#2539

So… Rohrlich’s argument that “retro-causality is intrinsic to QM”. For some details, see Section II of Rohrlich’s paper “A reasonable thing that just might work”, which is here:

http://physweb.bgu.ac.il/~rohrlich/Bell50titles.pdf

The basic idea is as follows. Consider three people, Alice, Bob, and Jim, who share a bunch of GHZ-state particle trios:

|GHZ> = ( |+z>|+z>|+z> – |-z>|-z>|-z> ) / sqrt(2).

Rohrlich points out that if Jerry measures the spin of his particle along the z-axis, the remaining two particles (held by Alice and Bob) will be left in a product state (one of two possible product states, really, depending on the outcome of Jerry’s measurement). Whereas if instead Jerry chooses to measure the spin of his particle along (say) the x-axis, the remaining two particles will be left in an entangled state. So the idea is that Jerry can *control* whether Alice’s and Bob’s particles are entangled, or not. And this of course has observable consequences: Jerry can control whether (at least once the data are later appropriately binned according to Jerry’s outcomes) Alice’s and Bob’s measurements are, on the one hand, totally uncorrelated — or, on the other hand, so strongly correlated that they violate a Bell inequality.

So far so good.

But then Rohrlich goes on to point out that the above remains true even if Jerry’s measurement on his particle happens *after* Alice and Bob have already made their measurements and recorded their data. So it seems that Jerry can control whether or not Alice’s and Bob’s particles were entangled (at the earlier time of Alice’s and Bob’s measurements) — i.e., Jerry can control whether or not Alice’s and Bob’s outcomes violate a Bell inequality — by his free and *later* choice about whether to *later* measure his particle in the z- or instead the x- direction.

Here is Rohrlich: “[All this] nicely illustrates the fact that quantum mechanics is retrocausal…. On the one hand, there is no reason to doubt that Alice, Bob, and Jim have free will. Indeed the results of Alice and Bob’s measurements are consistent with whatever Jim chooses right up to the moment when he decides to measure [spin along z] or [spin along x] on each of his particles and record the results. On the other hand, there is no doubt about the effect (in Jim’s past light cone) of Jim’s choice. After Alice and Bob obtain the results of Jim’s measurements (within his forward light cone) they can reconstruct from their data whether their particles were entangled or not at the time they measured them. Thus quantum mechanics is retrocausal….”

It’s maybe not clear whether Rohrlich means to claim that the particular candidate theory “orthodox quantum mechanics” is inherently retro-causal, or instead the more general claim that *any* empirically viable quantum theory will have to be retro-causal. If the latter claim is intended, though, it is definitely false, and we can see that it is false by considering what is going on in this experiment according to Bohmian mechanics.

I don’t want to write out all the technical details (which are trivial anyway), so here’s the gist of it. The important thing to consider is the “effective [or here, equivalently, conditional] wave function” of Jerry’s particle at the time he ends up making his measurement. This is the “one-particle wave function” that, in Bohmian mechanics, can be understood as guiding the particle in question along its deterministic trajectory through the Stern-Gerlach apparatus (or whatever). See, e.g., my paper on “The pilot-wave perspective on spin” if these concepts are unfamiliar:

http://arxiv.org/abs/1305.1280

Anyway, in the case that Jerry is the first to make a measurement, the conditional wave function of his particle is such that (for the standardly-assumed statistical distribution of possible Bohmian particle positions within the wave) his outcome is 50/50 random, no matter what axis he measures the spin along. As a result of his measurement, though, the conditional wave function(s) associated with the other two particles change (non-locally, to be sure) and hence the subsequent trajectories of Alice’s and Bob’s particles are different from what they would have been had Jerry instead made a different (or no) measurement.

In the other case, though, where Alice and Bob perform their measurements first, Jerry’s outcome turns out *not* to be 50/50 random for all measurement directions. For example: suppose Alice measures her particle along the x-axis and Bob measures his particle along the n-axis (60 degrees toward the y-axis from the x-axis… just the kind of measurements we’d expect them to be making if they planned on seeing later if a Bell inequality was violated) and suppose that Alice and Bob both find their particles to be “spin up” along the measured direction (this will happen, according to Bohmian mechanics, some of the time, depending on the exact initial positions of the various particles). Then it turns out that the conditional wave function of Jerry’s particle is such that, if Jerry measures along the z-direction, his outcome is 50/50 random… *but*… if Jerry measures along the x-direction, there is a 25% probability that his particle will be “spin up along x” and a 75% probability that his particle will be “spin down along x”. It’s not 50/50 random at all.

To be sure, there is nonlocality here: Alice’s and Bob’s measurements change the state (meaning, here, the conditional wave function) of Jerry’s particle and hence (sometimes, maybe) cause it to emerge in a different direction from the Stern-Gerlach device than it would have (even for the same initial position of Jerry’s particle!) had Alice’s and Bob’s measurements been different (or not been done at all or had they come out differently). So there is non-local (faster than light) causation, but no retrocausality. So the claim that the phenomena in question require retrocausality is simply wrong. The feeling that there is somehow something retro-causal going on, is in fact just a result of the false assumption that Jerry’s measurement outcome is “really 50/50 random” when, in fact, in the relevant cases, it’s not. What maybe vaguely looks like retro-causation from some perspective is instead, from the perspective of Bohm’s theory, just a matter of biased post-selection.

Some random notes about all this:

* Ordinary QM, with nonlocal collapse, also provides a (rather parallel) way of understanding why retrocausality isn’t required in this kind of situation. I say it’s “rather parallel” because what I said about the Bohmian conditional wave function of Jerry’s particle above is just exactly the same as what ordinary QM would say happens to “Jerry’s particle’s wave function” after Alice’s and Bob’s measurements collapse the overall 3-particle state. Of course, I prefer the Bohmian account, even for this polemical purpose, because, well, Bohmian mechanics might actually be true.

* The same exact ideas apply (in almost exactly the same exact way) to the delayed choice quantum eraser. There too, there is (at least according to Bohm’s theory) no backwards-in-time causation. There may be faster-than-light causation (depending on exactly what quantum eraser setup one is talking about) but really the appearance of retrocausation is fully explained in terms of biased post-selection.

* And finally note that the basic idea here is really quite simple. Roderich Tumulka linked above to Bell’s nice article about Wheeler’s delayed choice (thought) experiments, explaining how there’s really nothing retrocausal (or, frankly, nothing the least bit weird *at all*) going on, according to Bohm’s theory. That’s definitely worth reading/reviewing. And here’s another simple example that I think brings out (what is, from the Bohmian point of view) the error in Rohrlich’s reasoning quite simply. Consider the EPRB situation — a pair of spin-1/2 particles in the singlet state. Suppose Alice and Bob are just both measuring their particles spins along the z-axis. The outcomes are of course perfectly (anti-) correlated. Well, you might argue as follows: “suppose Alice measures first and Bob measures second; Bob’s measurement outcome is 50/50 random (because QM); but the results are perfectly (anti-) correlated; so it must be that Bob’s result retro-causally affects the state of Alice’s particle, prior to her measurement, and hence affects the outcome of Alice’s earlier measurement!!” But of course that is silly. I mean, that’s one logically possible story, I suppose, but it’s hardly required. It’s perfectly possible to explain everything without retrocausality — by just allowing that Alice’s measurement (which happens first!) non-locally influences Bob’s particle (and hence the outcome of his subsequent measurement). See my paper on spin, linked above, for details about this case. I think, when the dust clears, it really is equivalent to Rohrlich’s more complicated case — the error in both arguments comes down to assuming (unjustifiably and wrongly, from the Bohmian point of view) that some later measurement is “really 50/50 random”.