1 / 24

Bayesian Quantum Tomography

Bayesian Quantum Tomography. Byron Smith December 11, 2013. Outline. What is Quantum State Tomography? What is Bayesian Statistics? Conditional Probabilities Bayes ’ Rule Frequentist vs. Bayesian Example: Schrodinger’s Cat Interpretation Analysis with a Non-informative Prior

max
Download Presentation

Bayesian Quantum Tomography

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Bayesian Quantum Tomography Byron Smith December 11, 2013

  2. Outline • What is Quantum State Tomography? • What is Bayesian Statistics? • Conditional Probabilities • Bayes’ Rule • Frequentist vs. Bayesian • Example: Schrodinger’s Cat • 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

  3. What is Quantum (State) Tomography? • Tomography comes from tomos meaning section. • Classically, tomography refers to analyzing a 3-dimensional trajectory using 2-dimensional slices. • Quantum State Tomography refers to identifying a particular wave function using a series of measurements.

  4. Conditional Probability

  5. Bayes’ Rule is the Likelihood is the Prior Probability (or just prior) is the Posterior Probability (or just posterior)

  6. Frequentist vs. Bayesian • Inference on probability arises from the frequency that some outcome is measured and that measurement is random. • In other words, a measurement is random only due to our ignorance. • Inference on probability arises from a prior probability assumption weighted by empirical evidence. • Measurements are inherently random. Frequentist Bayesian

  7. Example: Interpretation of Schrodinger’s Cat 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. Bayesian: The probability a is a random variable (unknown) and therefore there is an inherent probability associated with each particular cat.

  8. Example: Schrodinger’s Frequentist Cat

  9. Example: Schrodinger’s Bayesian Cat

  10. A Graphical Comparison:Uniform Prior (α = 1, β= 1) True value of a = 0.3

  11. A Graphical Comparison:Beta Prior (α = 3.1, β = 6.9) • Flourine-18, half-life=1.8295 Hours • Cat has been in for 3 hours

  12. Sources of Error in Tomography • Error in the measurement basis. • Error in the counting statistics. • Error associated with stability. • Detector efficiency • Source intensity • Model Error

  13. Estimation Error Frequentist Bayesian (α=3.1, β=6.9)

  14. Density Estimation • There are several measures of “lack of fit.” • Likelihood • Infidelity • Shannon Entropy

  15. Adaptive Tomography • The number of measurements required to sufficiently identify ρ can be reduced when using a basis which diagonalizesρ. • 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.

  16. Conclusion • 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.

  17. References • 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. A85, 052120 (2012). • R. Blume-Kohout, New J. Phys.12, 043034 (2012).

  18. Stoke’s Parameters • For density operators which are not diagonal, we use a basis of spin matrices: • If choose an instrument orientation such that the first parameter is 1, we are left with the Stoke’s Parameters,Si: • The goal is then to identify the Si.

  19. Poincare Spheres • 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.

  20. Tomography with Poincare Spheres • 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.

  21. Imperfect Tomogrpahy • 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.

  22. Statistical Tomography • Because there is a finite sample size, states are not characterized exactly. • Each measurement constrains the sample space.

  23. Pitfalls of Maximum Likelihood Analysis • 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.

  24. Solutions through Bayesian Analysis • 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.

More Related