2. Effective Non-Hermitian Hamiltonian of a pre- and post-selected quantum system Lev Vaidman 12.7.2015

5. Conditioned evolution Weak values and weak measurements Evolution of pre- and post-selected system Plan Two state-vector Past of a quantum particle 3-box paradox Correlations of uncorrelated pre- and post-selected particles

6. Unitary evolution

7. no click Non-unitary evolution

8. no click Non-unitary evolution

9. no click Non-unitary evolution

10. Collapse of the wave function

11. What is the evolution conditioned on nondetection? no click

12. What is the evolution conditioned on nondetection?

13. What was time evolution before the particle was detected, given that it was detected? What is the evolution conditioned on detection?

15. What was the interaction Hamiltonian for (weak) interaction with other systems? What is the evolution conditioned on detection?

16. What was the interaction Hamiltonian for (weak) interaction with other systems?

17. Where were the pre- and post-selected photons?

18. B A Asking photons where have they been POWER SPECTRUM A. Danan, D. Farfurnik, S. Bar-Ad and L. Vaidman, Phys. Rev. Lett. 111, 240402 (2013)

19. POWER SPECTRUM B A Asking photons where have they been Photons were on the paths they could pass

20. B A Asking photons where have they been POWER SPECTRUM Photons were on the paths they could pass

21. B A Asking photons where have they been POWER SPECTRUM Photons were on the paths they could pass

22. C F E POWER SPECTRUM B A Asking photons where have they been Photons were on the paths they could pass

23. Asking photons where have they been B C A F E POWER SPECTRUM Photons were on the paths they could not pass! How to explain this?

24. The two-state vector formalism

25. The two-state vector

34. The two-state vector is a complete description of a system at time t ? The two-state vector is what we can say now ( ) about the pre- and post-selected system at time t So, what can we say?

35. The Aharonov-Bergmann-Lebowitz (ABL) formula: described by the two-state vector: Strong measurements performed on a pre- and post-selected system

36. The outcomes of weak measurements are weak values Weak value of a variable C of a pre- and post-selected system described at time t by the two-state vector

37. Weak value of a variable C of a pre- and post-selected system described at time t by the two-state vector The outcomes of weak measurements are weak values

38. The weak value If the pre- and post-selected system is coupled to other systems through C, then its coupling at time t is described (completely) by the weak value

39. Effective non-Hermitian Hamiltonian Y. Aharonov, S. Massar, S. Popescu, J. Tollaksen, and L. Vaidman, PRL 77, 983-987 (1996 )

40. Asking photons where have they been B C A F E POWER SPECTRUM Photons were on the paths they could not pass! How to explain this?

41. B A The two-state vector formalism explanation

42. B A

43. B A POWER SPECTRUM The two-state vector formalism explanation

44. B C A F E D

45. B C A F E D

46. B C A F E D

47. B C A F E D POWER SPECTRUM

48. B C A F E D The two-state vector formalism explanation POWER SPECTRUM

49. B C A F E D The two-state vector formalism explanation

50. B C A F E D

51. B C A F E D POWER SPECTRUM K.J. Resch,, J.S. Lundeen,, A.M. Steinberg, PLA 324, 125 (2004) Experimental realization of the quantum box problem

52. Where is the ball? ? Aharonov and Vaidman, JPA 24, 2315 (1991) Aharon and Vaidman, PRA 77, 052310 (2008) The 3-boxes paradox Vaidman, Found. Phys. 29, 865 (1999)

53. The three box paradox It is in always !

54. The three box paradox It is always in

55. Two useful theorems: The three box paradox For dichotomic variables:

56. Correlation between separable pre- and post-selected particles Aharonov and Cohen, arXiv:1504.03797

57. Failure of the product rule for pre- and post-selected particles

58. Pre- and post-selected quantum systems are described best by two-state vector and weak values of observables Evolution of systems coupled to pre- and post-selected quantum systems is described by non-Hermitian Hamiltonians Conclusions

61. B C A F E D The one-state vector formalism explanation

62. C F E POWER SPECTRUM B A Photons: Wheeler is right!

65. For dichotomic variables Connection between strong and weak measurements

67. Pointer probability distribution ! Weak measurement of Pre-selection Post-selection The outcomes of weak measurements are weak values

68. Pointer probability distribution Weak Measurement of 20 particles pre-selected 20 particles post-selected Robust weak measurement on a pre- and post-selected single system The system of 20 particles !