|As Contributors:||Alba Cervera-Lierta|
|Submitted by:||Cervera-Lierta, Alba|
|Submitted to:||SciPost Physics|
|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 are a first step towards understanding the connections between maximal entanglement and the fundamental symmetries of high-energy physics.
We'd like to thank the second referee for her/his comments and suggestions for improvements.
We shall address each of the observations in turn.
a) The referee says: "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."
The referee asked for a more elaborated discussion on the idea of maximal entanglement as a fundamental principal. We have added further arguments in the introduction that include a discussion of Wheeler's "it from bit" philosophy and we have expanded the discussion.
b) The referee says: "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."
True, but low entanglement can be simulated classically in an efficient way. Maximal entanglement is not. We have addressed this point in the text (in particular, in Section IV).
c) The referee says: "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)."
Indeed, entanglement can appear on any set of quantum numbers. Yet, in particle physics
states are prepared and measured on specific momenta. The discussion of entanglement is
still open to consider not only helicities, but also angular dependences, as we stress further in new text. A note is made that a full analysis should consider other quantum numbers, such as flavor and color.
d) The referee says: "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."
This is right. What is remarkable is that in the case of e- e- scattering, maximal entanglement is generated for any energy. This includes the standard physics of electrons at low energies. However, it is important to notice that this is not the case in other processes such as e+ e- scattering. We have added a comment in the text along these lines.
e) The referee says: "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)."
This is a possibility, though as mentioned by the referee it is quite a complicated process. In other cases where principles are put forward, extremization of some quantity is called for. In the same spirit, we analyzed the consequences of extremization of entanglement.
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 role of new parameters from the weak sector is, a priori, completely unpredictable.
Amplitudes such as Bhabha scattering add new diagrams with intermediate Z parameters. This spoils the QED balanced between different outgoing helicity states. It turns out that the parameters in the standard model are such that entanglement is still possible. Then, when all processes are included (that is when Z decays are in place) the solution to MaxEnt is again an isolated point. We have added comments to emphasize this point.
We have also amended some clumsy sentences in the text and corrected several typos. We hope that having addressed the points raised by the referee the revised version of our work can be considered suitable for publication in SciPost.
-We have modified the last sentence of the abstract, which now reads "Our results are a first step towards understanding the connections between maximal entanglement and the fundamental symmetries of high-energy physics."
-Intro. Par2: we have removed ‘more’ from the last sentence in the paragraph.
-Intro. Par7: we have changed "systems" into "subsystem".
-Section III. Last Par: we have changed a sentence to"In two cases, MaxEnt is generated independently of the scattering angle".
-Table 1: we added a column with the name of the processes and added in the caption that a dash indicates that MaxEnt cannot be reached for any value of the scattering angle theta.
-Section IV. First par: the paragraph is now reformulated, and we added a discussion in the conclusions.
-Section IV, around equation 9: we have clarified the notation, and amended the sentences.
-Section V. Par 1: we have modified the sentence as suggested by the referee
-Section V, Penultimate paragraph: we have changed the paragraph as suggested by the referee
-Fig. 3: the horizontal axis now is in multiples of pi
-Fig 3. we have expanded the caption as suggested by the referee.
-Section VI. we have added further discussion
- we have moved Figures 2 and 3 to the main text
- we have moved the Figure to the main text (current Fig. 3, right panel). We have also modified the discussion taking into account the reorganization of the text.
-finally, we have done minor amendments in some sentences and corrected a few typos.