Midterm 5 Solutions

# Therefore p a 2 1 2 1 2 p a 2 trb

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Unformatted text preview: e or Bob that measures in the ± n\ basis. ü Method 2: Using a reduced state We can also take p A, + — ` = TrB +n\ X+n rA F 2 where ` rA = TrB YH-L ] XYH-L = = ` rA = 1 TrB @ +z, 2 1 @ +z\ X+z 2 1` 1ཽ 2 -z\ X+z, -z - +z, -z\ X-z, +z - -z, +z\ X+z, -z + -z, +z\ X-z, +z D + -z\ X-z D ` (and by symmetry, we may argue that the same is true of rB ). Therefore p A, + — = 2 1 2 1 2 = p A, + — 2 = ` TrB +n\ X+n 1ཽF ` X+n 1ཽ +n\ 1 2 and p A, - — =1-p 2 A, + — 2 = 1 2 ` Using our symmetry argument above and the fact that Z+z 1ཽ +z^ = 1, we may therefore conclude that the probabilities that Bob — will measure the outcomes ± 2 are also Printed by Mathematica for Students p B, + — 2 =p B, - — 2 = 1 2 p A, + — = 1 2 = p A, + — = 2 TrB +n\ X+n 1ཽF 2 2 ` X+n 1ཽ +n\ 1 2 and p A, - — =1-p 2 A, + — = 2 MidtermExam5Solutions.nb 1 2 9 ` Using our symmetry argument above and the fact that Z+z 1ཽ +z^ = 1, we may therefore conclude that the probabilities that Bob — will measure the outcomes ± 2 are also p B, + — =p 2 B, - = — 2 1 2 ü Part b From Method 2 of (a), ` 1ཽ 2 ` ` r A = rB = which is independent of the q8A, B&lt; , so regardless of which qB Bob measures along, his measurement outcomes will always be consistient with the probabilities given by this state, from which he can extract no information about qA . Therefore, Bob cannot determine the orientation of Alice’s device by changing his orientation if Alice and Bob do not communicate their outcomes. ü Part c If Alice and Bob share the classical mixture ` r= 1 @ 2 +z, +z\ X+z, +z + -z, -z\ X-z, -z D then we again have ` rA = ` rB = 1 @ 2 1` 1ཽ 2 +z\ X+z + -z\ X-z D = 1 2 ` 1ཽ which are the same marginals they would have if they had instead shared the entangled state YH-L \. Thus, if they cannot commu` nicate their outcomes, then they cannot determine whether they share the mixed state r or the entangled state YH-L \. If they can communicate their outcomes, then without loss of generality, assume Alice measures her degree of freedom in the z-direction — first and obtains the result + 2 . If...
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