PHYS 1051W: Quantum Worlds Michel Janssen and Michael Janas, Spring 2023 Pr(mamBlab) = Pr(mA |ab) x Pr(mg | mAab) = Pr(mg |ab) x Pr(malmBab) When Alice goes first, we need the expression on the first line (when Bob goes first we need the expression on the second line). The Born rule tells us that: Pr(mg |mAab) = la(mal - mA)6|? Here is how we arrive at that formula. As soon as Alice, peeling in the a direction, tastes her banana and finds either yummy or nasty (m, = 1), we know for sure that Bob, if he were to peel in a direction, would find the opposite taste (+). This means that we need to calculate the probabilities of Bob finding yummy or nasty when peeling in the b direction assuming that the state vector for Bob's banana is the eigenvector for "taste when peeled in the a direction" with the eigenvalue that is the opposite of what Alice found (i.e., -m). Since Bob peels in the b direction, we expand this eigenstate in the b-basis. 1-mala = (6(+1 -mA)a ) It)8 + (6(-1 -ma)a ) |-)b According to the Born rule, the squares of the coefficients of the two basis vectors give the probabilities that Bob's banana tastes either yummy or nasty (m; = 1) (when Bob goes first, expand eigenvectors of "taste when peeled in the b direction" in the a-basis). 1+ ) b 1-)2 cos ( Pab 1 Pab Pab 2 (Pab) (b) [16 points] Use the figure above (cf. Understanding Quantum Raffles, p. 32, Fig. 2.16) to express the four inner products in these expressions (two for -m, = + and two for -ma = -) in terms of sines and cosines of half the angle between the peeling directions a and bPHYS 1051W: Quantum Worlds Michel Janssen and Michael Janas, Spring 2023 MA = + : 6(+ 1-MA)a = 6(+|-)as 6(-1-mA)a = 6(-[-). MA = - : 6 (+ 1-MA)a = (+|+)as 6(-1-MA)a = (- |+)a (c) [16 points] Use your answers to (b) to express the conditional probability Pr(mB |mA ab) in terms of sines and cosines of 4./2 for all four combinations of (m, = 1, mg = #) (d) [16 points] Combine your answers to (a) and (c) to express the four probabilities in the correlation array on p. 1 in terms of sines and cosines of @../2. (2) Bob peels his banana first (2a)-(2d) [4, 8, 8, 8 points, respectively, for a total of 28 points] Go through the analogue of the steps under (1a) (1d) when Bob peels his banana first. Use the same notation and the same kind of explanatory prose I used under (1). Note that there are some hints in parentheses under (1) on what to do differently when Bob rather than Alice peels first. (3) What can we learn from the fact that it doesn't matter which banana gets peeled first? The standard story to go with the argument you went through under (1) and (2) goes something like this. Consider scenario (1) in which Alice peels first. When Alice peels the banana in the a direction and tastes it, the story goes, the entangled state vector collapses, leaving only one of the two terms that initially were there, the term with + for banana 1 if Alice finds that that banana tastes yummy, the term with - for banana 1 if Alice finds that that banana tastes nasty. This collapse of the state vector, the story continues, instantaneously affects Bob's banana, the state of which is now represented by the a-basis eigenvector with the opposite eigenvalue (taste) of what Alice found. The Born rule then tells Bob what taste he'll find when he peels the banana in that state in any direction he chooses. Now we also know the following: (1) Alice cannot exploit the collapse of the state vector that she causes by tasting her banana and that instantaneously affects Bob's banana to instant message Bob (i.e., send him a signal faster than the speed of light). (2) We can tell the same story starting with Bob causing the collapse by tasting his banana and thereby instantaneously affecting Alice's banana. For the probabilities we find it does not make any difference whatsoever which version of the story we choose.PHYS 1051W: Quantum Worlds Michel Janssen and Michael Janas, Spring 2023 [16 points] Use these observations to argue that this talk about one observer causing a collapse of the state vector that then spreads instantaneously to the other observer who could be light years away is probably the wrong way to conceptualize what is happening in our banana tasting experiment