Logical inconsistency in standard model of entanglement

[rcoreilly]

I'm confused about a premise implicit in the standard QM model of entanglement, which seems logically inconsistent.

I understand that entanglement arises when two or more particles interact in some way to become synchronized in their quantum states, which also must be indeterminate in terms of conjugate variables (e.g., spin or polarization), such that they share a common wave function that represents a superposition. When these particles then go their separate ways, the standard entanglement model states that you must continue to treat each particle as though it were a single particle with a common wave function until a measurement is made, thereby collapsing the wave function and removing the entanglement between the two particles. Until the wave function collapses, the quantum state is indeterminate, and because it started out with the unknowns all mixed up between the two particles (entangled), you can't treat them as separate particles, but rather as an undivided whole.

Of course, the critical implication of this entanglement scenario is that you can actually perform a measurement on one of the two particles, and thereby make a prediction about the state of the other, because they share this common entangled state. For example, if you measure the spin on one particle and it is "up", then the other must be "down".

Here's my question: when we say that the particles have moved away from each other, then we seem to be saying something definite about the spatial decoupling of their wave functions: in a simple one dimensional Schrodinger wave propagation model, you would see two wave packets moving away from each other in opposite directions, leaving nothing in between. At this point, it would seem that even the standard model would have to predict that there is no way for these separate wave packets to produce any kind of single-particle entangled behavior: you are explicitly measuring two separate wave functions.

The very presumption of separation into separate particles seems to imply knowledge of the relative locations of the two particles. How can this happen in the absence of a measurement? If you truly did not know whether the two particles were still spatially intermingled or not, this would be represented by a superposition of the two particle wave functions over the shared range of uncertainty in their locations (providing plenty of opportunity for the wave functions to remain entangled). And then you would have to first conduct a measurement to determine if the two particles had in fact separated or not, before you could say anything specific about having performed a measurement on one particle or the other.

And once you did that measurement, if it indicated that you did in fact have two separate particles, then you would represent that state of the world with two separate wave functions, and entanglement would be broken.

So either way, it seems like the idea that physically separated particles could remain entangled is logically inconsistent with the standard QM formulation. In the first case you've setup the problem wrong if you know for sure that the two particle states will be physically separate, and in the second case you destroy the entanglement with the first measurement of position.

Concretely, in the case of photon polarization experiments used to test the EPR paradox, the photodetectors are placed some distance on either side of the direction vector of the incident photon that triggered the pair creation, so you definitely know that the two photons have nonoverlapping spatial locations. Why isn't this fact included in the structuring of the wave functions describing the setup, and why wouldn't that known spatial separation break the entanglement?

What am I missing?

 

[ZapperZ]

Here's my question: when we say that the particles have moved away from each other, then we seem to be saying something definite about the spatial decoupling of their wave functions: in a simple one dimensional Schrodinger wave propagation model, you would see two wave packets moving away from each other in opposite directions, leaving nothing in between. At this point, it would seem that even the standard model would have to predict that there is no way for these separate wave packets to produce any kind of single-particle entangled behavior: you are explicitly measuring two separate wave functions.

What "spatial decoupling"?

In a simple, entanglement description for a bipartite system, you have something generic such as this:

[tex]\Psi = |u_1, v_2> + |v_1, u_2>[/tex]

I don't see the "spatial decoupling" there. u and v could easily be spin states that do not depend on how much it has traveled as long as one maintains coherence. These are NOT wavepackets.

Zz.

 

[rcoreilly]

Why aren't they wave packets? How else would you represent the definite knowledge that the particles now exist in separate physical locations (without conducting a measurement)? Isn't there something fundamentally different between a system that you know to have physically separate states, vs. one where this is truly unknown? Logically, it seems you would need to actually make a measurement to know this difference, and if you actually did make a measurement, entanglement would be destroyed. So it seems that these two very different situations are being treated the same. Perhaps the shorthand formalism without using actual Schrodinger wave functions is leading to an oversimplification??

 

[ZapperZ]

But the whole system isn't entangled, just that particular property! What I wrote is a standard, typical description of the entangled property. So what you need to figure out is, how can an entangled system get by with only just that description. Because once you accept that as a valid description (and most people have), then your question is answered.

Zz.

 

[rcoreilly]

I know it is standard. Can you confirm that if you didn't know if the two particles were in separate physical locations, and you performed a measurement of position, it would also destroy the entanglement with respect to spin? If that would not destroy the spin entanglement, then I guess there is no logical inconsistency. If it does, then it seems like there is still an inconsistency there. I had thought that any measurement collapses the wave function and destroys entanglement, but perhaps it is property specific collapse only?

More generally, is it the case that there are scenarios like this for which it is impossible to provide a concrete Schrodinger wave picture? I'm having trouble imagining how you would setup a Schrodinger wave configuration for physically separate particles that nevertheless maintain spin entanglement?

 

[ZapperZ]

Back up a bit. Even before we deal with a measurement of position and how it affects the spin, let's get back to the very basic QM. You have two operators, A and B, and they don't commute, i.e. [A,B] is not equal to zero.

Do you think if I measure A, thus "collapsing" the wavefunction and obtaining a value for that observable, I have also destroyed all the superposition associated with observable B?

Zz.

 

[kith]

I know it is standard. Can you confirm that if you didn't know if the two particles were in separate physical locations, and you performed a measurement of position, it would also destroy the entanglement with respect to spin? If that would not destroy the spin entanglement, then I guess there is no logical inconsistency. If it does, then it seems like there is still an inconsistency there. I had thought that any measurement collapses the wave function and destroys entanglement, but perhaps it is property specific collapse only?

Yes. The state vector collapse occurs only for the properties you measure. If your spins are entangled, performing a position measurement doesn't collapse the spin state.

To really understand entanglement, you should be familiar with the mathematics of combining two states (which is the tensor product). In this formalism, the answer to your questions is quite obvious.

More generally, is it the case that there are scenarios like this for which it is impossible to provide a concrete Schrodinger wave picture?

Yes. Simple wave functions do not take spin into account. If you want a complete representation of state vectors which is analogous to wave functions, you need objects with two components, called Pauli-spinors (at least for the nonrelativistic case of a spin-1/2 particle). From them you then get the probability of finding your particle at position x with spin y.

 

[rcoreilly]

Thanks for the input. After quite a bit of googling around, I happened upon the Bialynicki-Birula first quantized wave function of the photon, and a paper that uses it to describe the time evolution of two entangled photons, but unfortunately not in the case of the EPR-like experiments. This work is all surprisingly recent, and it seems like the critical model using wave functions to simulate the actual EPR experiments according to the explicit wave-based QM formalism has yet to be developed! I'm just skeptical of the temporal abstractions present in the Hilbert space framework..

BJ Smith & Raymer (2007): http://iopscience.iop.org/1367-2630/9/11/414

 

[kith]

I haven't read your paper, but if it is not common to describe multi-photon experiments via photonic wave functions, this is probably due to the relativistic and bosonic nature of photons or simply historic. Entanglement of two systems/properties itself is very well understood. For a start, I recommend the electron case. This is simpler and yields the same results for the questions you posed.

Consider a state |Ψ> = |φ> x |χ> (where |φ> is the state of the external degrees of freedom and |χ> the spin state). How do |φ> and |χ> look like, if you want to describe two electrons with only their spins entangled? And how do you get a wavefunction from this expression?

 

[ZapperZ]

Thanks for the input. After quite a bit of googling around, I happened upon the Bialynicki-Birula first quantized wave function of the photon, and a paper that uses it to describe the time evolution of two entangled photons, but unfortunately not in the case of the EPR-like experiments. This work is all surprisingly recent, and it seems like the critical model using wave functions to simulate the actual EPR experiments according to the explicit wave-based QM formalism has yet to be developed! I'm just skeptical of the temporal abstractions present in the Hilbert space framework..

BJ Smith & Raymer (2007): http://iopscience.iop.org/1367-2630/9/11/414

You still didn't address the example I gave you.

Zz.

 

[rcoreilly]

Here's what I think I've learned so far:

There is no guarantee that the QM formalism for photons will tell us something that is physically real about an experiment, because the QM notion of a photon is a mathematical abstraction that does not automatically describe any given physical situation accurately. Thus, it seems entirely possible that the standard QM framework applies in a perfectly valid and physically sensible way to massive particles like electrons etc, but is quite capable of producing nonsensical results when applied to EM radiation using the photon model.

In the context of entanglement, drawing a bright line between photons and everything else would nicely resolve the nonlocality issue in a quite satisfying way it would seem: as we know from special relativity, the speed of light relative to a massive particle moving at any speed is a constant due to time dilation and Lorentz contractions, and thus it is *always* possible for two entangled massive particles (which must have initially been physically interacting in the same location) to continue to interact physically, in a completely relativistically consistent fashion, via the EM field for example. So, entanglement phenomena associated with massive particles "mediated" via relativistic mechanisms is entirely plausible, and does not violate locality. Presumably most of the unambiguous evidence for the validity of QM entanglement that everyone always talks about is obtained in this massive case. Also, as a practical matter, it appears that maintaining entanglement while achieving any kind of significant spatial separation of massive particles is rather difficult.

However, one could argue that entanglement simply does not apply to photons, because of the non-physical representation of photons in QM. QM can only provide approximations of EM phenomena when applied very carefully, often by incorporating lots of constraints from what we know from classical EM. There are no guarantees that you can just take a generic Hilbert space formalism and apply it to photons and come up with a sensible result. Given that the speed of light constraint is so fundamental to our understanding of every single other EM phenomenon we know, validated by a massive volume of solid data, it would seem that one might be a bit more suspicious of just blithely applying generic QM to this case, and thereby tossing out this extremely well-validated physical fact about relativistic constraints on EM propagation.

Of course, to adopt this position, one would need to account for the actual experimental tests, but there are models of photon entanglement experiments that show how simple localist dynamics, combined with realistic assumptions about the detection properties of actual polarizers, can produce the observed results (e.g., Adenier & Krennikov, 2003; Aschwanden, 2004 -- see http://challengingbell.blogspot.com/ for code to download). And Santos (2004) argued that the detection/fair sampling "loophole" for these cases is theoretically likely to persist: if you try to increase the detection rate it will require higher energy "photons", which have less well defined polarization, and lower energy photons are fundamentally hard to detect with high efficiency. So it may be that a loophole-free test of photon QM is impossible, perhaps validating that the QM model was bogus to begin with for this case.

Anyway, I haven't come across this exact argument in anything I've read so far -- anyone know of some publications making this point? Seems too sane to be novel. Or maybe it is too unorthodox to have been considered?

 

[rcoreilly]

Back up a bit. Even before we deal with a measurement of position and how it affects the spin, let's get back to the very basic QM. You have two operators, A and B, and they don't commute, i.e. [A,B] is not equal to zero.

Do you think if I measure A, thus "collapsing" the wavefunction and obtaining a value for that observable, I have also destroyed all the superposition associated with observable B?

Zz.

Yep, this makes sense -- resolving A definitively should not affect B -- I just had an overly-general picture of the wavefunction collapse.