Midterm 5 Solutions

3 that the same protocol had bob and alice shared the

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Unformatted text preview: every measurement Alice communicates this result to Bob, so Bob is left with the state ` `£ ` BX+x Äȩ 1ཽ F r B +x\ Äȩ 1ཽ F 2 ` ` = BX+x Äȩ 1ཽ F@ +z, -z\ X+z, -z + -z, +z\ X-z, +z DB +x\ Äȩ 1ཽ F = = `£ rB after A = 2 X+x +z\ 1 B 2 -z\ X-z -z\ X-z + X+x -z\ 2 +z\ X+z ¬༼ X+x ± z\ 2 = 1 2 + +z\ X+z F ` 1ཽ 2 But from part (b), we know that - since the form of YH-L \ has the same structure in all choices of local bases (from the homework problem 5.3) - that the same protocol had Bob and Alice shared the entangled state YH-L \ would leave Bob’s state as ` rB after A = -x] X-x So using such a protocol, they can determine the difference between this classical mixture and an entangled state. In the former 1 case, Bob’s result upon measuring along the x-direction is only correlated with Alice’s with probability 2 , whereas in the latter, his result is always perfectly correlated with Alice’s. In general, so long as q ¹Ȧ 0 (Alice measures in the basis ± n\), we have, following the same steps as above `£ rB after A = X+n +z\ 2 -z\ X-z + = cos2 Hq ê 2L -z\ X-z = `£ rB after A = ` 1ཽ 2 ` 1ཽ 2 - cos q X+n -z\ 2 +z\ X+z + sin2 Hq ê 2L +z] X+z ` Sz — - cos q BHn ÿ zL ` Sn — + ...F The probability that Bob’s result is correlated withPrinted by Mathematica for Smanner as if they had shared the entangled state is thus Alice’s in the same tudents given by `£ Y-n r -n ] = 1 + 1 cos q Hn ÿ zL ¬༼ dots go away in inner product. `£ rB after A = X+n +z\ 2 -z\ X-z + = cos2 Hq ê 2L -z\ X-z = `£ rB after A = ` 1ཽ 2 ` 1ཽ 2 - cos q X+n -z\ 2 +z\ X+z + sin2 Hq ê 2L +z] X+z MidtermExam5Solutions.nb 11 ` Sz — - cos q BHn ÿ zL ` Sn — + ...F The probability that Bob’s result is correlated with Alice’s in the same manner as if they had shared the entangled state is thus given by `£ Y-n rB after A -n] = `£ Y-n rB after A -n] = 1 2 1 2 + + 1 2 1 2 cos q Hn ÿ zL ¬༼ dots go away in inner product. cos2 q so in the general case, Bob’s result will only be correlated with Alice’s with this probability when they share the mixed state. This is 1 (they can never tell the difference) for q = 0 and minimal at 1 2 (they can most quickly tell the difference) for q = ü Statistics mean = 5.72 median = 6 stddev = 1.64 Printed by Mathematica for Students p . 2 12 MidtermExam5Solutions.nb Total mean = 11.28 median = 11.5 stddev = 3.25 Printed by Mathematica for Students...
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This note was uploaded on 02/01/2014 for the course PHYC 491/496 taught by Professor Akimasamiyake during the Fall '13 term at New Mexico.

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