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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 YHL ] XYHL
=
=
`
rA = 1
TrB @ +z,
2
1
@ +z\ X+z
2
1`
1ཽ
2 z\ X+z, z  +z, z\ Xz, +z  z, +z\ X+z, z + z, +z\ Xz, +z D
+ z\ Xz 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,  — =1p 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,  — =1p 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< , 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\ Xz, z D then we again have
`
rA =
`
rB = 1
@
2
1`
1ཽ
2 +z\ X+z + z\ Xz D = 1
2 `
1ཽ which are the same marginals they would have if they had instead shared the entangled state YHL \. Thus, if they cannot commu`
nicate their outcomes, then they cannot determine whether they share the mixed state r or the entangled state YHL \. If they can
communicate their outcomes, then without loss of generality, assume Alice measures her degree of freedom in the zdirection
— first and obtains the result + 2 . If...
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 Fall '13
 AkimasaMiyake

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