Midterm 5 Solutions

# If they instead shared the mixed state r then alices

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Unformatted text preview: they share the entangled state YH-L \, this leaves Bob’s state as ` rB after A = 1 p A, + — ` ` BX+z Äȩ 1ཽ F YH-L \ XYH-L B +z\ Äȩ 1ཽ F ¬༼ p 2 A, + — = 2 1 2 ` = BX+z Äȩ 1ཽ F@ +z, -z\ X+z, -z - +z, -z\ X-z, +z ` - -z, +z\ X+z, -z + -z, +z\ X-z, +z DB +z\ Äȩ 1ཽ F ` rB after A = -z] X-z ` since in this situation, Bob knows the outcome of Alice’s measurement. If they instead shared the mixed state r, then Alice’s measurement would leave Bob’s state as ` rB after A = 1 p A, + — `` ` BX+z Äȩ 1ཽ F rB +z\ Äȩ 1ཽ F ¬༼ p 2 A, + — 2 = 1 2 ` ` = BX+z Äȩ 1ཽ F@ +z, +z\ X+z, +z + -z, -z\ X-z, -z DB +z\ Äȩ 1ཽ F ` rB after A = +z] X+z Printed by Mathematica for Students ` So in this case, they can determine whether they share the entangled state YH-L \ or the mixed state r. In the former case, they are ` since in this situation, Bob knows the outcome of Alice’s measurement. If they instead shared the mixed state r, then Alice’s measurement would leave Bob’s state as 10 MidtermExam5Solutions.nb ` rB after A = `` ` BX+z Äȩ 1ཽ F rB +z\ Äȩ 1ཽ F 1 p A, + — ¬༼ p A, + 2 — 2 = 1 2 ` ` = BX+z Äȩ 1ཽ F@ +z, +z\ X+z, +z + -z, -z\ X-z, -z DB +z\ Äȩ 1ཽ F ` rB after A = +z] X+z ` So in this case, they can determine whether they share the entangled state YH-L \ or the mixed state r. In the former case, they are guaranteed to have opposite outcomes when they both measure along z, whereas in the latter case, they are guaranteed to have the same outcome. ü Part d Here, Alice and Bob instead share the classical mixture `£ r= 1 @ 2 +z, -z\ X+z, -z + -z, +z\ X-z, +z D ` In comparison to the previous example r in part (c), Alice and Bob cannot tell the difference, even with communication, as far as they measure along the z-direction. However, without loss of generality, assume Alice measures her degre of freedom first, this p ), 2 time along the x-direction (q = — and that she observes the outcome + 2 with probability direction when Alice has the maximally mixed state `£ rB after A = 1 p Ax , + — ` 1ཽ ). 2 1 2 (this is true for...
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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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