Bell Inequality

Question 1

Bell inequality rules

Answer 1

To prove disprove EPR paradox: Information does not travel faster than light, rather information is exchanged before.
Play a game
Information is shared before hand in a superposition
2 random bits given to the players
P1 receive x, P2 receive y
P1 respond a, P2 respond b
win the game if a(+)b=x and y
they can win if
x = y then x^y=0, for a(+)b to =0, a must = b (one must change bit)
x=y=0, then x^y=0, for a(+)b to =0, a must = b (dont need to change anything)
x=y=1, then x^y=1, for a(+)b to =1, a must /=b (one must change bit)
Thus 3 out of 4 cases a must = b, if they decide to always output 0, they will win with proba 3/4

Question 2

a(+)b

Answer 2

xor
1 1 = 0
0 0 = 0
1 0 = 1
0 1 = 0

Question 3

Bell Inequality > 0.75

Answer 3

Can increase the proba by sharing an entangled state
|phi> = 1/sqrt(2) |00> + 1/sqrt(2) |11>
P1 receive x, P2 receive y
P1 respond a, P2 respond b
if x=0, don’t do anything
if y=0, don’t do anyhting
if x=1, rotate base by pi/8
if y=1, rotate base by -pi/8
angle between the 2 base : pi/4
thus if x=y=1 angle pi/4 and thus proba cos^2(pi/4)=1/2 to have a=b. ( a bigger angle would make the base change too often)
=> x=y=0 P(W)=1 of winning
x/=y 1 keep 0 the other rotate, angle pi/8 between bases, proba cos^2(pi/8)=0.8 to change output to 0. P(W) = 0.8
x=y=1 both rotate, angle pi/4, proba cos^2(pi/4)=1/2 to have a=b. P(W) = 1/2
==> sum probas then divide by 4 for total proba of Win
> 0.75 Entangled state increased the proba of winning

Question 4

Game probability

Answer 4

Either p=1 to play the bit they receive if decide to not do anything
or rotation theta of base and proba becomes cos^2 theta of changing bit
if x=y and both rotate then put theta together, cos^2 theta of having the same