ  n ) 2 
exp  

2
ˆ
2




1  F ( ˆ ,  )  1  Tr 

 ˆ
H    p i log( p i )
i
 

2

The number of measurements required to sufficiently
identify ρ can be reduced when using a basis which
diagonalizes ρ.

To do so, measure N0 particles first, change the
measurement basis, then finish the total
measurement.

Naturally this can be improved with Bayesian by using
the first measurements as a Prior.

Quantum State Tomography is a tool used to identify a
density matrix.

There are several metrics of density identification.

Bayesian statistics can improve efficiency while
providing a new interpretation of quantum states.
1.
2.
3.
4.
J. B. Altepeter, D. F. V. James, and P. G. Kwiat, Qubit
Quantum State Tomography, in Quantum State
Estimation (Lecture Notes in Physics), M. Paris and J.
Rehacek (editors), Springer (2004).
D. H. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R.
Blume-Kohout, and A. M. Steinberg, Phys. Rev. Lett.
111, 183601 (2013).
F. Huszár and N. M. T. Houlsby, Phys. Rev. A 85,
052120 (2012).
R. Blume-Kohout, New J. Phys. 12, 043034 (2012).

For density operators which are not diagonal, we use a
basis of spin matrices:
 
a
i
i
i

If choose an instrument orientation such that the first
parameter is 1, we are left with the Stoke’s
Parameters, Si:
 

1

  0   S i i 
2
i

The goal is then to identify the Si.

Used to visualize a superposition of photon
polarizations.

The x-axis is 45 degree polarizations.
The y-axis is horizontal or vertical polarizations.
The z-axis is circular polarizations.



Using a matrix basis similar to that for Stoke’s
Parameters, one can exactly identify a polarization with
three measurements.

Note that an orthogonal basis is not necessary.

Errors in measurement and instrumentation will
manifest themselves as a disc on the Poincare Sphere:

(a) Errors in measurement basis. (b) Errors in intensity
or detector stability.

Because there is a finite sample size, states are not
characterized exactly.

Each measurement constrains the sample space.



Optimization can lead to states which lie
outside the Poincare Sphere.
Unobserved values a registered as zeros in
the density matrix optimization. This can be
unrealistic.
There is no direct measure of uncertainty
within maximum likelihood estimators.



Parameter space can be constrained through
the prior.
Unobserved values can still have some small
probability through the prior.
Uncertainty analysis can come directly from
the variance of the posterior distribution,
regardless of an analytical form.
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