Bohmian Mechanics of DCES (swaps), featuring several papers

[DrChinese]

NOTE: This is more or less a continuation of another closely related thread on Entanglement Swapping, especially Delayed Choice Entanglement Swapping (DCES) versions. More specifically, we are examining how DCES is viewed from the perspective of Bohmian Mechanics (BM).

Earlier thread: Entanglement swapping and Bohmian mechanics

The primary sources for experimental implementation are two important papers, plus a host of supporting work by top teams:

a) Ma et al, (2012): Experimental delayed-choice entanglement swapping
This paper demonstrates entanglement of Photons 1 and 4, produced from different PDC crystals, whereby the choice to entangle them via BSM on Photons 2 and 3 (i.e. a swap) or not is made randomly, and subsequent to their polarization measurements. Product State statistics are produced when no swap is executed.

b) Megidish et al, (2012): Entanglement Between Photons that have Never Coexisted
This paper demonstrates entanglement of Photons 1 and 4, produced from the same PDC crystal but at different times. Photon 1 is measured and ceases to exist before Photon 4 is created. Product State statistics are produced when Photon 3 is delayed so that it does not overlap with Photon 2 to enable a BSM (Bell State Measurement).

c) A number of other papers are available describing the underlying setups, here are a few well known ones (I may add some here from time to time):
High-fidelity entanglement swapping with fully independent sources
Experimental loophole-free violation of a Bell inequality using entangled electronspins separated by 1.3 km

We are fortunate to have some excellent resources in our PF members, many of whom are quite versed in Bohmian theory. We have also been referring to several papers helping us to refine some of the key tenets of BM as it relates to spin. Here are some additional papers we have been discussing:

d) Norsen, (2013): The Pilot-Wave Perspective on Spin
See his opening paragraph containing (1) and (2), and also his Fig. 6 and related text.

e) Huggett (2009): Entanglement Exchange and Bohmian Mechanics
His Fig. 1 is a schematic of what he calls a "Bell-ometer". This is the same thing as a Bell State Measurement (BSM) device as labeled in most papers. He labels his inputs to the BSM (Bell-ometer) as Particles 1 and 3, where in others they are marked Photons 2 and 3. Otherwise, the setup is essentially the same. This makes it convenient for comparison purposes with actual experiments. Note that this paper was written prior to the advent of the a) and b) papers above. In other words, the Delayed Choice option is not being presented. This paper analyzes the more traditional "swap-first" style.

I will use a follow-on post to "bring us up to speed".

-DrC

 

[DrChinese]

In our previous thread:

We discussed how Delayed Choice Entanglement Swapping is explained/described/modeled in Bohmian Mechanics. We managed to come to some common ground - and some disputed ground. I will try to state where we are and what it is I am asking as best as I can.

Common ground:
-If it is a specific axiom of Bohmian Mechanics - such as nonlocality or definite particle positions - I accept it as valid for purposes of the discussion.

Disputed ground:
i) For some, simply saying BM makes the same predictions as orthodox QM is sufficient and no further discussion is to be expected. I don't accept this as adequate, as there are specific issues around DCES that are not reasonably explained in the literature. In fact, comparatively speaking, there is almost nothing written as of now regarding details of how BM treats DCES. Huggett is one.
ii) What is orthodox QM (oQM)? I look at the works of experimentalists in the area of DCES, and assume their written presentation (theory and results) would be considered orthodox. Others look at oQM as an Interpretation of Quantum Mechanics. So typically I tend to compare oQM to BM to discover differences. Whereas some might not start where I do.

@Demystifier (and others) has generously taken some time to explain some of the points he sees as relevant. Specifically, he sees the Measurement apparatus for Photon 1 as becoming entangled with Photon 2 when Photon 1 is measured and ceases to exist. While this explanation is not exactly common to BM descriptions, I accept it for purposes of discussion.

I am going to attempt to walk through the logic of Huggett's paper, since it is something we can all look at and follow.

What I am asking:
a) I am trying to understand how BM explains both nonlocality in space (something that is built into the very premise of BM) and temporal nonlocality (which is not).
b) I am also trying to understand how BM explains deterministic outcomes in either swap direction - i.e. you get the same result regardless of whether the BSM occurs early or occurs later.
c) What is the nature of "collapse" in BM? Some say there is effective collapse (FAPP), some say no, some say yes. Somehow that seems relevant to the ordering of operations in a swap.

Thanks in advance,

-DrC

 

[Matterwave]

Note that AFAIK Huggett's paper is about "regular" entanglement swapping and does not discuss the delayed choice version.

 

[DrChinese]

Note that AFAIK Huggett's paper is about "regular" entanglement swapping and does not discuss the delayed choice version.

You are correct. Somewhere in my mishmash above I thought I mentioned that. But I am hoping the principles I learn as we go through that will allow us to proceed to the DCES option. Also, he does use an approach where he labels t=0, 1, etc. similar to @Demystifier so hopefully we can get some added benefit from a comparison.

 

[Morbert]

Also note that Huggett's eq (1.3) and (1.4) are the same initial state. They correspond to Ma's eq (1) and (2), though the preparation slightly differs (Huggett's preparation has the form $\psi^+_{12}\psi^+_{34}$ while Ma's has $\psi^-_{12}\psi^-_{34}$).

 

[Morbert]

Taking Huggett's paper and formalism as a base, but reversing measurement ordering, we have Huggett's initial state $$\alpha_{12}\alpha_{34}\phi_1\phi_2\phi_3\phi_4\,\psi_0$$Instead of the BSM on 1 and 3, we'll look at the scenario where 2 and 4 are measured, correlating the spatial pointer states with the internal states of 2 and 4, yielding the state$$\begin{aligned}\frac12\Big(&
a_1b_2a_3b_4\,\phi_2^b\phi_4^b\\
&+a_1b_2b_3a_4\,\phi_2^b\phi_4^a\\
&+b_1a_2a_3b_4\,\phi_2^a\phi_4^b\\
&+b_1a_2b_3a_4\,\phi_2^a\phi_4^a
\Big)\phi_1\phi_3\psi_0.
\end{aligned}$$Each of these terms is itself an effective wavefunction. Let's say the pointer states $\phi_2^b,\phi_4^b$ are recorded. Then the effective wavefunction is $$a_1b_2a_3b_4\,\phi_2^b\phi_4^b\phi_1\phi_3\psi_0$$A BSM on 1 and 3 can only yield pointer outcomes $\psi_\gamma,\psi_\delta$ and so this effective wavefunction evolves to $$b_2b_4\,\phi_2^b\phi_4^b\frac{1}{\sqrt{2}}(\gamma_{1,3}\psi_\gamma + \delta_{1,3}\psi_\delta)\phi'_1\phi'_3$$My understanding of the Bohmian implications (and @Demystifier please critique): Say the initial configuration of 1 and 3 was one that was deterministically destined for a Bell outcome $\psi_\alpha$. Now, the choice to measure 2 and 4 with outcomes $\phi_2^b \phi_4^b$ has nonlocally forbidden that outcome, and changed that destination to either $\psi_\gamma$ or $\psi_\delta$.