Nonlinear Diffusion and Entanglement

Quadratic stochastic motion can prevent exact entanglement death while driving an entire continuous-variable quantum channel exponentially close to a measure-and-prepare channel. Independent position diffusion restores a sharp finite-time entanglement-breaking threshold.

On one noncompact canonical degree of freedom, with $[Q,P]=i$ and $H=P^2/2$, study the normal random-unitary semigroup with formal generator

$$\mathcal L(\rho)=-i\omega[H,\rho] -D[Q,[Q,\rho]]-\kappa[H,[H,\rho]] -\eta[P,[P,\rho]],\quad D,\kappa>0,\ \eta\ge0.$$

The exact entanglement-breaking time is

$$\boxed{t_{\mathrm{EB}}=\begin{cases} (2\sqrt{D\eta})^{-1},&\eta>0,\\\ \infty,&\eta=0. \end{cases}}$$

For every positive time strictly below this boundary, a normalizable finite-energy system–qubit input yields a negative-partial-transpose (NPT) output; at and beyond the finite boundary, the channel is entanglement breaking. The threshold includes its endpoint and does not depend on positive $\kappa$ or finite drift $\omega$. Exact theorem and proof.

When $\eta=0$, let $e(t)$ be the infimum of the half-diamond distance to all entanglement-breaking channels, with no input-energy restriction. For fixed $D,\kappa>0$ and $\omega\in\mathbb R$,

$$-\log e(t)=\Theta(D^2\kappa t^3)\quad(t\to\infty).$$

The upper bound applies uniformly to every input and reference; the lower bound is certified by compact, exactly orthogonal finite-energy momentum packets. The exponent's leading constant is not optimized. Cubic-distance theorem.

Sister project: rank-two spin diffusion

Spin-Diffusion-Classicalization supplies a controlled growing-excitation contraction of rank-two spin noise to this nonlinear channel. At spin time $t=s n^{-2/3}$, the local limit has $D=b/2$, $\kappa=c$ and $\eta=\omega=0$. Transferring this repository's finite-energy NPT witnesses proves a positive limiting distance from EB for every fixed finite $s>0$. Consequently $n^2t^3\to\infty$ is necessary for vanishing EB distance; no global sufficiency at this scale is established.

The application bridge gives the parameter dictionary, precise local quantifiers and the remaining global gap. The spin sister owns the contraction and finite-spin comparison; this repository owns the nonlinear conditional covariance, exceptional-path bound and oscillator converse. Both use Holevo's 2019 second edition as their common educational anchor.

The singular Gaussian boundary

At $\kappa=0$, the channel instead has Gaussian additive noise (up to a Gaussian unitary), with entanglement-breaking threshold determined by

$$4D\eta t^2+\frac{D^2\omega^2t^4}{3}=1.$$

With $\eta=0$ and $\omega\ne0$, it becomes entanglement breaking after the finite time $3^{1/4}/\sqrt{D|\omega|}$. Every $\kappa>0$, however small, removes this exact finite cutoff. This is a singular statement about the exact entanglement-breaking property in an unbounded-noise model, not a claim that a diamond-small perturbation dramatically enhances usable quantum memory. Physical interpretation.

A single fixed, finitely squeezed two-mode squeezed vacuum also has NPT output at all finite times for $\eta=0$; this follows from normal-state filtering and yields no lower bound on useful surviving entanglement or quantum capacity. Fixed-input corollary.

Scientific reading route

Question	Read
Precise Hilbert-space model, allowed resources, conventions and propositions	Model and definitions
Why does conditioning on quadratic noise produce a mixture of Gaussian channels?	Conditional Gaussian representation
Where is the exact entanglement-breaking cutoff proved?	Sharp threshold and finite-energy witness
Why is the channel uniformly close to classical but never exactly classical?	Matching cubic logarithmic bounds
What transfers to a growing spin, and what requires a separate global proof?	Rank-two spin application
Does one fixed finite-energy entangled input survive?	Faithful-state corollary
Which conclusions matter physically and which do not follow?	Interpretation and limits
What is inherited and what distinguishes this result?	Primary-literature comparison
How are the formulas and finite witnesses checked?	Verification and reproducibility

The science index gives a connected proof route. The LLM-oriented retrieval guide points to each claim's authoritative home.

Learn the project

The single educational anchor is Alexander S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, second revised and expanded edition (2019). The learning entrypoint offers a route through the physical question before the full proofs.

The Holevo bridge identifies the relevant sections, translates covariance, channel and energy conventions, and supplies the steps from Gaussian theory to the nonlinear mixture. The worked primer begins with an exactly solved two-qubit example, then develops the stochastic shear, sharp threshold, operational meaning and finite-spin application. The 30-milestone map contains 14 background and 16 narrative targets, with explicit dependencies and scientific sources. It is a common understanding map, not a personalized assessment record.

Reproduce

python3 -m pip install -r verification/requirements.txt
python3 verification/run_checks.py
python3 verification/check_learning.py

The scientific tests check the Mehler kernel, exact packet integrals, rigorous bounds, Gaussian-channel cutoff and Brownian quadratic-functional identities. The learning checks validate the map, local navigation and worked examples. Finite tests inspect algebra and numerical conventions; the universal statements rely on the written proofs. No Monte Carlo data are needed for the principal result. The Sanity Check record states the checks actually completed and separates them from hosted visual rendering.

Attribution and scope

The position/kinetic-energy-dephasing model, Grushin geometry and exact propagator come from Speed, Lyu and Schubert (2023). Gaussian entanglement-breaking criteria and finite-time Gaussian Brownian measurement descriptions are inherited from Holevo (2008) and Riedel (2016). The present account derives the precise non-Gaussian entanglement-breaking boundary, its singular dependence on quadratic noise, normalized NPT witnesses and the operational cubic-distance law. See the related-work comparison; no exhaustive priority claim is made.

The dynamics act on $L^2(\mathbb R)$, not a bounded-energy device or an irreducible finite spin. The environment is unobserved; the reference is arbitrary; no intermediate correction or noiseless quantum bypass is allowed. The lack of finite-time entanglement breaking does not imply positive quantum capacity, finite-error memory, or experimentally useful survival. Statements that vary $\kappa$ cannot be interpreted as unconstrained-diamond-small perturbations because $H$ is unbounded.

Purpose and contact

This repository serves as a record of the work and a guide for the author’s self-directed learning. For discussion or potential collaboration, please contact Ruge Lin at gogoko699@gmail.com.