1. “counterfactual communication”?
    
What is nonlocal in
“counterfactual communication”?
ICQF-17
BGU
Y. Aharonov, T. Landsberger and D. Rohrlich (2017)
drawings by Tom Oreb © Walt Disney Co.

2. What is nonlocal in
quantum mechanics?

3. What is nonlocal in
quantum mechanics?
Nonlocal quantum correlations

4. What is nonlocal in quantum mechanics? Nonlocal quantum correlations
Dynamical quantum nonlocality

5. Changing the nonlocal quantum phase

6. Changing the nonlocal quantum phase
locally
V(x) x

7. Changing the nonlocal quantum phase
nonlocally
V(x) x

9. Each charged particle diffracts around
the plates as they charge and discharge.

10. Outline
Dynamical quantum nonlocality and modular variables
The paradox of “counterfactual communication”
Nonlocality and counterfactual communication

11. Outline
Dynamical quantum nonlocality and modular variables
The paradox of “counterfactual communication”
Nonlocality and counterfactual communication

12. Dynamical quantum nonlocality is based on relative phases, and relative phases are modular: A relative phase φ is defined only up to 2π, so we can define a modular phase such that
φmod ≡ φ mod 2π
and
0 ≤ φmod < 2π

13. x Three examples of modular variables:
Example 1: modular momentum, pmod. Consider a wave function ψ(x) that is periodic with period L, with a relative phase eiα from one peak to the next.
ψ(x)
einα
ei(n+1)α x L

14. x Three examples of modular variables:
This wave function ψ(x) is an eigenfunction of eipL/ħ with eigenvalue e–iα. But if we add h/L to p, we get the same eigenfunction and eigenvalue: ei(p+h/L)L/ħ = eipL/ħ. So ψ(x) is
an eigenfunction of exp(ipmodL/ħ) where pmod ≡ p mod h/L.
ψ(x)
einα
ei(n+1)α x L

15. x Three examples of modular variables:
Example 2: modular position, xmod. The same wave function ψ(x) is an eigenfunction of xmod ≡ x mod L = δL (if we can neglect the spread of the gaussian wave packets).
ψ(x)
ei(n+1)α
L+δL
einα x L

16. x Three examples of modular variables:
But how can ψ(x) be simultaneously be an eigenfunction of xmod and of pmod? Doesn’t that violate the uncertainty principle?
ψ(x)
ei(n+1)α
L+δL
einα x L

17. x Three examples of modular variables:
But how can ψ(x) be simultaneously be an eigenfunction of xmod and of pmod? Doesn’t that violate the uncertainty principle? No! Exercise: prove [xmod , pmod ] = 0.
ψ(x)
ei(n+1)α
L+δL
einα x L

18. Three examples of modular variables:
Example 3: modular angular momentum, Lzmod. The operator
generates a rotation of angle α about the z-axis. Thus
generates a rotation of π about the z-axis. A state Φ(ϕ) that is antisymmetric under a π rotation about the z-axis has
but since adding 2ħ to Lz in the exponent does nothing, the Lz in the exponent is essentially Lzmod = Lz mod 2ħ.

19. A. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993)
An application of modular angular momentum:
A. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993)

20. A. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993)
An application of modular angular momentum: the bomb makes the modular angular momentum of the system completely uncertain.
A. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993)

21. A. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993)
An application of modular angular momentum: z
A. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993)

22. Outline
Dynamical quantum nonlocality and modular variables
The paradox of “counterfactual communication”
Nonlocality and counterfactual communication

27. 𝑈 𝜖 = cos 𝜖 𝑖 sin 𝜖 𝑖 sin 𝜖 cos 𝜖

28. 𝑈 𝜖 = cos 𝜖 𝑖 sin 𝜖 𝑖 sin 𝜖 cos 𝜖

29. 𝑈 𝜖 = cos 𝜖 𝑖 sin 𝜖 𝑖 sin 𝜖 cos 𝜖
Let n𝜖 = π/2 at time T. Then the ball is in Bob’s side.

30. 𝑈 𝜖 = cos 𝜖 𝑖 sin 𝜖 𝑖 sin 𝜖 cos 𝜖
Let n𝜖 = π/2 at time T. Then the ball is in Bob’s side.
At time 2T, the ball is back on Alice’s side, multiplied by –1.

31. If Bob’s end is open (no mirror) then the chance of finding the ball on Bob’s side at time T vanishes (amplitude 𝜖 and probability 𝜖2 per event, number of events proportional to n = π/2𝜖).

32. If Bob’s end is open (no mirror) then the chance of finding the ball on Bob’s side at time T vanishes (amplitude 𝜖 and probability 𝜖2 per event, number of events proportional to n = π/2𝜖).
Thus the quantum Zeno effect makes the interaction-free measurement 100% reliable, and it looks as if Bob can signal to Alice with no exchange of matter.

33. “Counterfactual communication”
H. Salih et al., Phys. Rev Lett. 110, (2013)
Y. Cao et al., Proc. Nat. Acad. Sci. 114, 4920 (2017)

34. “Counterfactual communication”
H. Salih et al., Phys. Rev Lett. 110, (2013)
Y. Cao et al., Proc. Nat. Acad. Sci. 114, 4920 (2017)

35. BL BR
If Bob’s end is open (no mirror), the right barrier (BR) acts
as a mirror, and the ball, at time T, is in the middle cavity. Alice checks for the ball in the left and middle cavities, finds it in the middle cavity, and concludes that Bob removed the mirror at his end.
Call this case Logical 1.

36. BL BR
If Bob’s end is closed, the ball accumulates slowly between the barriers BL and BR and some penetrates through BR. When the ball is most likely to be to the right of BR, Bob empties the right chamber. But the net loss via the right chamber vanishes. At time T, Alice checks for the ball, finds it in the left cavity, and concludes that Bob put the mirror on his end.
Call this case Logical 0.

37. BL BR
The probability for the ball to penetrate BR and reach Bob vanishes either way, yet Bob signals to Alice a single bit,
Logical 1 or Logical 0!

38. Outline
Dynamical quantum nonlocality and modular variables
The paradox of “counterfactual communication”
Nonlocality and counterfactual communication

39. Alice finds the particle at her end after time T.

40. Alice does not find the particle at her end after time T.

41. z
Alice does not find the particle at her end after time T. Now let’s define (modular spin) states
|+>𝑚 = [ ] | –>𝑚 = [ – ]

42. |+>𝑚 z
So now if Bob prepares his mirror in the state |+>𝑚 , after time 2T it turns into | –>𝑚 and vice versa – not because of “counterfactual communication” but because the ball acquires a local phase along the way.

43. | –>𝑚 z
Conclusion: Bob may prepare the mirror only in the state [|+>𝑚 + | –>𝑚] and thus only see “counterfactual communication”, but measurement in any other basis such as |±>𝑚 shows that the ball does reach Bob’s end.

44. | –>𝑚 z
Choosing the [|+>𝑚 ± | –>𝑚] basis is like placing a bomb in an arm of a Mach-Zehnder interferometer – it destroys the phase relation between the two paths of the ball.

45. Thank you!

46. z

47. z

48. Ψ 0 = 1 2 (|↑>𝑏+|↓>𝑏)(|↑>𝑚+|↓>𝑚)
Ψ 0 = 1 2 (|↑>𝑏+|↓>𝑏)(|↑>𝑚+|↓>𝑚) z
Ψ 0 is an eigenstate at time t = 0 of the modular angular momenta Lbmod and Lmmod of the ball and mirror, respectively, which are separately conserved.

49. Ψ 𝑇 = 1 2 [|↑>𝑏|↓>𝑚+|↓>𝑏|↑>𝑚]
Ψ 𝑇 = [|↑>𝑏|↓>𝑚+|↓>𝑏|↑>𝑚] z
Then at time t = T Alice finds the ball on her side.

50. Ψ 𝑇 = 1 2 [|↑>𝑏|↓>𝑚+|↓>𝑏|↑>𝑚]
Ψ 𝑇 = [|↑>𝑏|↓>𝑚+|↓>𝑏|↑>𝑚] z
Then at time t = T Alice finds the ball on her side.

51. Ψ 𝑇 = 1 2 [|↑>𝑏|↓>𝑚+|↓>𝑏|↑>𝑚]
Ψ 𝑇 = [|↑>𝑏|↓>𝑚+|↓>𝑏|↑>𝑚] z
The ball and the mirror never met (since the ball never crossed the barrier). But the ball and mirror exchanged modular angular momentum!