1. Likelihood and entropy for quantum tomography Z. Hradil, J. Řeháček Department of Optics Palacký University,Olomouc Czech Republic Work was supported by the Czech Ministry of Education.

2. Collaboration SLO UP ( O. Haderka) Vienna: A. Zeilinger, H. Rauch, M. Zawisky Bari: S. Pascazio Others: HMI Berlin, ILL Grenoble

3. Motivation Inverse problems Quantum measurements vs. estimations MaxLik principle MaxEnt principle Several examples Summary Outline

4. Diffraction on the slit as detection of the direction Motivation 1:

5. Measurement according to geometrical optics: propagating rays Measurement according to the scalar wave theory: diffraction

6. Estimation: posterior probability distribution Fisher information: width of post. distribution Uncertainty relations

7. registered mean values j = 1,..M desired signal i= 1,..N N number of signal bins (resolution) M number of scans (measurement) Motivation 2: Inversion problems

8. Over-determined problems M > N (engineering solution: credible interpretation) Well defined problems M = N (linear inversion may appear as ill posed problem due to the imposed constraints) Under-determined problems M < N (realm of physics)

9. Inversion problems: Tomography Medicine: CT, NMR, PET, etc.: nondestructive visualization of 3D objects Back-Projection (Inverse Radon transform) ● ill-posed problem ● fails in some applications

10. Motivation 3: All resources are limited!

11. Elements of quantum theory Probability in quantum mechanics Desired signal: density matrix Measurement: positive-valued operator measure (POVM)

12. Complete measurement: need not be orthogonal Generic measurement: scans go beyond the space of the reconstruction

13. Quantum observables: q-numbers Stern-Gerlach device

14. Mach-Zehnder interferometer

15. Maximum Likelihood (MaxLik) principle selects the most likely configuration Likelihood L quantifies the degree of belief in certain hypothesis under the condition of the given data. Principle of MaxLik

16. MaxLik principle is not a rule that requires justification. Mathematical formulation: Fisher Bet Always On the Highest Chance! Philosophy behind

17. MaxLik estimation Measurement: prior info posterior info Bayes rule: The most likely configuration is taken as the result of estimation Prior information and existing constraints can be easily incorporated

18. Likelihood is the convex functional on the convex set of density matrices Equation for extremal states

19. MaxLik Linear MaxLik inversion: Interpretation

20. Various projections are counted with different accuracy. Accuracy depends on the unknown quantum state. Optimal estimation strategy must re-interpret the registered data and estimate the state simultaneously. Optimal estimation should be nonlinear.

23. MaxLik = Maximum of Relative Entropy Solution will exhibit plateau of MaxLik states for under-determined problems (ambiguity)!

24. Laplace's Principle of Insufficient Reasoning: If there is no reason to prefer among several possibilities, than the best strategy is to consider them as equally likely and pick up the average. Principle of Maximum Entropy (MaxEnt) selects the most unbiased solution consistent with the given constraints. Mathematical formulation: Jaynes Philosophy behind Maximum Entropy

25. MaxEnt solution Lagrange multipliers are given by the solution of the set of nonlinear constraints Entropy Constraints

26. MaxLik: the most optimistic guess. Problem: Ambiguity of solutions! MaxEnt: the most pesimistic guess. Problem: Inconsistent constraints. Proposal: Maximize the entropy over the convex set of MaxLik states! Convexity of entropy will guarantee the uniqueness of the solution. MaxLik will make the all the constraints consistent.

27. Implementation Parametrize MaxEnt solution Maximize alternately entropy and likelihood MaxEnt assisted MaxLik inversion

28. MaxEnt assisted MaxLik strategy Searching for the worst among the best solutions!

29. Interpretation of MaxEnt assisted MaxLik The plateau of solutions on extended space Regular part“Classical” part

30. MaxLik strategy Specify the space (arbitrary but sufficiently large) Find the state Specify the space Specify the Fisher information matrix F

31. Phase estimation Reconstruction of Wigner function Transmission tomography Reconstruction of photocount statistics Image reconstruction Vortex beam analysis Quantification of entanglement Operational information Several examples

32. (Neutron) Transmission tomography Exponential attenuation

33. Filtered back projection Maximum likelihood J. Řeháček, Z. Hradil, M. Zawisky, W. Treimer, M. Strobl: Maximum Likelihood absorption tomography, Europhys. Lett. 59 694- 700 (2002).

34. MaxEnt assisted MaxLik Numerical simulations using 19 phase scans, 101 pixels each (M=1919) Reconstruction on the grid 201x 201 bins (N= 40401) Object MaxLik1 MaxLik2 MaxEnt+Lik

35. Fiber-loop detector Commercially available single- photon detectors do not have single-photon resolution Cheap (partial) solution: beam splitting Coincidences tell us about multi- photon content J.Řeháček et al.,Multiple-photon resolving fiber-loop detector, Phys. Rev. A (2003) 061801(R)

36. Fiber loop as a multi-channel photon analyser

37. Example: detection of 2 events = 4 channels Inversion of Bernouli distribution for zero outcome

38. Results of MaxLik inversion: True statistics : (a) Poissonian (b) Composite (d) Gamma (d) Bose-Einstein

39. True statistics: 50/50 superposition of Poissonian statistics with mean numbers1 and 10 Data: up to 5 counted events (= 32 channels) Mesh: 100 Original MaxLik MaxLik & MaxLik

40. Thank you!