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Maximal Entanglement in High Energy Physics

by Alba Cervera-Lierta, José I. Latorre, Juan Rojo, Luca Rottoli

Submission summary

As Contributors: Alba Cervera-Lierta
Arxiv Link:
Date submitted: 2017-06-07
Submitted by: Cervera-Lierta, Alba
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: High-Energy Physics - Theory


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.

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Anonymous Report 1 on 2017-8-16


1. A striking observation that could open up new avenues of investigation into the organising principles of nature.
2. It's technically very simple and readable, using not much more knowledge than can be gained from one course in QFT, but still surprising.


1. "MaxEnt" is not a very good choice of name, since it traditionally means something quite different.
2. It's not clear to what extent the observation is independent of angular momentum conservation.
3. Fails to get to the mathematical core of the effect (this can be easily recitifed).
4. The phenomenological result is far off even though the math is exact, which is probably a weakness of the method.


This paper considers spin entanglement in the product of 2->2 scattering in QED and Z-boson-mediated processes.
It makes the striking observation in the case of QED that it is one of only 8 theories of interactions between Dirac fermions and photons that are allowed by the requirement that the spin entanglement in the products is maximal for some scattering angles; further, the other 7 theories don't seem to be allowed by Lorentz-invariance.
It then demands a similar saturation from interactions mediated by a massive vector boson that can have both vector and axial couplings and shows that in the limit where the coupling is purely vectorial we recover QED and in the other limit we obtain a Weinberg mixing angle that is close to the experimental value.

This is a striking observation, and could lead to new avenues of exploration, provided my criticisms below are satisfactorily answered.

Before getting into the meat of the matter, a relatively minor quibble: MaxEnt is the principle that one must choose a prior to be the maximum entropy probability distribution consistent with one's knowledge; it is also used in other contexts than Bayesian probabilty, but the sense is always that one is maximising the entropy consistent with one's knowledge (an example is choosing density matrices according to this principle).
The authors call the principle that interactions should create spin-space Bell pairs at some scattering angle MaxEnt, which is a confusing choice of name since there's no sense in which one is "spreading out maximally in uncertainty-space" here.

Now for the main questions I feel it is important for the authors to answer.

Firstly, the authors focus exclusively on entanglement in spin-space, creating a rather cumbersome scattering-angle dependence.
This seems a bit artificial, because even in a similar scattering experiment of four scalars (where there's no spin-space) entanglement between the particles is generated, just by the fact that the scattering amplitude is non-zero for multiple outgoing momentum configurations.
So, the first question is: how are these results modified if one takes into account the full entanglement?
Regardless of the answer to this question, I feel the authors should make this point clear.

Second, it seems to me that the effect found in the paper, mathematically, is a rather trivial consequence of the structure of the $\gamma$ matrices.
At the core, the point is that every Bell pair has the form $\sum_{\alpha=0}^3 a_\alpha \sigma^{\alpha}_{ij} |i,j\rangle,\ i,j \in {1,2}$, where the $\sigma^\alpha$s are Pauli matrices and $\sigma^0$ is the identity. Since the $\gamma$ matrices are made out of the Pauli matrices, maximal spin-entanglement is mostly a matter of momentum degrees of freedom not getting in the way.
Consider, for example, a four-Fermi sort of interaction, where the interaction vertex is $j_1^\mu j_{2,\mu}$. Suppose $j_1$ corresponds to the incoming particles, and the incoming momenta are such that the only non-zero component of $j_1$ is $\mu=3$ (never mind that this isn't actually allowed), the outgoing state turns out to be proportional to $|RL\rangle + |LR\rangle$ in the centre-of-momentum frame always (assuming I got my calculations right; my mathematica code is at
The reason this works is that I engineered this example so that the momenta wouldn't interfere with the spins (the mechanism for this non-interference was the contribution of only one component of the outgoing current operator).
While this isn't a criticism of the paper, it would read much better and be even simpler than it already is if written from this point-of-view.

The most important point is that it's not clear to what extent the concrete results of this paper, the foundation on which the speculation is built, aren't just conservation of angular momentum.
This is related to the fact that the authors completely ignore momentum-space degrees of freedom, because for a particular configuration of ingoing and outgoing momenta only some spin configurations are allowed by angular momentum conservation.
To illustrate this, consider the $e^- e^+ \to \mu^- \mu^+$ scattering that is the first example in the paper. Let's stick to the frame where the incoming particles are moving along the z-axis, in opposite directions. The transformation of the incoming state under $L_z$, then, is completely determined by the spins. Consider, further, a singlet incoming state, $|RR\rangle - |LL\rangle$.
Angular momentum conservation, then, dictates that for scattering angle $\theta=0$ the outgoing state is also a singlet $|RR\rangle - |LL\rangle$. This is confirmed by the formulas provided by the authors themselves in appendix A.
Another example of this is that when the incoming state is $L_z=\pm 1$, which are the $|RL\rangle$ and $|LR\rangle$ states, and the scattering angle is $\pi/2$ so that the orbital part of the outgoing particles' angular momentum is a pure vector, the outgoing spin state becomes a singlet in both cases.
I have not been able to find any counter-examples to these naive considerations in these simple limits of forward and right-angle scattering. However, it is also true that I have not spent the time required to be sure that there are in fact no counterexamples.
Therefore, I would like to request the authors to seriously consider the question of how much of their results are a consequence of angular momentum conservation and how much are something genuinely new.

Finally, and this may show my own inadequacy, I'm a little worried that the result they claim for the Weinberg mixing angle is $10\%$ off from the measured value, when nothing in their formalism is approximate.
Unless you can hope to make the calculations more precise with more work, how can we tolerate deviation of predictions from correct answers? Or is it that there is some way in which the calculations in the final section are approximate despite looking exact?

Requested changes

1. Change name of constraining principle (minor, not necessary).
2. Clearly address role of momentum-space entanglement.
3. Consider simplifying presentation by exploiting structure of $\gamma$ matrices in the interactions considered.
4. Clearly explain difference from and relation to angular momentum conservation (absolutely vital, cannot be published without adequate discussion of this point).

  • validity: ok
  • significance: good
  • originality: good
  • clarity: top
  • formatting: good
  • grammar: reasonable

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