PHYS 1051W: Quantum Worlds Michel Janssen and Michael Janas, Spring 2023 The notation used for the entries of this cell is explained in Understanding Quantum Raffles, p. 25, note 14. In this homework assignment we will make heavy use of this notation and of some basic rules of probability theory (in part in preparation for Ch. 3 of Understanding Quantum Raffles). Here is a two-page handout on elementary probability theory in which I introduce these rules using simple examples of drawing cards from complete and incomplete decks of cards. Even if you have encountered these rules-the addition rule for Pr(A or B) and the multiplication rule for Pr(A and B)- before, I strongly recommend that you review this handout before proceeding any further. Our pairs of quantum bananas are all prepared in the entangled state below (see Understanding Quantum Raffles, p. 59, Eq. (2.6.38); 1 refers to the banana given to Alice, 2 to the banana given to Bob): [V)12 = (1+)1al-)20 - 1->lal+)20) Here we used combinations of the two eigenvectors of the operator for "taste of a banana when peeled in the a-direction" that form an orthonormal basis of the one-banana Hilbert space as our orthonormal basis for the two-banana Hilbert space. Let's call this the a-basis. We saw that this entangled state has the exact same form in the b-basis, constructed in the same way as the a-basis but now with the eigenvectors of the operator for "taste of a banana when peeled in the b-direction" (Understanding Quantum Raffles, p. 60, Eq. (2.6.41): [V) 12 = V2 ( 1+)161-)26 - 1-)161+)26) Both in Understanding Quantum Raffles (p. 61, Eq. (2.6.43)) and in lecture we wrote this state in yet another basis to analyze what happens when Alice peels banana #1 in the a direction and Bob peels banana #2 in the b direction: {It)1al+)26, 1+)1al-)26, 1-)lal+)26, 1-)ial-)26) In this basis the state has the form: 14/12 = 2 ( sin ( 20) 1+)zal+) 26 + cas (42 ) It)ial-)26 - cos (2 1-)lal+)26 + sin (Pab (2 ) 1-Dial-)26) 2PHYS 1051W: Quantum Worlds Michel Janssen and Michael Janas, Spring 2023 We then applied the Born rule to find the four probabilities in the correlation array at the bottom of p. 1. In this homework assignment I'll guide you through a different way of finding these probabilities, using the expressions for |> > in just the a-basis or just the b-basis. We will consider two scenarios: (1) Alice peels her banana first; (2) Bob peels his banana first. I'll walk you through the details of (1). I'll then have you do (2) on your own (though in going through scenario (1) I'll indicate in a few places how you should modify things for scenario (2)). Finally, I'll ask you to think about what quantum mechanics is telling us is happening in this banana tasting experiment (3). (1) Alice peels her banana first If Alice goes first, we use the expression for |4'> , in the a-basis (when Bob go first we use the expression in the b-basis): [V)12 = (1+)al-)20 - 1-)ial+)20) (a) [8 points] What are the probabilities that Alice finds yummy or nasty? In other words what are the probabilities Pr(+A lab) = ? and Pr(-A |ab) = ? Pr(+Ajab) = 1/2 Pr(-Ajab) = 1/2 Now we need to calculate the probability Pr(mamBlab) where ma and ma refer to the outcomes found by Alice and Bob respectively. Both mA and me can thus take on two values, yummy (+) or nasty (-). We can use the general multiplication rule for this probability:" See the handout on probability theory. The general multiplication rule is: Pr(A and B) = Pr(A) x Pr(B|A) = Pr(B) x Pr(A|B). This still looks a little different from the formulae given here. (1) we omit the "and" in "m, and me-" (2) All probabilities are conditional on the peelings used by Alice and Bob. In a sense, all probabilities are conditional probabilities: they are always conditional on some background knowledge that is tacitly assumed. So a more accurate statement of the general multiplication rule would be: Pr(A and B|b) = Pr(A|b) x Pr(BJA, b) = Pr(B|b) x Pr(A|B, b), where "b" stands for "background knowledge." In the examples used in the handout, "b" would be that we are drawing from some specific deck of cards.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.PHYS 1051W: Quantum Worlds Michel Janssen and Michael Janas, Spring 2023 Problem set #4: Using the Born rule to calculate the probabilities of Alice and Bob finding yummy and nasty upon tasting one of a pair of entangled quantum bananas using different peelings Due Monday, March 20, 9 am (upload pdf scan in canvas) As long as you write legibly, you can submit (a scan of a handwritten version of your answers to questions (1)-(3) below. Please use the free adobe scanning app to scan your assignment. Save your scan as a pof and upload it via Canvas. In this assignment, I'll guide you through a quantum-mechanical analysis of our by now familiar quantum banana tasting experiment in the Mermin setup (see below) Alive Bob We'll consider the case where the triplet of angles (9.b, Pan , ( ) between the three peeling directions can be any triplet satisfying the elliptope inequality, not just the extremal case where (@, Par , P.) = (120", 120', 120") shown in the figure. Our goal is to derive the probabilities for cells in the 3x3 correlation array for this experiment. One of these cells, the ab-cell, is shown below. Bob Alice Pr( + + 156) Pr(+-156) Pr(-+ 136) Pr(-- 136)