1. What is the origin of complex probability amplitudes? William K. Wootters Williams College

2. A simple quantum experiment

3. Computing probabilities in a sensible way (½)(½)+(½)(½) = ½ ½ ½

4. ½ ½ Computing probabilities in a sensible way

5. The experimental result 0% 100%

6. A quantum explanation of this result

8. 0% 100% A quantum explanation of this result

9. Questions (i)Why do we have to work with “square roots” of probability? Is there a deeper explanation? (ii)And why are these “square roots” complex? I will try to answer the first question—why square roots? But my answer will make the second question worse. Then I address the second question—why complex amplitudes?—and ask in particular whether the real-amplitude theory could conceivably be correct.

10. Photon polarization polarizing filter

11. Measuring photon polarization polarizing filter

12. 42° Measuring photon polarization

13. polarizing filter 42° There is no such device. Measuring photon polarization

14. polarizing filter Measuring photon polarization polarizing beam splitter

15. polarizing filter Measuring photon polarization polarizing beam splitter

16. polarizing filter Measuring photon polarization By measuring many photons, we can estimate the probability of the vertical outcome. This tells us about the angle.

17. The standard account of probability vs angle amplitude for vertical Squaring the amplitude gives the probability:  p 

18. polarizing filter The angle varies continuously. But the measurement is probabilistic with only two possible outcomes. Is the communication optimal? A completely different explanation for that curve: Optimal information transfer?

19. A Communication Puzzle Alice is going to think of a number  between 0 and  /2. 

20. A Communication Puzzle Alice is going to think of a number  between 0 and  /2. She will construct a coin, with her number encoded in the probability of heads. 

21. A Communication Puzzle Alice is going to think of a number  between 0 and  /2. She will construct a coin, with her number encoded in the probability of heads. She will send the coin to Bob. 

22. A Communication Puzzle Alice is going to think of a number  between 0 and  /2. She will construct a coin, with her number encoded in the probability of heads. She will send the coin to Bob. To find , Bob will flip the coin... 

23. A Communication Puzzle Alice is going to think of a number  between 0 and  /2. She will construct a coin, with her number encoded in the probability of heads. She will send the coin to Bob. To find , Bob will flip the coin, but it self-destructs after one flip. 

24. The Goal: Find the optimal encoding p (   )  Maximize the mutual information: Here n is the number of heads Bob tosses ( n = 0 or 1), and  is distributed uniformly between 0 and  /2.

25. An Optimal Encoding (1 flip) (Information-maximizing for a uniform a priori distribution.) 0 0 1 probability of heads Alice’s number   /2

26. Modified Puzzle—Bob Gets Two Flips The coin self-destructs after two flips. (It’s like sending two photons with the same polarization.) 

27. An Optimal Encoding (2 flips) 0 0 1 probability of heads Alice’s number   /2

28. New Modification—Bob Gets 25 Flips The coin self-destructs after 25 flips. (It’s like sending 25 photons with the same polarization.) 

29. An Optimal Encoding (25 flips) 0 0 1 probability of heads Alice’s number   /2

30. Taking the limit of an infinite number of flips For any given encoding p heads (   ), consider the following limit. We ask what encodings maximize this limit.

31. An optimal encoding in the limit of infinitely many flips 0 0 1 probability of heads Alice’s number  This is exactly what photons do!  /2

32. Why this works: Wider deviation matches greater slope 0 0 1 n/Nn/N Alice’s number n/Nn/N  /2 N = number of tosses.

33. Another way of seeing the same thing Δ(n/N) pictured on the probability interval. same size p1p1 p2p2 p1p1 p2p2 Using square roots of probability equalizes the spread in the binomial distribution. 0 1 0 1 0 1 0 1

34. A Good Story In quantum theory, it’s impossible to have a perfect correspondence between past and future (in measurement). But the correspondence is as close as possible, given the limitations of a probabilistic theory with discrete outcomes. This fact might begin to explain why we have to use “square roots of probability.”

35. But this good story is not true! Why not?

36. But this good story is not true! Why not? Because probability amplitudes are complex.

37. No information maximization in the complex theory.  |  An orthogonal measurement completely misses a whole degree of freedom (phase). p vertical = cos 2 (  /2), but  is not uniformly distributed.

38. Quantum theory with d orthogonal states: With real amplitudes, information transfer is again optimal. a1a1 a2a2 a3a3 p1p1 p2p2 p3p3 The rule p k  a k 2 again maximizes the information gained about a, compared with other conceivable probability rules, in the limit of an infinite number of trials.

39. Making statistical fluctuations uniform and isotropic p2p2 p1p1 p3p3 p2p2 p1p1 p3p3 In this sense real square roots of probability arise naturally.

40. From Am. J. Human Genetics (1967).

41. In d dimensions, a pure state holds 2( d  1) real parameters, but there are only d  1 independent probabilities for a complete orthogonal measurement. But again, information transfer is not optimal in standard quantum theory with complex amplitudes. Why complex?Why this factor of 2?

42. Conceivable answers to “Why complex amplitudes?”  Want an uncertainty principle (Stueckelberg; Lahti & Maczynski)  Want local tomography (Hardy; Chiribella et al; Müller & Masanes et al; Dakić & Brukner; me)  Want complementarity (Goyal et al)  Want square roots of transformations (Aaronson)  Want algebraic closure (many people)

43. A Different Approach: Maybe the real-amplitude theory is correct. Require that every real operator commute with Then the real theory in 2d dimensions becomes equivalent to the complex theory in d dimensions. where

44. Our take on Stueckelberg’s idea: The ubit model (with Antoniya Aleksandrova and Victoria Borish) Assume:  Real-amplitude quantum theory  A special universal rebit (ubit)—doubles the dimension  The ubit interacts with everything ubit environment local system We can get an effective theory similar to standard quantum theory.

45. Roughly, the ubit plays the role of the phase factor. The ubit’s state space: |1|1 |1|1 |i|i |i|i But treating it as an actual physical system makes a difference.

46. The dynamics Generated by an antisymmetric operator S :  is the ubit’s rotation rate. s is the strength of the ubit-environment interaction. B EU is chosen randomly. We do perturbation theory, with s /  as our small parameter. ubit environment E local system A

47. Effect of the environment on the ubit The state of UA has components proportional to J U and X U and Z U. ubit environment E coefficient of J U coefficient of X U time We assume (i) rapid rotation of the ubit and (ii) strong interaction with the environment, so that the above processes happen much faster than any local process. local system A

48. What we assume in our analysis:  Both s (ubit-environment interaction strength) and  (ubit rotation rate) approach infinity with a fixed ratio.  Infinite-dimensional state space of the environment; thus, infinitely many random parameters in B EU. What we find for the effective theory of the UA system:  We recover Stueckelberg’s rule: all operators commute with J U.  There is no signaling through the ubit (for a UAB system).  As s /  approaches zero, we seem to recover standard quantum theory.  Short of this limit, we see deviations from the standard theory. We see three distinct effects:

49. Precessing qubit x component of Bloch vector time s /  = 0.3 x z (i) Retardation of the evolution (second order in s /  )

50. Precessing qubit x component of Bloch vector time Not sure how to look for this effect. There’s another special axis that is not special in standard quantum theory. x z (i) Retardation of the evolution (second order in s /  )

51. Precessing qubit s /  = 0.1 x component of Bloch vector time x z (ii) Flattening of the Bloch sphere (second order in s /  )

52. Precessing qubit x component of Bloch vector time This effect vanishes if we include part of the environment in an “effective ubit.” x z (ii) Flattening of the Bloch sphere (second order in s /  )

53. length of the Bloch vector time in precession periods, for s/  = 0.1 Precessing qubit (iii) Decoherence (third order in s /  )

54.  coh ≈ (period)(   / s ) 3 To be consistent with experimental results we must have s/  < 10  6 (Chou et al, 2011) length of the Bloch vector time in precession periods, for s/  = 0.1 Precessing qubit (iii) Decoherence (third order in s /  )

55. Summary If probability amplitudes were real, we could tell a nice story: information is transferred optimally from preparation to measurement, and this fact could begin to explain “square roots of probability.” But there is no such optimization if amplitudes are complex. Many ideas have been proposed to explain complex amplitudes. We may use the real-amplitude theory, if we supplement it with Stueckelberg’s rule: all operators commute with J. Or we may assume a universal rebit. In a certain limit we seem to recover standard quantum theory, but short of this limit the model predicts spontaneous decoherence.