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arXiv:2511.15919 · Physical Review Research

Reversing Quantum Noise: Quantum Reverse Diffusion for Pauli Channels

Einar Gabbassov

quant-phmath-phSubmitted Nov 19, 2025• Announced Nov 21, 2025• 1 citation
v1 · Nov 19, 2025v2 · Nov 26, 2025v3 · Dec 10, 2025v4 · Dec 19, 2025v5 · Jun 3, 2026
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Abstract

The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are quantum reverse diffusion stochastic differential equations, along with corresponding stochastic master equations. These equations describe the exact and approximate reverse diffusion dynamics for continuously monitored Pauli channels, including time-dependent depolarizing noise. Our results bridge the gap between highly nonlinear classical reverse diffusion, prominent in generative modelling, and linear quantum mechanics. Consequently, this establishes a theoretical framework for diffusion-driven quantum gates, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.

Journal reference
Phys. Rev. Research 8, 023329 (2026)
DOI
10.1103/f7fd-2rb9

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Quantum state diffusion unravels a Lindblad master equation into stochastic trajectories of pure states. The ensemble of trajectories is described by a probability density on the manifold of pure states, which contains much more information than the density matrix, its first moment. We derive the time reversal of this diffusion. Writing the reversal of a general diffusion on a manifold in Stratonovich form, and passing through the manifold of pure states, we obtain a backward stochastic differential equation for the state vector, valid for any Hilbert space dimension and any Lindbladian. The backward stochastic equation provides a physical, nonlinear unravelling of the inverse Lindbladian that holds only for the ensemble from which it was constructed. Following score-based generative models, we then use the backward equation to generate a new ensemble close to the original. For the depolarizing channel the unravelling is a Brownian motion on complex projective space, so the uniform distribution of pure states is its stationary law in any dimension. Starting the backward equation from that prior, we bound the error of the resulting distribution compared to the original one in different settings: exact and learned scores, continuous and discrete time.
Quantum diffusion models typically parameterize forward corruption by raw channel strength, even though this parameter does not directly quantify how much ensemble information is erased or how difficult the corresponding reverse problem is. We introduce the classical--quantum information decrement Δt=I(X:Qt−1)−I(X:Qt)Δ_t=I(X{:}Q_{t-1})-I(X{:}Q_t)Δt​=I(X:Qt−1​)−I(X:Qt​) as an intrinsic diffusion coordinate that links forward noise allocation to reverse recoverability. Along depolarization, equalizing ΔtΔ_tΔt​ yields the unique minimax discretization of the forward information loss, while universal recoverability gives the same quantity an operational interpretation as a physically attainable local recovery budget. We further show that such local calibration is not sufficient for stochastic generation: models can satisfy the same recovery criterion while producing substantially different state distributions. This motivates a stochastic learner that combines information-calibrated recovery constraints with distribution matching. We establish finite-sample calibration and compositional trace-Wasserstein control for the resulting learner. Controlled quantum experiments validate the predicted information--recovery alignment, show that the recovery constraints improve local inversion, and confirm the complementary role of distribution matching in endpoint generation. The resulting framework also achieves stronger endpoint trace-Wasserstein performance than an official QuDDPM implementation with fewer trainable parameters. Overall, our work provides a unified information-theoretic principle for designing forward schedules, calibrating reverse steps, and separating physical recovery from generative coverage in quantum diffusion.
Stochastic unravelings are a widely used tool to solve open quantum system dynamics, in which the exact solution is obtained via an average over a stochastic process on the set of pure quantum states. Recently, the generalized rate operator unraveling formalism was derived, allowing not only for the engineering of stochastic realizations, but also to unravel without reverse jumps even for some dynamics in which P-divisibility is violated, thus greatly improving the simulation efficiency. This is possible because the unraveling depends on an arbitrary nonlinear transformation which can incorporate the memory effects. In this work, a stochastic Schrödinger equation for this formalism is derived, both for the cases with and without reverse jumps. It is also shown that a failure of this method can be used to witness master equations leading to unphysical time evolutions, independently of the particular nonlinear transformation considered.
Positivity preservation is naturally guaranteed in exact non-Markovian master equations for open quantum system dynamics. However, in many approximated non-Markovian master equations, the positivity of the reduced density matrix is not guaranteed. In this paper we provide a general class of time-local, perturbative, and positivity-preserving non-Markovian master equations generated from stochastic Schr\"odinger equations, particularly quantum-state-diffusion equations. Our method has an expanded range of applicability for accommodating a variety of non-Markovian environments. We show the positivity-preserving master equation for a three-level system coupled to a dissipative bosonic environment as a particular example to exemplify our general approach. We illustrate the numerical simulations with an analysis explaining why the previous approximated non-Markovian master equations cannot guarantee positivity. Our work provides a consistent master equation for studying the non-Markovian dynamics in ultrafast quantum processes and strong-coupling systems.
This paper proposes an interpretation of quantum mechanics, relying on the time-symmetric stochastic dynamics of quantum particles and on non-classical probability theory. Our main purpose is to demonstrate that the wave function and its complex conjugate can be interpreted as complex probability distributions in two complex diffusion equations related to non-real forward and backward in time stochastic motions respectively. We say non-real because Schroedinger forward and backward diffusions describe both reversible (real trajectories) and irreversible trajectories (non-real trajectories). The reversible trajectories are the only real trajectories and are given by the intersection of those forward and backward processes. It turns out that if we translate this intersection using set-theoretic language, we are led to a reversible diffusion described by Born rule probabilities. This proposal is useful also for explaining more about the role of complex numbers in quantum mechanics that produces this so-called "wave-like" nature of quantum reality. Our perspective also challenges the notion of physical superposition and aims at a derivation of superposition principle not based on the linearity of Schroedinger's equation but relying on pure probability theory. Moreover, it is suggested that, embracing the idea of stochastic processes in quantum theory, explains the reasons for the appearance of classical behavior in large objects, in contrast to the quantum behavior of small ones. In other words, we claim that a combination of a probabilistic and no-ontic view (neither epistemic though) of the wave function with a stochastic hidden-variables approach, may provide some insight into the quantum physical reality and potentially establish the groundwork for a novel interpretation of quantum mechanics.