Resource page:
Quantum Tomographic Reconstruction using Kalman Filtering
Extra Material for the following papers:
[1] “Quantum Tomographic Reconstruction with Error Bars: a Kalman Filter Approach”,
NJP 11, 023028 (2009)
[2] “Statistical inference from imperfect photon detection”
arXiv:0906.3668. Accepted for publication in NJP (2009).
(Both coauthored with Stefan Scheel)
Contents:
1) .mpg movies illustrating the action of the Kalman Filter on the Loopy optical POVM (data courtesy I. Walmsley, Oxford)
For explanations, read the relevant sections in [1].
The movies show the mean values along with their marginal error bars, of the diagonal matrix elements of
the first POVM element (the one corresponding to the optical vacuum).
Initial pass to process the tomographic data:
After this first pass, the algorithm to restrict the solution to physical space calls the Kalman Filter again a number of times.
During every pass, the error bars go down in size. Note that most of the action happens during the first few seconds of each pass.
The remaining time of each pass is spent on the other POVM elements, which are not shown in the movie.
Loopy1.mpg (pass 1)
Loopy2.mpg (pass 2)
Loopy3.mpg (pass 3)
[and so on…]
Loopy7.mpg (final pass)
2) Matlab Code for Loopy POVM reconstruction
Main file:
Subroutines (zipped):
3) Matlab Code for simulating two-qubit tomography and subsequent KF reconstruction
Utility random number generators used in simulation: RandomGenerators.zip
(Most routines in here are based on the algorithms in Luc Devroye’s book “Non-uniform random variate generation”. )
Simulation/Reconstruction code (bugs resolved, replaced 17/11/09): TwoQubitKalman.zip
Note: 2) and 3) require that you download and install the (free) Sedumi package, available here.
4) Improved routines for incomplete beta and incomplete gamma functions.
Based on formulas in Abramowitz and Stegun, Chapters 6 (gamma) and 26 (beta). See also the arxiv (!) version of [2].
Core parts are written in C; for Windows users the DLLs are included.
For other OS’s, please use Matlab’s MEX functionality to compile the .C files.
5) Routines for calculating moments of inverse statistics of imperfect optical detectors.
For all explanations, refer to [2], esp. its arxiv version.
This package contains routines for calculating/approximating first and second order moments of the truncated Dirichlet distribution.
Uses all betagammainc files.