DrChinese said:
Mjelva: "Assuming the projection postulate [can be applied here], Alice and Bob’s measurements have the effect of projecting the state |Ψ(0)〉into one of the following four product states, each with probability p=1/4: ... On any given run of the experiment, the total system will be in one of the these possible states."
If it is in one of the product states, as he says here, then according to your 1. it does not lead to entanglement swapping. You are taking both sides of the same question.
I'll try to show that we (Mjelva,
@Morbert and me) are not wrong. I really hope we all can agree. When I shared Mjelva's paper on the OP, I wished to discuss the implications the forward-in-time analysis has on how the different interpretations of QM deal with DCES. I understand we spend some time with the nuances of the forward-in-time analysis, but it's a bit frustating that we'are stuck still discussing basic linear algebra.
I believe that the problem lies in:
1. You confuse the state assigned to any given run of the experiment, which is a product state as you correctly showed Mjelva said in his eq. pre-(4), with the state of the whole system composed of all the runs, which is a mixture of product states, as Mjelva showed through the densitiy operator in his eq. (4).
2. You (wrongly) believe that a mixture of product states cannot reproduces the statistical data in Ma's paper.
To be clearer (one more time), let's do the math. Please,
@DrChinese, follow me on this calculation. I'll consider the case in Ma's experiment where Alice and Bob always measure in the ##L/R## basis. After preparation at ##t_0##, the state of the system is:
##\ket{\Psi(t_0)} = \ket{\psi^-}_{1,2} \otimes \ket{\psi^-}_{3,4}##
At time ##t_1##, Alice and Bob measure photons 1 and 4, respectively. They could obtain four different combinations: RR, RL, LR, LL. According to the projection postulate as in Zweibach's textbook, Alice and Bob project the initial state onto four different product states with 25% chance:
##\ket{\Psi(t_1)}_A = \ket{R}_1 \otimes \ket{R}_4 \otimes \ket{L}_2 \otimes \ket{L}_3##
##\ket{\Psi(t_1)}_B = \ket{R}_1 \otimes \ket{L}_4 \otimes \ket{L}_2 \otimes \ket{R}_3##
##\ket{\Psi(t_1)}_C = \ket{L}_1 \otimes \ket{R}_4 \otimes \ket{R}_2 \otimes \ket{L}_3##
##\ket{\Psi(t_1)}_D = \ket{L}_1 \otimes \ket{L}_4 \otimes \ket{R}_2 \otimes \ket{R}_3##
So, if we want to define the state of the whole system including all the runs, it will be a mixture:
##\rho(t_1) = \frac{1}{4} (\ket{\Psi(t_1)}_A\bra{\Psi(t_1)}_A + \ket{\Psi(t_1)}_B\bra{\Psi(t_1)}_B + \ket{\Psi(t_1)}_C\bra{\Psi(t_1)}_C + \ket{\Psi(t_1)}_D\bra{\Psi(t_1)}_D)##
Something very important: the state ##\rho(t_1)## does not represent one of the four subsets ##A,B,C,D##. The state ##\rho(t_1)## is assigned to the whole system formed by all the runs taken together, so it is a mixture of product states. Now, the question is: This state is able to reproduce the statistical data in Ma's paper? Let's see.
First, we can rewrite the states ##\ket{\Psi(t_1)}_{A,B,C,D}## with photons 2&3 in the Bell-states basis:
##\ket{\Psi(t_1)}_A = \ket{R}_1 \otimes \ket{R}_4 \otimes \frac{1}{2} (\ket{\phi^-}_{2,3}-i\ket{\psi^+}_{2,3})##
##\ket{\Psi(t_1)}_B = \ket{R}_1 \otimes \ket{L}_4 \otimes \frac{1}{2} (\ket{\phi^+}_{2,3}+i\ket{\psi^-}_{2,3})##
##\ket{\Psi(t_1)}_C = \ket{L}_1 \otimes \ket{R}_4 \otimes \frac{1}{2} (\ket{\phi^+}_{2,3}-i\ket{\psi^-}_{2,3})##
##\ket{\Psi(t_1)}_D = \ket{L}_1 \otimes \ket{L}_4 \otimes \frac{1}{2} (\ket{\phi^-}_{2,3}+i\ket{\psi^+}_{2,3})##
What the states above predict if Victor decides to perform a BSM? To be short, take only the case when Victor measures ##\phi^-##. According to the Born's rule, Victor will measure ##\phi^-## 50% of the times Alice and Bob measured ##RR## and 50% of the times Alice and Bob measured ##LL##. So, if you took all the runs, i.e. the system whose state is a mixture of product states described by the density operator ##\rho(t_1)##, and construct a subset formed only by the runs where Victor measured ##\phi^-##, you could see perfect correlation between Alice's and Bob's measurements in the ##L/R## basis. This is the experimental result observed in Ma's paper! We could do the same calculations considering any other basis for Alice's and Bob's measurements.
So what did we proved above? Well, we can see that if we solve the time-dependent Schrödinger equation evolving the initial state of the system from time ##t_0## until time ##t_2##, unitarily between measurements and "collapsing" the state upon measurements, we can reproduce statistical data in Ma's paper with photons 1&4 never being in an entangled state even when Victor performed a BSM. If you like, you could say that these forward-in-time analysis does not involve entangled states for 1&4. This is the entire point of the forward-in-time treatment, you could reproduces all the data without 1&4 entangled states!
On the other hand, the experiment can be interpreted in a very different way by giving priority to Victor's measurements. In that case, when he decides to perform a BSM, he projects the state of the system and post-selects the runs where he obtained only one of the Bell states (let's say ##\phi^-##), the state of the remaining photons 1&4 will be in an entangled state ##\phi^-##, which correcly predicts 1&4 results even if they were already measured before the BSM (swap) was performed. This is how authors present the results in Ma's paper. I use the word "present", instead of "interpret" because they (Zeilinger's group) does not take the state of the system as something real. They don't favor a ##\Psi##-ontic interpretation, but a ##\Psi##-epistemic one.
What is really interesting about all these analysis is that both descriptions correctly predict the same data. If you argue against retrocausality, the forward-in-time analysis à la Mjelva (or Cohen, by the way) is perfect for you, because all the data can be explained by evolving the state of the system from past to future with 1&4 photons never being in an entangled state. In that sense, this allows to (re)-interpret the well-known conclusion by Peres "
each subset behaves as if it consisted of entangled pairs of distant particles". Some authors (Bacciagaluppi and others) interpret that as a "proof" against the reality of entanglement. In my view, it could be even saw as a hint against ##\Psi##-ontic interpretations or even against causation. Of course, these last thoughts are interpretation-dependent.
As I said at the beginning of this post, I really hope we can all agree on the math, so we can move on to the physical analysis of these results.
Lucas.
