How Does the Planck Scale Affect Qubits?

Round 1

Reviewer 1 Report

How Does the Planck Scale Affect Qubits?
by Matthew J. Lake

This work deals with generalised uncertainty relations (GURs) based on a new model of "nonlocal geometry" in which
smearing of classical points generates GURs for angular momentum at the Planck scale, implying an analogous generalisation of
the spin uncertainty relations. Previous models assume that deviations of the non-spin part of the wave function (from the standard
quantum statistics at the Planck scale) do not affect the spin part by the quantisation of the background space.
It is claimed in the paper that the extension of the model studied here successfully generates the generalised uncertainty principle (GUP), extended uncertainty principle (EUP) and the extended generalised uncertainty principle (EGUP) with the additional advantage of evading several problems [(i)-(iv) in the paper] and extrapolates its consequences for orbital angular momentum and spin, focussing on the structure of the modified spin-measurement operators, and their associated eigenstates, for one- and two-particle systems.

My overall impression of the work is that the theory underlying the model is somewhat confussing in some important aspects which I would like the author to clarify before I reach an opinion on recommending the work or not.  

Please, could you clarify and perhaps go deeper into the following?:

1. The Heisenberg microscope argument, which is employed to include the effects of the
gravitational attraction, motivates the generalised uncertainty principle (GUP), as well as incorporating
repulsive effects from the dark energy density, motivates the extended uncertainty principle (EUP), and both in turn to
motivates the extended generalised uncertainty principle (EGUP). In which way the new model you proposed (by the way, what is the status of you recent contributions in ArXiv?),
avoids problems (i)-(iv) and at the same time successfully generating the GUP, EUP and EGUP?. I understand you followed this model in order to
extrapolate for orbital angular momentum and spin, which is the aim of the paper, right?. And more important, you start from here to design a model of quantum nonlocality? to represent entangled states?

2. In which way the smeared space model is conceptually related to nonlocal aspects of quantum mechanics?. In my understanding a physical theory is called non-local when observers can produce instantaneous effects over distant systems. Non-local theories rely on two fundamental effects: local uncertainty relations and steering of physical states at a distance. Is in these aspects what you mean by quantum nonlocal geometry? This is perhaps my main concern about yor work... I really dont see a clear connection, other than spreading particles in physical space, fluctuations of the geometry producing scalable features with no new physics at all?. Or perhaps there is something deeper into the model which requires a much depper explanation on your side. It was Max Born who correctly (as far as we know) interpreted the ψ of the Schrödinger equation in terms of a probability amplitude, hence the square of the amplitude is not the charge density but is only the probability per unit volume of finding a particle there. The wave function ψ(r) of a particle does not, then, describe a smeared-out electron with a smooth charge density, i.e., the whole charge is there, from a given probability of finding the particle.

3. Moreover, it might be reasonable to thing that the simplicity of Eq. (27a) is not deceptive at all but a result of the subalgebra structure (26a)-(26e) which does not overlook that the presence of (h-bar + beta) rather than h-bar only represents a rescaling of your measurement units?, then what is the new physics?, please clarify this point

4. On the other hand, ever since the work of Bell, it has been known that entangled quantum states can produce non-local correlations between the outcomes of separate measurements. However, for a long time it has been assumed that the most non-local states would be the maximally entangled ones. Surprisingly it is not the case: non-maximally entangled states are generally more non-local than maximally entangled states for all the measures of non-locality proposed to date: Bell inequalities, the Kullback-Leibler distance, entanglement measures etc. Two questions arise: (i) in your model, do you assume that featuring smeared space, would produce nonlocality quantum aspects of particles that would be entangled by defect?. I doubt it very much but I would like to know of your thoughts on this matter. and (ii) How could you propose tio test your model in a way to measure the strength of nonlocal correlations to produce util qubits in the context of quantum information theory as you seemed to propose through spin?

5. Please explain in subsection 3.2. (Two-particle systems in smeared space), what is the physical background behind Eq (69), i.e., independent particles in a nonlocal smeared space which produces entangled particles?, I cant see the logic of it if you excuse me.

FInally, I may agree in that your analysis has shown an interesting algebraic and geometric structure of your model of quantum nonlocal geometry and have investigated it consequences for both external and internal symmetries. In position space representation, classical spatial points are ‘smeared’ over the Planck volume, whereas, in the momentum space representation, momentum space points are smeared over the volume associated with the de Sitter mass. You were able to show that the smearing of the canonical phase space may generate several uncertainty relations GURs, GUP, and else. However interesting it might be I dont see a clear physical understanding associated with quantum nonlocality phenomena nor entanglement. I recommend you clarify on these physical concepts in connnection with your theory or perhaps you may choose to present your manuscript within a more focused framework.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The author quickly reviewed the generalized uncertainty relations motivated from an effective gravitational correction to the quantum mechanics, and therefore can be expanded in the series of Planck scale. Using the `smeared space non-local geometry' formalism previously developed by the author himself, the author focus on the consequences on both the orbital and spin angular momentum and relates the results to the quantum information science where the spin degrees of freedom which play central roles. The author predicts measurable effects of single and two qubit system, and conclude that in most cases the small deviation from canonical quantization can hardly be experimentally measured; however, one exception might be possible by measuring the so-called noncanonical correlation which might be found in later works. 

The algebraic derivation seems mostly correct. However I have a conceptual question that I want the author to answer. To me, this framework can be simply understood as a renormalization of Planck constant. Could there be something more subtle happen if this formalism holds? I noticed the author has a section 3.3 about the Bell state in smeared space, but I cannot get the physical consequence of this state since the author claimed it to be `the smeared Bell states are considerably more complex than their canonical counterparts'. 

I also have some problems with the references in this manuscript. Out of the total 25 references, 3 are very basic concepts ([1-3]), 9 are relevant and good resources ([4-12]), and 7 are from this author ([13-16, 19, 20, 23]) include 3 in preparation manuscripts ([19, 20, 23]). One the one hand this indicates that the author has produced a series of original work in this particular direction, while on the other hand this means there are only very narrow interests in this topic. This also indicates that the content of this manuscript has a relatively weak significance. 

The author calculated the non-local geometry effect is at the order of \hbar \to \hbat + \beta is a 10^{-61} deviation and is definitely too small to observe. Currently the state-of-art atomic clock are in the 10^{-17} resolution in 1S, and in reasonable scale of month-long measurement this number can go to 10^{-20}, and entanglement can further reduce this number to lower (see the references A&B provided in the end of the report). I think if the author can propose some measurable effects especially in the presence of the spin GUR, and argue in what regime of parameters the effects could be measured, the significance of this paper can be improved by a lot. 

There are few typos that the author should take care of, such as

- in line 26, the `Heisenberg microscope argument has been extended'...
- the abbreviation EGUP has been introduced twice
- the author should change all `we' to `I' to avoid confusions
- ...

Also, the link of grant information cannot be opened. 

Overall, I recommend a major revision of this manuscript. I think the author should focus on improving the significance of this manuscript, considering other aspects are already acceptable in the current form. 

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References for the measurement: 

[A]: Oelker, E. et al. Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks. Nat. Photon. 13, 714–719 (2019).
[B]: Pedrozo-Peñafiel, E, et al. "Entanglement on an optical atomic-clock transition." Nature 588, 414-418 (2020). 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Though still not sure about whether this manuscript will attract most audiences,  I found the revised version along with the reply from the author address most of my concerns. So I recommend publication of the paper in its present form.