White Paper 05
Date: 2026-01-09
Status: Complete draft (self-contained)
A Quantum Positioning System (QPS) or any entanglement-assisted time-transfer scheme relies on maintaining phase coherence between distributed quantum systems. In curved or time-varying spacetime, different worldlines accumulate different proper times, creating an effective timing mismatch $\Delta\tau(t)$. A minimal engineering question follows:
Given a spacetime disturbance model (gravitational wave strain $h(t)$ and/or static gravitational redshift), what is the induced $\Delta\tau(t)$, how does it degrade entanglement visibility $V(t)$, and what refresh cadence is required to keep $V(t)$ above a floor?
This paper gives a concrete, computable pipeline:
1) map a GR perturbation (strain or metric potential) to a clock-offset signal $\Delta\tau(t)$, 2) map $\Delta\tau(t)$ to visibility $V(t)$ via a coherence-time model, 3) extract a refresh time (first threshold crossing) and an equivalent refresh rate.
We also highlight a key scaling behavior: for oscillatory high-frequency strain, $\Delta\tau \propto \int h\,dt$ can be strongly suppressed by cancellation, making fast gravitational waves comparatively harmless for timing-based decoherence metrics unless there is a DC component or “memory” effect.
In distributed quantum protocols (entanglement distribution, teleportation-based links, QPS), a common failure mode is loss of phase reference between nodes. In many physical implementations, the relevant mismatch can be modeled as a time/phase offset between two arms or stations.
This paper focuses on an intentionally minimal question: how much timing mismatch does a given gravitational disturbance produce, and what does that imply for refresh.
The goal is a clean mapping layer you can later embed into richer models.
We consider two stations (or arms) $A$ and $B$.
We use:
A simple engineering approximation is to treat the strain as producing a fractional time-rate perturbation:
\[\boxed{\ \frac{d\tau}{dt} \approx 1 + \frac{1}{2}h(t).\ }\]Then the induced timing offset relative to the undisturbed clock is
\[\boxed{\ \Delta\tau(t) \approx \int_0^t \frac{1}{2} h(t')\,dt'.\ }\]Notes:
If $h(t)$ is oscillatory with zero mean, then $\int h(t)dt$ can remain small.
Example: $h(t)=h_0\sin(\omega t)$ gives
\[\Delta\tau(t)=\frac{1}{2}\int_0^t h_0\sin(\omega t')dt' = \frac{h_0}{2\omega}\,(1-\cos(\omega t)).\]So the scale of the mismatch is
\[\boxed{\ \Delta\tau_{\max} \sim \frac{h_0}{\omega}.\ }\]Higher frequency ($\omega$ larger) yields smaller accumulated mismatch.
The integral suppression disappears when $h(t)$ has:
For slowly varying, weak gravitational potentials, one often uses
\[\boxed{\ \frac{d\tau}{dt} \approx 1 + \frac{\Phi}{c^2}\ }\]where $\Phi$ is Newtonian potential (units m\u00b2/s\u00b2) and $c$ is the speed of light.
Near Earth, for a small height difference $\Delta z$,
\[\Delta\Phi \approx g\,\Delta z,\]so the fractional rate difference is
\[\boxed{\ \frac{d}{dt}\Delta\tau \approx \frac{g\,\Delta z}{c^2}.\ }\]Integrating for duration $T$ gives
\[\boxed{\ \Delta\tau(T) \approx \frac{g\,\Delta z}{c^2}\,T.\ }\]This grows linearly with $T$, but the coefficient is tiny in terrestrial settings.
A simple and common model is that visibility decays as a Gaussian in timing mismatch:
\[\boxed{\ V(t) = \exp\!\left(-\Big(\frac{\Delta\tau(t)}{\tau_{\mathrm{coh}}}\Big)^2\right).\ }\]Interpretation:
Pick a minimum allowed visibility $V_{\mathrm{floor}}$.
Define the refresh time as the first threshold crossing:
\[\boxed{\ T_{\mathrm{refresh}} := \inf\{t\ge 0 : V(t) \le V_{\mathrm{floor}}\}.\ }\]If $T_{\mathrm{refresh}}$ exists, a simple equivalent “refresh rate” is
\[\boxed{\ f_{\mathrm{refresh}} \approx \frac{1}{T_{\mathrm{refresh}}}.\ }\]If $\Delta\tau(t)$ grows linearly: $\Delta\tau(t)\approx \rho\,t$ with drift rate $\rho$ (dimensionless), then
\[V(t)=\exp\!\left(-\Big(\frac{\rho t}{\tau_{\mathrm{coh}}}\Big)^2\right).\]Solving $V(t)=V_{\mathrm{floor}}$ gives
\[\boxed{\ T_{\mathrm{refresh}} = \frac{\tau_{\mathrm{coh}}}{\rho}\,\sqrt{\ln\!\left(\frac{1}{V_{\mathrm{floor}}}\right)}.\ }\]For static redshift, $\rho \approx g\Delta z/c^2$.
Assume we have sampled strain $h[n]$ at times $t[n]$.
Using trapezoidal integration:
\[\Delta\tau[n] \approx \sum_{k=1}^{n} \frac{1}{2}\,\frac{h[k]+h[k-1]}{2}\,(t[k]-t[k-1]).\]Find the smallest index $n$ such that $V[n]\le V_{\mathrm{floor}}$, then set
\[T_{\mathrm{refresh}}\approx t[n].\]If no such $n$ exists over the observation window, then no refresh is triggered by this disturbance under the model.
These are not fundamental constants; they are sanity-check outcomes of the mapping assumptions.
Because $\Delta\tau \propto \int h\,dt$, a bursty sinusoidal strain can yield very small integrated area even for large peak amplitude. In prior sweeps with a 200 Hz sine-burst and a visibility floor near $10^{-4}$ with $\tau_{\mathrm{coh}}\sim 10^{-3}\,\mathrm{s}$, visibility remained close to 1 for strain scales far above astrophysical values. The suppression is simply the $1/\omega$ scaling of the integral and envelope cancellation.
Practical takeaway:
Using $\Delta\tau(T)\approx (g\Delta z/c^2)T$ with $\Delta z=100\,\mathrm{km}$ and $T=1\,\mathrm{year}$ gives a differential offset on the order of $10^{-8}\,\mathrm{s}$, which is far below a millisecond-scale coherence window.
Practical takeaway:
To probe regimes where the integral does not cancel, include:
In real deployments, $\Delta\tau$ depends on geometry:
A next-step model can treat $h(t)$ as producing a differential phase between $A$ and $B$ directly.
If refresh is a resource, you can treat it like an adaptive control variable. For example:
This mirrors the “reinforce on need, decay otherwise” pattern used elsewhere in the archive.
1) Frequency suppression: for comparable peak strain amplitudes, higher-frequency components contribute less to $\Delta\tau$ (roughly $\propto 1/\omega$).
2) DC dominates: a small DC drift in $h(t)$ or a memory-like step dominates $\Delta\tau$ over a long window.
3) Static redshift linearity: $\Delta\tau$ from altitude differences grows linearly with duration $T$ and scales with $\Delta z$.
1) If measured decoherence correlates strongly with instantaneous $h(t)$ rather than with a time-integrated or low-frequency component, this mapping is missing the relevant coupling.
2) If high-frequency oscillatory strain (with near-zero integral) produces large timing mismatches in controlled experiments, the model’s integral suppression is incorrect for that geometry.
3) If visibility degradation vs $\Delta\tau$ is not well-fit by a Gaussian overlap model, the $V=\exp(-(\Delta\tau/\tau_{\mathrm{coh}})^2)$ ansatz is not appropriate; another lineshape (Lorentzian/exponential or a more detailed wavepacket model) is required.
This matches the computational structure used in the strain-mapper prototype:
import numpy as np
def strain_to_delta_tau(strain, t):
# delta_tau(t) = ∫ 0.5*h(t) dt
frac = 0.5 * strain
dt = np.diff(t)
# trapezoid cumulative integral
out = np.zeros_like(t)
out[1:] = np.cumsum(0.5 * (frac[1:] + frac[:-1]) * dt)
return out
def visibility(delta_tau, tau_coh):
return np.exp(- (delta_tau / tau_coh) ** 2)
def refresh_time(delta_tau, t, tau_coh, V_floor=1e-4):
V = visibility(delta_tau, tau_coh)
idx = np.where(V <= V_floor)[0]
return None if idx.size == 0 else float(t[idx[0]])
If you add static drift, you can add it directly into $\Delta\tau(t)$ as a linear term.