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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\ Xz, +z DB +x\ Äȩ 1ཽ F
=
=
`£
rB after A = 2 X+x +z\
1
B
2 z\ Xz z\ Xz + 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 YHL \ 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 YHL \ would leave Bob’s state as
`
rB after A = x] Xx 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 xdirection 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\ Xz + = cos2 Hq ê 2L z\ Xz
=
`£
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
`£
Yn r n ] = 1 + 1 cos q Hn ÿ zL ¬༼ dots go away in inner product. `£
rB after A = X+n +z\ 2 z\ Xz + = cos2 Hq ê 2L z\ Xz
=
`£
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
`£
Yn rB after A n] =
`£
Yn 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.
 Fall '13
 AkimasaMiyake

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