Here is the thought experiment I’ve come up with to celebrate my ignorance.
An electron-positron pair is emitted such that they are entangled on spin.
Case 1: The electron and positron are brought back together and they annihilate while the entanglement is still intact and a pair of gamma rays are emitted. Add everything up.
Case 2: A second entangled electron-positron pair is emitted and travels an energetically identical path to the first pair, except somehow “the entanglement is lost to the environment” in Case 2 before annihilation. Add everything up.
My understanding assumes:
a) The superposition of the two particles is lost to the environment in the second case.
b) But, that the wavefunction doesn’t “collapse” at instant the entanglement is lost.
That said, my knowledge of notation, wave equations and information theory is too limited to know if there is there a difference in entropy from results of the *isolated* entangled annihilation and *isolated* un-entangled annihilation.
1) Is there something different about the wave-equations of the gamma rays emitted in both cases?
2) Is the information and/or entropy of the *isolated* (electron, positron, gamma-pair) the same in both instances or do I have to account for the information in the wavefunction of the “environment” too?
3) From an information theory standpoint some kind of “half-bit” missing from the second instance that is somehow carried away by the wavefunction of the environment?
You don’t have to answer all of the above questions! I’m really just looking for a nudge in the right direction, since most papers I’ve read are on closing EPR loopholes, not on the information theory perspective on those experiments.
Hello Dean,
This is a good question!
I'm no expert in e-p annihillation, but I see no reason for this process not to conserve spin. If indeed so, the spin state of the electron and positron will just be transferred to the photon pair. That is, if the e-p pair was in a statistical mixture (your case #2), so wil be the spins (polarizations) of the outgoing photon pair. On the other hand, if the e-p wew spin entangled ( case #1), so will be the photons. In short, no information is either added or subtracted from the (isolated) system during the annihillation itself.
If you want to go a bit into the details of the process, search "spin in electron positron annihilation" on google, and follow the first result there (a pdf presentation from Cambridge).
Cheers,
Eilon.
Hello Dean,
This is a good question!
I'm no expert in e-p annihillation, but I see no reason for this process not to conserve spin. If indeed so, the spin state of the electron and positron will just be transferred to the photon pair. That is, if the e-p pair was in a statistical mixture (your case #2), so wil be the spins (polarizations) of the outgoing photon pair. On the other hand, if the e-p wew spin entangled ( case #1), so will be the photons. In short, no information is either added or subtracted from the (isolated) system during the annihillation itself.
If you want to go a bit into the details of the process, search "spin in electron positron annihilation" on google, and follow the first result there (a pdf presentation from Cambridge).
Cheers,
Eilon.
Eilon ... great suggestion for a reference. I was having difficulty locating a reference that showed the situation using Feynman diagrams and and that does even more than that. Now I just have to figure out what you said actually answers the question I was attempting to ask! :-)
Hello Dean,
I agree with Eilon Poem and add something else:
Case #1: S(e-, e+)=S(gamma, gamma)=0.
Case #2: S(e-, e+)=S(gamma, gamma)=2.
Wave-equations of the gamma rays emitted in both cases are the same, but wave functions are different. But more correctly to say about von Neumann equation with different density matrix.
But I have a question too. Are there experimentally observed entangled particle-antiparticle pairs? Sombody even believes that particle-antiparticle is forbidden.
Best,
Srgey
@Sergey. In he immortal words of Homer Simpson: "Doh!"
I am an independent researcher. I have no advisers or institutions to guide me. I read a great deal, but my motto is "Think Crazy, Prove Yourself Wrong." I should add, "I will miss important details!" :-)
When I first read about EPR I saw electron-positron pairs as examples in the thought experiments. I remember reading that eventually experiments were.done with photons to attempt to prove Bell' theorem, or at least close loopholes, so I read those. I then realized that photon experiments were at the crux of eliminating Bell loopholes and followed loop-hole elimination papers for years. I was intrigued by entanglement and didn't want to be a fool if it was eliminated as not-scientifically valid. That noose is tight, as technology is exploiting entanglement for quantum-cryptography at the very least.
After all that fruitful research, It never occurred to me to look to see if particle-anti-particle pairs where every proven to be entangled.. Your question was quite a shock to me!
I have found leads as to whether particle/anti-particle entanglements exist, but not yet solid enough to post. Thanks to both of you for the nudges!
The question I now know I am asking (partially answered above), is: "How does the quantity of information in an entangled pair differ from that of a entangled pair which loses its entanglement to the environment?
The follow-up question would be, "how much *information* is lost ot the environment, is it information and does it affect entropy?
@Sergy. After my philosophical rant above, could you answer, "what does S stand for in your equarions?" Spin? Psi? I a young daughter and wife and have to pick and choose the best times to research the deeper aspects of replies. In short ... I haven't had time to unpack yours or Eilon's suggestions fully, but that doesn't mean your questions didn't spark more questions I wanted to ask!
Thanks again to you both for taking my question seriously.
Dean
Dean,
S is the entropy of the whole system state (electron + positron, or two photons).
Mutual information I between its subsystems (electron and positron, or two photons):
Case #1: I=2Sr , where Sr - is the entropy of the reduced subsystem state. The whole state is pure, hence both the Sr are equal.
Case #2: Since entanglement has been lost, 0
Sergy,
Thank you for your clarification. I've got my work cut out for me to understand this properly, but I think you have provided me with a nudge in the right direction.
Dean.
Reading the above discussions I believe that the question was now clarified.
I have no more comments to do.
Best,
Cattani
Mauro Cattani -- I'm pretty sure I'll be back with further questions, because I have to learn the langue of entropy much better for I will fully understand the above. I hope to restate the above conclusions to see if I really do comprehend. That may take a while!
Dean,
In terms of information theory, your Case 1 and Case 2 differ in a nicely simple way: In Case 1, no information is transferred irreversibly from the e+/e- pair. In Case 2, information *is* transferred to some system outside of the e+/e- pair.
Case 1 thus remains quantum throughout. In contrast, in Case 2 the transfer of information causes the event to "lose coherence", or more simply, to stop being quantum. You just get two particles that are no longer entangled in the way you mean, which most likely is angular momentum (spin) entanglement.
One qualifier to that: To *keep* Case 2 non-quantum, the information transfer must be "pragmatically irreversible." By that I mostly mean that the information must touch and become entangled with some thermal object of non-trivial size. (And yes, I meant to be just that fuzzy about issues such as size.)
The thing is this: You *can* erase information, by which I mean return it "unopened" to the quantum systems from which it was taken. However, erasure is possible only if you are careful indeed about where and how you store that information while it sojourns beyond the boundaries of the original quantum system.
Or stated the other way around: There exist situations where you can lose information from the Case 2 e+/e- pair and thereby cause it to lose entanglement, yet you can still reverse that loss of information and thereby make the system entangled again. The trick to that is you have to extract, store, and return the information in a way that keeps it absolutely secret from the rest of the universe throughout its sojourn.
Once you get careless about where you store your sojourning information -- in particular, once you let it touch and interact with some kind of impossibly complex thermal system -- then the always-in-motion fabric of classical thermal time will capture your event in a way that will make the likelihood of that information ever returning to its source so unlikely that it would be difficult even to calculate the probability accurately. At such a point, Case 2 has become pragmatically classical.
(Notice also that no "conscious observer" is needed in any of this. All you really need is a bit of matter with a sufficiently complicated network of information exchanges to make the return or erasure of the lost information vanishingly unlikely.)
Hi Dean,
Great question, and very nice answers from @Eilon and @Terry.
The key here is in the preservation of the entanglement and the properties of the annihilation. As you correctly intuit, there really has to be a difference between the annihilation of entangled or un-entangled pairs. This arises in part due to the formation of positronium and the resultant change in the lifetime of the electron-positron pair.
But the key here is not in the details of the annihilation event IMO, but rather in the fact that the annihilation event is effectively performing an entangled basis measurement, so it is effectively (and destructively) projecting the electron-positron pair into a particular Bell state. In the case of perfect preservation of entanglement, this collapse will simply be the time reversal of the creation process, whereas in the un-entangled (decohered) version, this will be the result of a (stochastic) projection.
In principal, I would guess (as @Eilon mentions) that you should be able to reconstruct the state of the electron-positron pair by reconstructing the spins of the two photons.
Finally, I wanted to add that your problem really reminds me very much of the initial formulation of Hardy's paradox, which involves two electron-positron pairs. Given that connection and the fact that Hardy's paradox has been demonstrated (at least twice) using entangled photons, I'm pretty sure that you could come up with a purely photonic version of this that might illustrate what you are thinking of.
I would suppose that it would take the form of two entangled or unentangled photons entering a Bell state measurement device.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Quantum Physics
- Quantum Mechanics