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arxiv: 2605.22293 · v1 · pith:MXVYC5DNnew · submitted 2026-05-21 · 🪐 quant-ph

Modular Variables and the Limits of Phase Detectability in Open Quantum Systems

S. V. Mousavi This is my paper

Pith reviewed 2026-05-22 06:24 UTC · model grok-4.3

classification 🪐 quant-ph
keywords modular variablesquantum nonlocalitygravitational fieldopen quantum systemsBohmian mechanicsphase sensitivitywave packet superposition
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The pith

Gravitational acceleration produces a time-varying modular signal sensitive to relative phase in separated wave packets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates how modular variables can access relative phase information in spatially separated quantum wave packets that conventional local measurements cannot. Under a uniform gravitational field, the time evolution of Hermitian modular operators generates a signal that stays sensitive to this phase for both unitary and open-system dynamics. Using the Bohmian interpretation, local expectations along trajectories show that gravity induces phase-dependent variations in the modular observable, while probability density and current become phase-insensitive without overlap. Environment correlations in multi-particle cases affect local signals but limit phase transfer to distant particles, and standard coherence measures miss this information.

Core claim

The central discovery is that gravitational acceleration induces a time-varying modular signal, the expectation value of the modular observable, that remains sensitive to the relative phase between the separated wave packets. In contrast, standard local quantities such as the probability density and probability current become insensitive to the relative phase in the regime of negligible spatial overlap. This is shown for Gaussian wave-packet superpositions evolving under the Schrödinger equation and the Caldeira-Leggett master equation, with local values computed via Bohmian trajectories. For particles coupled to a shared environment, environment-induced correlations modify the local modular

What carries the argument

The modular observable, a Hermitian operator built from modular variables that captures nonlocal relative phase information between well-separated wave-packet components.

If this is right

  • Gravity induces a detectable time-varying variation in modular expectations that tracks relative phase.
  • Probability density and current lose phase sensitivity when wave packets have negligible spatial overlap.
  • Shared environments create correlations modifying one particle's modular signal without significant phase transfer to distant particles.
  • Conventional coherence and entanglement measures fail to capture relative phase in the non-overlapping regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could enable probing of gravitational effects on quantum superpositions using nonlocal observables in separated systems.
  • It suggests that phase information may persist in modular signals under decoherence longer than in standard observables.
  • Similar analyses in other external fields could test the generality of phase preservation in modular variables.

Load-bearing premise

That local expectation values of modular operators can be meaningfully computed along individual Bohmian trajectories and that these values continue to encode the relative phase information when the wave packets have negligible spatial overlap.

What would settle it

A direct computation or measurement demonstrating that the modular expectation value loses its phase sensitivity under gravity when spatial overlap is negligible, contrary to the predicted time-varying signal.

Figures

Figures reproduced from arXiv: 2605.22293 by S. V. Mousavi.

Figure 1
Figure 1. Figure 1: FIG. 1: Density plots of the probability density for the Schr¨odinger dynamics (left) and the CL dynamics (right) at [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Local expectation value of the modular variable along Bohmian trajectories for a relative phase [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Local expectation value of modular variable along the Bohmian trajectory with initial position [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Expectation value of the modular variable of the first particle [Eq. (59)] for relative phases [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

Modular variables serve as a striking example of quantum nonlocality, particularly in superpositions of wave packets that are spatially well separated, where the relative phase between components cannot be accessed through conventional local measurements. In this work, we explore the time evolution of Hermitian modular operators for Gaussian wave-packet superpositions under the influence of a uniform gravitational field. We consider both unitary dynamics governed by the Schr\"odinger equation and open-system dynamics described by the Caldeira-Leggett master equation in the high-temperature limit. Adopting the Bohmian interpretation of quantum mechanics, we compute local expectation values of these modular operators along individual particle trajectories. Our analysis shows that gravitational acceleration induces a time-varying modular signal, the expectation value of the modular observable, that remains sensitive to the relative phase between the separated wave packets. In contrast, standard local quantities such as the probability density and probability current, while modified by gravity, become insensitive to the relative phase in the regime of negligible spatial overlap. For a pair of particles coupled to a shared environment, we find that environment-induced correlations can modify the local modular expectation value observed for one particle, yielding a clear signature of environmental influence. However, the transfer of phase sensitivity via environment-generated entanglement to the modular signal of the distant particle remains negligible within the regime considered. We further demonstrate that conventional measures of coherence and entanglement do not capture the relative phase information in this non-overlapping regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that modular variables can detect the relative phase between spatially separated Gaussian wave packets in a gravitational field even when their spatial overlap is negligible. Using the Bohmian interpretation, local expectation values of Hermitian modular operators are computed along particle trajectories for both closed (Schrödinger) and open (Caldeira-Leggett) dynamics. These modular signals show time-varying behavior induced by gravity that retains phase sensitivity, in contrast to the probability density and current. For two particles in a shared environment, environment-induced correlations affect the modular signal but phase sensitivity transfer is negligible.

Significance. If the central claim holds, this work illustrates how modular observables combined with Bohmian trajectories can access phase information that is inaccessible to standard local measurements in the non-overlapping regime. This has potential significance for testing quantum effects in gravity and understanding decoherence in open systems. The explicit use of Gaussian packets and master equation provides a concrete, calculable example.

major comments (2)
  1. [Bohmian trajectories section (around the sentence beginning 'Adopting the Bohmian interpretation...')] The key assumption that local expectation values of modular operators along Bohmian trajectories encode the relative phase when spatial overlap is negligible requires clarification. Specifically, since the Bohmian velocity field depends on the global ψ, it is not clear if the modular expectation is computed in a way that uses only local information or retains access to the full wave function. This is load-bearing for the claim that it provides a genuinely local yet phase-sensitive signal.
  2. [Two-particle open system analysis] The statement that 'the transfer of phase sensitivity via environment-generated entanglement to the modular signal of the distant particle remains negligible' should be supported by explicit parameter values or plots showing the dependence on coupling strength or temperature to make the conclusion robust.
minor comments (2)
  1. Check for consistency in notation for the modular operators throughout the text.
  2. The abstract could benefit from a brief mention of the specific modular operators considered, e.g., periodic functions of position or momentum.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below, providing clarification where needed and indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [Bohmian trajectories section (around the sentence beginning 'Adopting the Bohmian interpretation...')] The key assumption that local expectation values of modular operators along Bohmian trajectories encode the relative phase when spatial overlap is negligible requires clarification. Specifically, since the Bohmian velocity field depends on the global ψ, it is not clear if the modular expectation is computed in a way that uses only local information or retains access to the full wave function. This is load-bearing for the claim that it provides a genuinely local yet phase-sensitive signal.

    Authors: We thank the referee for highlighting this important point regarding the locality of the computation. Although the Bohmian velocity field is determined by the global wave function, the local expectation value of the modular operator along each trajectory is evaluated by integrating the operator against the local probability density and phase gradient in the immediate vicinity of the particle position. This uses only information accessible locally at that point on the trajectory, while the phase sensitivity to the distant packet arises from the structure of the superposition encoded in the wave function. We will revise the Bohmian trajectories section to include an explicit derivation of this local evaluation and a discussion of how it differs from standard local observables, thereby clarifying that the signal is genuinely local in its measurement while retaining phase information. revision: yes

  2. Referee: [Two-particle open system analysis] The statement that 'the transfer of phase sensitivity via environment-generated entanglement to the modular signal of the distant particle remains negligible' should be supported by explicit parameter values or plots showing the dependence on coupling strength or temperature to make the conclusion robust.

    Authors: We agree that providing explicit support will make this conclusion more robust. In the revised manuscript, we will include additional plots of the modular signal for the distant particle as a function of the system-environment coupling strength and temperature. We will also report the specific numerical parameter values employed in our Caldeira-Leggett simulations to demonstrate that the phase sensitivity transfer remains negligible across the relevant range of these parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of modular phase signals

full rationale

The paper solves the time-dependent Schrödinger equation and the high-temperature Caldeira-Leggett master equation for Gaussian wave-packet superpositions in a uniform gravitational field, then evaluates Hermitian modular operators along Bohmian trajectories generated by the standard guidance equation v = (ħ/m) Im(∇ψ/ψ). The reported time-varying modular expectation value emerges directly from this computation and retains sensitivity to the relative phase precisely because the pilot-wave field encodes global phase information even at negligible overlap; this is a standard consequence of the chosen interpretation rather than a definitional equivalence or fitted input. No self-citation chains, uniqueness theorems, or ansatzes are invoked to force the central result, and the contrast with probability density and current follows immediately from the same equations without additional assumptions that presuppose the outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on two established domain assumptions rather than new free parameters or invented entities: the validity of the Bohmian interpretation for local modular expectations and the applicability of the high-temperature Caldeira-Leggett master equation.

axioms (2)
  • domain assumption Bohmian interpretation of quantum mechanics permits computation of local expectation values of modular operators along individual particle trajectories that encode relative phase even without spatial overlap
    Explicitly adopted to obtain local modular signals in the non-overlapping regime.
  • domain assumption High-temperature limit of the Caldeira-Leggett master equation adequately models environment-induced decoherence for the open-system dynamics considered
    Used to describe the shared-environment case for two particles.

pith-pipeline@v0.9.0 · 5782 in / 1640 out tokens · 74432 ms · 2026-05-22T06:24:15.937633+00:00 · methodology

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Reference graph

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