# Can double entanglement preserve correlations?

We have 2 EPR experiments running in parallel, with Alice having one leg of each (a1,a2) and Bob the other leg of each (b1,b2). Thus (a1,b1) are anticorrelated, as are (a2,b2). Thus also (a1,a2) are uncorrelated as are (b1,b2). Now Alice locally entangles (a1,a2), and Bob measures b1 and b2. After repeating the entire experiment (including setting up the initial entanglements) many times, does Bob see consistent correlation or anticorrelation between his measured b1 and b2?

How Alice accomplishes this final entanglement (a1,a2) is either via entanglement swapping, or via the method described in Yurke and Stoler, Phys. Rev. A46, 2229 (1992): "Bell’s-inequality experiments using independent-particle sources".

Put another way, it's clear that b1 and b2 will show no correlation if Alice does not entangle (a1,a2), since the two EPR experiments are independent. Will this situation change for Bob as a result of Alice's entanglement of (a1,a2)?

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It's clear from no signalling--- by entangling $a_1$ and $a_2$, Alice uses local operators which necessarily commute with the spin operators on $b_1$ and $b_2$, so the reduced density matrix for $b_1$ and $b_2$ stays completely random. It makes no difference what method Alice uses, unless it involves mucking around with Bob's electrons.

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Well spotted; I was trying to sneak by no-signalling. The only loose end for me is to see the maths that demonstrates your assertion. A link that makes this clear mathematically would be great. –  Andrew Palfreyman Aug 27 '12 at 13:07
I tried this because, since (a1,a2) are now correlated - i.e. are distinguishable from random - I inferred that (b1,b2) would behave similarly, and thus would be distinguishable from random. –  Andrew Palfreyman Aug 27 '12 at 13:15