by gill1109 » Tue Apr 08, 2014 8:12 am
Michel has not realized that I was deliberately confusing things in my initial posting, in order to imitate his way of arguing.
He argues that CHSH is meaningless.
I reproduced his argument, but replacing CHSH by Tsirelson.
So if Michel's argument is correct, Tsirelson is meaningless too.
Yet Joy Christian thinks that Tsirelson's bound is correct, the CHSH bound is incorrect.
This little conflict should have made Michel think. But no ...
My initial statement was deliberately false. A correct statement should be:
According to QM:
E(a, b) - E(a, b') - E(a', b) - E(a', b') <= 2 sqrt 2 = 2.828...
where E(a, b) etc are correlations predicted by quantum theory.
Therefore in a CHSH type experiment on quantum systems, the following bound will be true with large probability:
F(a, b) - F(a, b') - F(a', b) - F(a', b') <= 2 sqrt 2 plus a few standard deviations
where F(a, b) etc are empirically observed correlations (each based on a different, finite sample of pairs of particles).
minkwe wrote:gill1109 wrote:According to QM, in a CHSH type experiment the following bound holds:
E(a, b) - E(a, b') - E(a', b) - E(a', b') <= 2 sqrt 2 = 2.828...
A clear violation of this bound (i.e., taking account of experimental error) has almost never been reported, though many attempts were made.
As already explained,
1. that is not a CHSH-type experiment. A CHSH-type experiment produces 4 columns of single-sided data which are recombined in pairs to calculate paired-correlations.
2. There is nothing wrong with 2.828 it is fully consistent with the upper bound of 4 for experiments of that type which produce 8 columns of data in 4 pairs.
3. An upper bound is never violated, never, not even by experimental error, never!!!
What should we learn from all this?
A
CHSH experiment, and the
CHSH inequality, are two very, very different things.
A
CHSH-type experiment generates at the end of the day one data set of four columns, where the first column contains the setting being used by Alice (identified by names or labels, e.g. "1", "2"), the second column contains the setting being used by Bob (identified by names or labels, e.g. "1", "2"), the third column contains the outcome observed by Alice (+/-1), the fourth column contains the outcome observed by Bob (+/-1). Alice and Bob chose their settings by tossing fair coins, independently, again and again. The N rows of the data set correspond to N pairs of particles. Four empirical correlations are then computed or observed, by splitting the data set into four disjoint subsets, according to the pair of values of the settings. Each empirical correlation is just the
average of the product of the outcomes within the relevant subset. Obviously an empirical correlation comes with a statistical error, an error bar, or a standard deviation. We get to see some numbers, e.g. 0.72 +/- 0.01, and we conclude that the true (population or ensemble) correlation is, with large probability, within 0.72 plus or minus a few multiples of 0.01. But we might have been very unlucky and our "best guess" might be far from the truth.
Traditionally, for an experiment on spin half particles, Alice's settings "1" and "2" might correspond to directions 0 degrees, 90 degrees; while Bob's settings "1" and "2" might correspond to directions 45 degrees, 135 degrees.
The
CHSH inequality is an inequality constraining four theoretical correlations - i.e. population or ensemble averages - if the mathematical model from which those correlations are determined leads to a formula something like E(a, b) = integral A(a, lambda) B(b, lambda) rho(lambda) d lambda). (The formula indeed only makes sense within a LHV theory).
Notice that a
CHSH experiment, and the
CHSH inequality, are two very, very different things.
The four empirical correlations observed in a CHSH-type experiment can all be anything from -1 to +1, since there is no constraint at all connecting the numbers involved in one of the correlations with those in any of the others. Hence one of the four correlations minus the sum of the other three can be anything between -4 and +4.
Perhaps people who talk of bounds being violated should be a bit more careful in what they say. What do they mean, precisely?
For a very precise statement, see Theorem 1 in my paper
http://arxiv.org/abs/1207.5103.
Michel has not realized that I was deliberately confusing things in my initial posting, in order to imitate his way of arguing.
He argues that CHSH is meaningless.
I reproduced his argument, but replacing CHSH by Tsirelson.
So if Michel's argument is correct, Tsirelson is meaningless too.
Yet Joy Christian thinks that Tsirelson's bound is correct, the CHSH bound is incorrect.
This little conflict should have made Michel think. But no ...
My initial statement was deliberately false. A correct statement should be:
According to QM:
E(a, b) - E(a, b') - E(a', b) - E(a', b') <= 2 sqrt 2 = 2.828...
where E(a, b) etc are correlations predicted by quantum theory.
Therefore in a CHSH type experiment on quantum systems, the following bound will be true with large probability:
F(a, b) - F(a, b') - F(a', b) - F(a', b') <= 2 sqrt 2 plus a few standard deviations
where F(a, b) etc are empirically observed correlations (each based on a different, finite sample of pairs of particles).
[quote="minkwe"][quote="gill1109"]According to QM, in a CHSH type experiment the following bound holds:
E(a, b) - E(a, b') - E(a', b) - E(a', b') <= 2 sqrt 2 = 2.828...
A clear violation of this bound (i.e., taking account of experimental error) has almost never been reported, though many attempts were made.
[/quote]
As already explained,
1. that is not a CHSH-type experiment. A CHSH-type experiment produces 4 columns of single-sided data which are recombined in pairs to calculate paired-correlations.
2. There is nothing wrong with 2.828 it is fully consistent with the upper bound of 4 for experiments of that type which produce 8 columns of data in 4 pairs.
3. An upper bound is never violated, never, not even by experimental error, never!!!
[/quote]
What should we learn from all this?
A [b]CHSH experiment[/b], and the [b]CHSH inequality[/b], are two very, very different things.
A [b]CHSH-type experiment[/b] generates at the end of the day one data set of four columns, where the first column contains the setting being used by Alice (identified by names or labels, e.g. "1", "2"), the second column contains the setting being used by Bob (identified by names or labels, e.g. "1", "2"), the third column contains the outcome observed by Alice (+/-1), the fourth column contains the outcome observed by Bob (+/-1). Alice and Bob chose their settings by tossing fair coins, independently, again and again. The N rows of the data set correspond to N pairs of particles. Four empirical correlations are then computed or observed, by splitting the data set into four disjoint subsets, according to the pair of values of the settings. Each empirical correlation is just the [b]average of the product of the outcomes within the relevant subset[/b]. Obviously an empirical correlation comes with a statistical error, an error bar, or a standard deviation. We get to see some numbers, e.g. 0.72 +/- 0.01, and we conclude that the true (population or ensemble) correlation is, with large probability, within 0.72 plus or minus a few multiples of 0.01. But we might have been very unlucky and our "best guess" might be far from the truth.
Traditionally, for an experiment on spin half particles, Alice's settings "1" and "2" might correspond to directions 0 degrees, 90 degrees; while Bob's settings "1" and "2" might correspond to directions 45 degrees, 135 degrees.
The [b]CHSH inequality[/b] is an inequality constraining four theoretical correlations - i.e. population or ensemble averages - if the mathematical model from which those correlations are determined leads to a formula something like E(a, b) = integral A(a, lambda) B(b, lambda) rho(lambda) d lambda). (The formula indeed only makes sense within a LHV theory).
Notice that a [b]CHSH experiment[/b], and the [b]CHSH inequality[/b], are two very, very different things.
The four empirical correlations observed in a CHSH-type experiment can all be anything from -1 to +1, since there is no constraint at all connecting the numbers involved in one of the correlations with those in any of the others. Hence one of the four correlations minus the sum of the other three can be anything between -4 and +4.
Perhaps people who talk of bounds being violated should be a bit more careful in what they say. What do they mean, precisely?
For a very precise statement, see Theorem 1 in my paper http://arxiv.org/abs/1207.5103.
