Coherence & Entanglement Fluctuation Distances

• Coherence and entanglement fluctuation distances are metrics that quantify the persistence of quantum superposition and nonlocal correlations under decohering processes.
• They combine geometric measures in state space with operational analyses, such as thermodynamic fluctuation metrics, to assess quantum resource robustness.
• These distances are applied across systems—from photonic propagation to emergent spacetime models—informing advances in quantum communication, metrology, and error correction.

Coherence and entanglement fluctuation distances quantify the degree to which quantum superposition and nonlocal correlations persist, degrade, or fluctuate in quantum systems as they evolve through physical processes or propagate through media. These concepts provide both geometric and operational measures of quantum resource robustness: geometric in terms of distances in state space or Hilbert space, and operational in terms of observable impact on thermodynamics and quantum information tasks.

1. Core Definitions and Foundational Metrics

Coherence distance refers to how far quantum superpositions survive under a given set of dissipative, scattering, or measurement processes. Entanglement fluctuation distance generalizes this to nonlocal quantum correlations, probing their spatial or protocol-dependent resilience. Key metrics include:

• Photon mean free path ($\ell_{mfp}$): For single photons in a medium, $\ell_{mfp} = 1/\Gamma(\nu)$ with $\Gamma(\nu) = \alpha(\nu) + n \sigma_s(\nu)$, where $\alpha$ is the absorption coefficient, $n$ the scatterer density, and $\sigma_s$ the scattering cross-section. This establishes the 1/e-length for both coherence and entanglement decay in photonic propagation (Berera, 2020).
• Resource-resolved fluctuation distances: For general quantum channels, the coherence fluctuation distance (CFD) and entanglement fluctuation distance (EFD) are defined as Kullback-Leibler divergences between operational probability distributions (e.g., energy-change histograms arising from end-point measurement schemes, EPM). CFD is $$D_C(\rho) := \min_{\sigma \in \mathbb{I}} D_{KL}(P^{\rho} \| P^{\sigma})$$ over incoherent reference states; EFD is $$D_E(\rho_{AB}) := \min_{\sigma \in \mathbb{S}} D_{KL}(P^{\rho_{AB}} \| P^{\sigma})$$ over separable references (Mondkar et al., 17 Dec 2025).
• Geometric distances in state space: Trace-norm, Hilbert-Schmidt, and Bures distances to the sets of maximally entangled or maximally coherent states have well-characterized averages and concentration properties for Haar-random states (Bu et al., 2016).

2. Physical Models and Propagation in the Interstellar Medium

Coherence and entanglement distances take on macroscopic significance in photonic propagation through astrophysical media. Using mean free path analysis (Berera, 2020):

• Radio/microwave band ($\nu \lesssim 10^{11}$ Hz, $\hbar \omega \lesssim 10^{-3}$ eV): $\Gamma \sim 10^{-32}$ m$^{-1}$, $\ell_{mfp} \gtrsim 10^{32}$ m, vastly exceeding galaxy scales.
• Soft x-ray ($\hbar \omega \sim 100$ eV): $\ell_{mfp} \sim 10$ pc.
• Hard x-ray ($\hbar \omega \sim 10$ keV): $\ell_{mfp} \gtrsim 10^5$ pc.

For both single-photon coherence and two-photon Bell pair entanglement, the statistical decay is exponential in path length $L$:

$$P_{surv}(L) = e^{-L/\ell_{mfp}}, \quad F(L) \simeq C(L)/C(0) \simeq e^{-L/\ell_{mfp}}$$

Thus the canonical "fluctuation distance" for coherence or entanglement loss is $L_{1/e} = \ell_{mfp}$ (Berera, 2020). These results indicate that, in select bands, coherence persists on scales comparable to or exceeding the diameter of the Milky Way.

3. Distance Dependence, Coupling, and Non-Markovian Effects

In continuous variable and quantum field-theoretical settings, the fate of entanglement as a function of physical separation is controlled by direct and indirect coupling, along with non-Markovian memory (Hsiang et al., 2015). For two harmonic oscillators, entanglement (as quantified by the smallest symplectic eigenvalue $\eta_<$ of the partially transposed covariance matrix) exhibits regimes:

• Strong direct coupling ($\sigma r \gg 2\gamma$): Entanglement persists for $r > r_1 \approx 2\gamma/\sigma$.
• Weak coupling (non-Markovian regime, $\sigma r \ll 2\gamma$): Entanglement is present only for $r < r_2 \sim \mathcal{O}(1/\omega_0)$.
• Critical separations $r_1$, $r_2$: Mark the boundaries for entanglement survival or revival. Non-Markovian field-mediated oscillations can enhance or suppress entanglement at certain distances.

Coherence (and thus entanglement) decays with rates $$\Gamma_\pm = \gamma[1 \pm \sin(\omega_\pm r)/(\omega_\pm r)]$$, introducing spatial modulation via retardation and memory effects. This framework establishes that for macroscopic but finite distances, quantum correlations may persist or be regenerated, depending critically on the coupling structure and spectral attributes of the bath (Hsiang et al., 2015).

4. Thermodynamic Fluctuation Distances under Quantum Resources

The operational sensitivity of thermodynamic fluctuation theorems to quantum coherence and entanglement is precisely characterized by fluctuation distances (Mondkar et al., 17 Dec 2025):

• CFD $D_C(\rho)$ quantifies the minimal KL-divergence between the observed process outcome distribution and any diagonal (incoherent) reference, capturing the "thermodynamic shift" attributable to quantum coherence.
• EFD $D_E(\rho_{AB})$ is the minimal KL-divergence versus separable states. Large EFD indicates strong thermodynamic signatures of entanglement in energy and entropy change statistics.
• Key properties: Nonnegativity, additivity for independent processes, operational vanishing iff the resource has zero impact on all EPM process trajectories. Upper bounds relate these distances to resource measures such as the relative-entropy of coherence and relative-entropy of entanglement.

Illustrative calculations for qubits under unitaries, and bounds such as $$D_C(\rho) \leq D_{KL}(P^\rho \| P^{\Delta[\rho]}) \leq 2 C_r(\rho)$$ (with $C_r$ the relative-entropy coherence), give precise analytical control (Mondkar et al., 17 Dec 2025).

5. Concentration of Measure and Typicality in High Dimension

The distances between random quantum states and maximally coherent (or maximally entangled) manifolds are highly predictable in high dimension, exemplifying "concentration of measure" (Bu et al., 2016):

Distance Metric	Average for large $d$	Typical Fluctuation (variance)
Trace-norm (entang.) $D_1$	$\sim 1.058$	$\sim O(1/d)$
Trace-norm (coherence) $\Delta_1$	$\sim 0.962$	$\sim O(1/d)$
Bures $D_B$, $\Delta_B$	$\rightarrow 2,\,0.2275$	$\sim O(1/d)$

Almost every bipartite pure state is at a definite, finite distance from the maximally entangled manifold, with negligible variance for large $d$. Analogous concentration results hold for coherence. The mean $l_1$-norm of coherence scales as $\sim (d-1)\pi/4$, and its scaled version $C_{sc}=C_{l_1}/(d-1)$ concentrates at $\pi/4$ for all $d$. These findings demonstrate that while random pure states are "almost maximally entangled/coherent" in terms of marginal spectra, the actual geometric distances remain nonzero and tightly peaked (Bu et al., 2016).

6. Emergent Geometry, Mutual Information, and Fluctuations

In approaches to emergent spacetime, entanglement and mutual information establish operationally meaningful distances between subsystems. The emergent distance $d(A,B)$ is defined via a monotonic function $\Phi$ of the normalized mutual information, typically:

$$d(A,B) = -l_{RC} \ln \left(\frac{I(A:B)}{I_0}\right)$$

with $I_0$ the maximum mutual information across pairs, and $l_{RC}$ setting the length scale (Franzmann et al., 2022). Sector-specific decoherence (e.g., loss of momentum vs. spin entanglement) selectively increases the mutual-information-derived distance, providing a physical mechanism for fluctuating "entanglement distances" in emergent spacetime scenarios. Decoherence in one sector can open up the emergent distance, even while coherence in another sector is preserved.

Experimental proposals include modulating subsystem correlations and measuring corresponding induced distance fluctuations via advanced interferometric techniques, providing direct laboratory tests of emergent geometry hypotheses (Franzmann et al., 2022).

7. Open Problems and Physical Implications

Critical questions include:

• Robust modeling of multipartite entanglement decay, including collective and correlated scatterings (Berera, 2020).
• The effect of inhomogeneous or turbulent environments (magnetohydrodynamic turbulence, variable scatterer densities) on coherence and entanglement distances.
• Protocol design for in situ error correction or post-selection to recover quantum resources after unavoidable decoherence events (Berera, 2020, Mondkar et al., 17 Dec 2025).
• Laboratory realization of operationally accessible distance fluctuations in emergent geometry experiments (Franzmann et al., 2022).

The ability to sustain or control coherence and entanglement distances is central to quantum communication, metrology, and fundamental tests of the quantum-classical transition. The rigorous formalism provided by geometric, information-theoretic, and thermodynamic fluctuation distances underpins both theoretical advances and experimental realizations across quantum information science and foundational physics.