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Maximal Entanglement in High Energy Physics
by Alba Cervera-Lierta, José I. Latorre, Juan Rojo, Luca Rottoli
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|Authors (as registered SciPost users):||Alba Cervera-Lierta|
|Preprint Link:||http://arxiv.org/abs/1703.02989v3 (pdf)|
|Date submitted:||2017-09-27 02:00|
|Submitted by:||Cervera-Lierta, Alba|
|Submitted to:||SciPost Physics|
We analyze how maximal entanglement is generated at the fundamental level in QED by studying correlations between helicity states in tree-level scattering processes at high energy. We demonstrate that two mechanisms for the generation of maximal entanglement are at work: i) $s$-channel processes where the virtual photon carries equal overlaps of the helicities of the final state particles, and ii) the indistinguishable superposition between $t$- and $u$-channels. We then study whether requiring maximal entanglement constrains the coupling structure of QED and the weak interactions. In the case of photon-electron interactions unconstrained by gauge symmetry, we show how this requirement allows reproducing QED. For $Z$-mediated weak scattering, the maximal entanglement principle leads to non-trivial predictions for the value of the weak mixing angle $\theta_W$. Our results illustrate the deep connections between maximal entanglement and the fundamental symmetries of high-energy physics.
Author comments upon resubmission
We would like to thank the referee for his/her detailed report on our manuscript "Maximal Entanglement in High Energy Physics". We shall take in turn all the comments by the referee and provide appropriate answers, indicating, when relevant, the corresponding modifications in the manuscript.
1) The referee suggests that "MaxEnt" is not a good choice of name since this acronym is used for maximum entropy distribution, and in that specific case it relates to maximum uncertainty.
Indeed, maximal entanglement in 2-particle processes amounts to having the reduced density matrix for any of the two particles proportional to the identity, which is the maximally entropic state, and is also maximally uncertain. That is, maximal entanglement in two-party systems is tantamount to maximal entropy of one of the subsystems. Beyond 2 particles, the classification of entanglement becomes richer and goes beyond the idea of entropies in subsystems.
We have added a comment in the manuscript to clarify this issue. We have chosen however to retain the name MaxEnt, since we believe it is appropriate choice to describe the phenomena we are studying.
2) The referee mentions that the study is made for the spins, but not for the full dependency of the out-going particles.
Here we would like to argue that this is not the case. Indeed, we do analyze the figure of merit of helicity as a function of both the out-going angles and momenta. This is a large phase space, and therefore we have divided the analysis into the high energy and low energy limits, finding different results in each case.
In order to clarify the interplay between the results at different energies, we have to the revised version of the manuscript added a new table (TABLE I) summarizing all the cases considered.
3) The most important comments by the referee are related to whether angular momentum conservation is equivalent to the principle of maximal entanglement.
In our analysis, angular momentum conservation was never assumed.
It remains the spirit of the paper to see to what extent MaxEnt dictates symmetries. We found that at the level of 2-particle scattering processes, some extra solutions are acceptable which do not correspond to Lorentz invariant interactions. As a matter of fact, no 2-particle scattering processes can determine the sign of each gamma matrix. It is only at higher orders in perturbation theory where this (unphysical) solution could be discarded by MaxEnt.
Another example where angular momentum plays a different role than maximal entanglement can be found in the decay of positronium to three photons (see quant-ph/0007080). There angular momentum conservation on three outgoing particles gives a much richer range of possibilities to analyze entanglement. So in general angular momentum conservation and the principle of maximal entanglement are physically inequivalent.
In this respect, the situation is fairly comparable to the one of Bell Inequalities. The simplest CHSH is maximally violated by a singlet in a given basis. As more local systems are included, Mermin inequalities for many-particle systems are violated by GHZ-states which are not related to simple representations of angular momentum.
Therefore, keeping the ideas of maximal entanglement and angular momentum conservation separate appears to be the right way to present the ideas explored in our work.
We have added a number of comments to make this point more transparent in the revised version of our manuscript.
4) Finally, the referee makes the observation that MaxEnt leaves no space to account for the theoretical uncertainties in the predictions of the weak decays of the Z boson.
Our answer to this point is two-fold. First of all, the computation presented in this work is just a tree-level one. The full MaxEnt principle should be applied at the level of physical observables (cross-sections), rather than tree-level amplitudes. This is now mentioned explicitly in the new text.
The second comment is that there is in our work no aim to claim that MaxEnt is necessarily fixing all parameters in the Standard Model. We simply wanted to point out the somewhat surprising fact that the weak angle is pretty close to the MaxEnt prediction at tree level. Whether or not this points to a deeper principle or it is just a numerical coincidence, only further research can tell, involving in particular extending the present analysis to include the effects of higher-order corrections.
We hope that having addressed the comments raised by the referee, and updated the manuscript accordingly, the revised version of our work can be considered suitable for publication in SciPost.
List of changes
- Section I: Introduction. We added paragraph five.
- Section III: MaxEnt generation in QED. We modified the last paragraph and added Table I.
- Section IV: MaxEnt as a constraining principle. We added paragraph four.
- Section V: MaxEnt in the weak interactions. We modified paragraph three.
- Appendix B: Unconstrained QED. We addedd last paragraph, including equations (B8) and (B9).
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:1703.02989v3, delivered 2017-10-09, doi: 10.21468/SciPost.Report.256
How entanglement is generated in elementary particle scattering processes is a natural question, which connects two fields (quantum information and high energy physics), and which has not been asked before. The paper thus has a high degree of novelty.
1) It is not clear whether this approach will lead to important new developments in either particle physics or quantum information.
2) Taking the generation of Maximal Entanglement as a fundmental principle is questionnable.
3) the discussion of the results should be significantly expanded.
Report on Maximal Entanglement in High Energy Physics
Entanglement is maybe the key feature that differentiates classical from quantum physics. The question asked in this paper “what processes in the fundamental interactions create entanglement, and how much do they create” is very natural. The authors therefore make a new contribution which should be published.
On the other hand the authors take the stance that generating Maximal Entanglement in helicity degrees of freedom could be taken as a fundamental principle of nature. This is much more speculative, and even questionable.
I strongly recommend that some changes be made to the manuscript. Most of these concern the discussion (either in the introduction, or the conclusion) which should both be expanded and be more nuanced. In addition there are a few more minor issues of presentation that should be addressed (garbled sentences, etc.).
Overall, given that SciPost publishes “publishes outstanding-quality research articles”, I am not entirely sure whether this work should be published in SciPost. By taking a provocative position, the authors seem to be giving their work more importance than it most likely deserves. Probably with a more nuanced and extensive discussion, the paper could be accepted. In the end it depends on how selective the editors want to be.
I strongly recommend that the authors expand the discussion and interpretation of their results. Not only because it will put better in perspective the results themselves, but also because it will help frame the context in which any follow up work will be presented. The following points should/could be discussed, either in the introduction, or the conclusion.
a) Why choose MaxEnt as a fundamental principle? The statement at the beginning of section IV, that it allows “ Bell-type experiments to be carried out violating the bounds set by classical physics.” Is wrong, as Bell type experiments can be carried out also with non maximally entangled states. This needs reformulation and more discussion. Possibly one could say that easy generation of maximal entanglement facilitates non classical experiments, such as strong violation of Bell inequaltiies, or generation of multipartite states appropriate for quantum computation.
The introduction states: "Taking a step further, one can ask what are the consequences
of imposing that the laws of Nature must be able to realize maximally entangled states. Can this requirement be
promoted to a principle?" But there is no discussion of why it would interesting/relevant to take this as a principle. Points b,c,d,e below all suggest that it need not be taken as fundamental. Please discuss in more detail.
b) Essentially all interaction Hamiltonians will generate entanglement (this is because there are (infinitely) more entangled states than non entangled ones). So the fact that entanglement is generated in elementary scattering processes is not surprising. In fact it’s the opposite (absence of generated entanglement) that would be surprising.
c) Entanglement can occur not only in helicity degrees of freedom, but also in position-momentum, energy-time. The authors should mention this, and explain why they decide to focus on helicity. (Note that it could be that little entanglement is generated in helicity, but a lot in momentum).
d) Most experiments to generate entanglement take place at low energies, and are not perturbative (e.g. parametric downconversion mentioned by the authors is a photon-photon scattering process mediated by a non centro symmetric material). Therefore even if maximal entanglement was not generated in high energy scattering processes, it could very well be generated in low energy non perturbative processes.
e) It is possible to transform weak entanglement into maximal entanglement (see Bennett et al Phys. Rev. A 53, 2046) by local operations and classical communication. This also implies that if some (non maximal) entanglement can be generated, then maximal entanglement could be in principle generated also (albeit by a much more complicated process).
f) The above points all suggest that generating entanglement in particle collisions is generic. Nevertheless there is one point that puzzles me. Namely one would naively think that generating maximal entanglement would occur only at isolated points in parameter space. Indeed the maximum of a function should only be reached at isolated points in parameter space. But this does not seem to be the case, see Table 1 and most strikingly Figure 1, where the maximum lies on a curve. A conservative interpretation is not that this is a fundamental feature of the standard model, but rather a generic (geometric) feature of interactions between two systems with Hilbert space dimension 2. This point deserves to be raised.
The following points are minor points of presentation. Correcting them will make the manuscript easier to read.
-last sentence of abstract “ Our results illustrate the deep connections between
maximal entanglement and the fundamental symmetries of high-energy physics.”. I would modify this sentence. See the points raised above.
-Intro. Par2. “ This implies that entanglement must be generated by quantum unitary evolution at the fundamental level.” (Remove “more”).
-Intro. Par4. “ For systems with more than two particles, the classication of entanglement becomes richer and does
not necessarily correspond to the entropy of the SUBsystems.” (change to subsystems).
-Section III. Last Par. “ In two cases, MaxEnt is generated independently of the center of mass angle:”
Change to “ In two cases, MaxEnt is generated independently of the scattering angle:”
-Table 1. Possibly add a column with the name of the scattering process.
Table 1. Caption. Add that a dash indicates that MaxEnt cannot be reached for any value of the scattering angle theta.
-Section IV. First par. Please reformulate and discuss also in introduction and/or conclusion (see above).
-Section IV. Sentences around eq. 9 beginning with “Full consistency..” are not comprehensible. Also notation in eq. 9 is not defined.
-Section V. Par 1. Correct English of sentence beginning: “In the Standard Model…”
-Section V. Par beginning “ We have also studied how this result is modied if we include the contribution from…” is difficult to understand. It needs rewriting and most likely expansion. The authors use “Max Ent” as shortcut for something like “the principle that MaxEnt should be reached for some scattering angles”. I would avoid such shortcuts which may be misleading, and spell out the principle in full detail someplace in the text.
-Fig. 1. I would prefer a horizontal axis in multiples of pi.
-Fig 1. Caption. I would expand a bit to make it clearer to the reader that looks only at the figure what is plotted. (Captions should be selfcontained).
-Section VI. “In this work we have explored the relationship between GENERATION OF maximally entangled states
and high energy scattering amplitudes in QED and the weak interactions.”
-Figs 2,3,4 should appear earlier, and be referenced in the main text.
-Fig. 5. Consider moving this figure to the main text.
-Fig. 5. Axes. Better put scale in multiples of pi.
-Fig 5. Caption. Try to improve the caption. For instance sentence rewrite “If the e-(e+) has a right(left)-handed helicity,
MaxEnt occurs when theta= arccos(-1/3), whereas if the have left(right)-handed helicity MaxEnt occurs along a more complex curve”
1) Expand discussion
2) correct typos.