2. If is an element of reality then

3. If then is an element of reality For dichotomic variables:

4. Two useful theorems: If is an element of reality then If then is an element of reality For dichotomic variables: The three box paradox Tunneling particle has (weak) negative kinetic energy

6. Pointer probability distribution Weak measurements performed on a pre- and post-selected ensemble ! Weak Measurement of The particle pre-selected The particle post-selected

7. 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 !

8. Properties of a quantum system during the time interval between two measurementsProperties of a quantum system during the time interval between two measurements Y. Aharonov and L. Vaidman PRA 41, 11 (1990 ) Another example: superposition of positive shifts yields negative shift A. Botero Superposition of Gaussians shifted by small values yields the Gaussian shifted by the large value

10. Generalized two-state vector protection

11. PRL 58, 1385 (1987)

13. What is the past of a quantum particle?

14. The “past” and the “Delayed Choice” Double-Slit Experiment J.A. Wheeler 1978 The present choice of observation influences what we say about the “past” of the photon; it is undefined and undefinable without the observation. The “past” of the photon is defined after the observation Wheeler: No phenomenon is a phenomenon until it is an observed phenomenon. My lesson:

15. Wheeler delayed choice experiment Wheeler: The photon took the upper path It could not come the other way

16. Wheeler delayed choice experiment Wheeler: The photon took both paths Otherwise, the interference cannot be explained

17. Interaction-free measurement Did photon touched the bomb? Wheeler: The photon took the upper path It could not come the other way

18. The past of a quantum particle can be learned by measuring the trace it left

19. Wheeler delayed choice experiment Wheeler: The photon took the upper path It could not come the other way The trace shows Wheeler’s past of the photon

20. Wheeler delayed choice experiment Wheeler: The photon took both paths Otherwise, the interference cannot be explained The trace shows Wheeler’s past of the photon

21. Interaction-free measurement Did photon touched the bomb? Operational meaning: Nondemolition measurements show NO! Yes No

22. Wheeler delayed choice experiment Nondemolition measurements show that the photon took the upper path Operational meaning: Yes No Yes

23. Nondemolition measurements show that the photon took one of the paths Operational meaning: Yes No Yes Where is the photon when it is inside a Mach-Zehnder interferometer? But nondemolition (strong) measurements disturb the photon

24. Weak measurementsOperational meaning: Where is the photon when it is inside a Mach-Zehnder interferometer? The information is obtained from weak measurements on an ensemble of identically prepared photons “Half a photon” or half the times the photon passes each path (no disturbance at the limit) Yes No Yes or No or Half a photon Yes or No or Half a photon

25. Wheeler delayed choice experiment Weak measurementsOperational meaning: (no disturbance at the limit) Yes No Yes No The information is obtained from a pre- and post-selected ensemble

26. Interaction-free measurement Did photon touched the bomb? Weak measurements Operational meaning: The information is obtained from a pre- and post-selected ensemble Yes No Yes No

27. Interaction-free measurement Yes No Strong measurements Did photon touched the bomb? Operational meaning:

28. Interaction-free measurement Did photon touched the bomb? Weak measurements Operational meaning: No Yes (no disturbance at the limit) The information is obtained from a pre- and post-selected ensemble

29. Wheeler delayed choice experiment Weak measurementsOperational meaning: (no disturbance at the limit) The information is obtained from a pre- and post-selected ensemble Yes No

30. Interaction-free measurement Did photon touched the bomb? Weak measurements Operational meaning: No Yes (no disturbance at the limit) The information is obtained from a pre- and post-selected ensemble

31. The best measuring device for pre-and post-selected photon is the photon itself Strong measurements Yes

32. The best measuring device for pre-and post-selected photon is the photon itself Strong measurements No

33. The best measuring device for pre-and post-selected photon is the photon itself Weak measurements Yes

34. The best measuring device for pre-and post-selected photon is the photon itself Weak measurements No

35. Wheeler’s argument: “The photon took the upper path because it could not come the other way” seems to be sound. The presence of the bomb can be found without anything passing near the bomb Can we find that the bomb or anything else is not present in a particular place without anything passing near this place? Hosten,…Kwiat, Nature 439, 949 (2006) Yes! Its validity is tested in a best way by weak measurements using external system or the photon itself.

36. Kwiat’s proposal

40. Wheeler: We know that the bomb is not there and the photon was not there since it could not come this way. Weak measurements: the photon was there!

41. Kwiat’s proposal Weak measurements: the photon was there! Yes No But it was not on the path which leads towards it!

42. Kwiat’s proposal Weak measurements: the photon was there! But it was not on the path which leads towards it! Yes

43. Kwiat’s proposal Weak measurements: the photon was there! But it was not on the path which leads towards it! No

44. Kwiat’s proposal Weak measurements: the photon was there! But it was not on the path which leads towards it! No

45. Kwiat’s proposal Weak measurements by environment

46. Kwiat’s proposal Weak measurements by environment

47. Kwiat’s proposal Weak measurements: the photon was there! But also in another place

48. Kwiat’s proposal Weak measurements: the photon was there! But also in another place. The effects are equal! Yes

49. Kwiat’s proposal Weak measurements: the photon was there! But also in another place. The effects are equal! Yes

50. The pre- and post-selected particle is described by the two-state vector The outcomes of weak measurements are weak values The two-state vector formalism expalnation

53. Where Is the Quantum Particle between Two Measurements?

54. The two-state vector formalism expalnation

57. A B C The two-state vector formalism explanation

58. A B C

59. A B C Yes

60. A B C The two-state vector formalism explanation Yes

61. A B C The two-state vector formalism explanation ?

63. Interaction-free measurement

65. In IFM the photon was not near the bomb