- #1
gespex
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Hello everybody,
I've been trying to understand the CHSH proof as it is listed on Wikipedia:
http://en.wikipedia.org/wiki/CHSH_inequality
I got to this without any problem:
[itex]E(a, b) - E(a, b^\prime) = \int \underline {A}(a, \lambda)\underline {B}(b, \lambda)[1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)d\lambda - \int \underline {A}(a, \lambda)\underline {B}(b^\prime, \lambda)[1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)d\lambda[/itex]
However, now it mentions two things to get to the next step:
- The fact that [itex][1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)[/itex] and [itex][1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)[/itex] are non-negative (easy enough to see).
- The triangle inequality "to both sides" (how?)
And the next equation is:
[itex]|E(a, b) - E(a, b^\prime)| \leq \int [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)d\lambda + \int [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)d\lambda[/itex]
I don't understand how it gets to this last equation from the one before. Could somebody please explain?Thanks in advance,
Gespex
I've been trying to understand the CHSH proof as it is listed on Wikipedia:
http://en.wikipedia.org/wiki/CHSH_inequality
I got to this without any problem:
[itex]E(a, b) - E(a, b^\prime) = \int \underline {A}(a, \lambda)\underline {B}(b, \lambda)[1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)d\lambda - \int \underline {A}(a, \lambda)\underline {B}(b^\prime, \lambda)[1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)d\lambda[/itex]
However, now it mentions two things to get to the next step:
- The fact that [itex][1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)[/itex] and [itex][1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)[/itex] are non-negative (easy enough to see).
- The triangle inequality "to both sides" (how?)
And the next equation is:
[itex]|E(a, b) - E(a, b^\prime)| \leq \int [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)d\lambda + \int [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)d\lambda[/itex]
I don't understand how it gets to this last equation from the one before. Could somebody please explain?Thanks in advance,
Gespex
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