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arXiv:2610.38059

Exact Positive-P Simulation of Open Quantum Spin Systems with Generalized Density Matrix Formalism

Alex Ferrier, Yi Shi, Marzena H. Szymańska, Piotr Deuar

quant-phSubmitted Oct 27, 2026• Announced Oct 29, 2026• 0 citations
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Abstract

The positive-P representation for bosons allows scalable exact simulations of many-body open quantum systems, provided dissipation is strong enough to stabilise stochastic trajectories. Designing analogous methods for spin systems, relevant to quantum technologies, remains challenging, as previous spin positive-P formulations can exhibit systematic errors even when simulations are numerically stable. Here, we develop a phase-space representation that maps spin-$\frac{1}{2}$ degrees of freedom onto parameters of a local generalized density matrix (GDM), rather than SU(2) coherent states. In all stable regimes, we observe no deviations from exact solutions, restoring the positive-P property: stable solutions are exact up to stochastic sampling noise. The GDM representation combines seamlessly with bosonic positive-P, enabling highly scalable treatment of coupled spin-boson systems. We map simulation stability versus spin and boson dissipation in the Jaynes-Cummings model and apply the method to a Jaynes-Cummings-Hubbard quasi-Lieb lattice, showing how spin dissipation and driving can destroy or restore excitation incompressibility.

AI Summary

Positive-P simulations of dissipative spin systems can exhibit systematic errors even when trajectories appear numerically stable. This paper investigates whether replacing SU(2) coherent-state kernels with a nonsingular local kernel can avoid this problem while retaining compatibility with bosonic positive-P simulations. The proposed generalized density matrix kernel is $\Lambda_S=\begin{pmatrix}x&z\\w&1-x\end{pmatrix}$, with three independent complex variables per spin. Operator correspondences yield stochastic differential equations for driven-dissipative Jaynes–Cummings–Hubbard models, using five complex variables per spin–boson site. A diffusion gauge redistributes noise between spin and boson variables without changing the formal ensemble means. Single-site benchmarks against direct master-equation solutions use 1,000 trajectories per parameter set, generally through $gt=100$ and sometimes $gt=1000$. No statistically significant discrepancies are observed before noise instability. For gauge $G=1$, simulations broadly remain stable over the tested times when spin decay satisfies $\Gamma_\downarrow>g$; choosing $G=10$ stabilizes some cases with stronger bosonic loss, while destabilizing others. A lattice demonstration treats 20 periodic unit cells containing 40 bosonic modes and 20 spins. Strong spin decay removes a previously reported excitation plateau versus bosonic drive; direct spin driving produces a low-bosonic-drive plateau and correlations absent from mean field. Accessible times remain limited by stochastic noise amplification, and sustained stability requires sufficient dissipation. Accuracy in stable regimes is supported by benchmarks rather than a general guarantee. Pure interacting spin lattices and optimized, variable-dependent gauges are proposed extensions, not demonstrated applications.

Little sign of AI-generated textOverclaimsStrong assumptionsPossible error

Central claim may not hold: The claim that numerical stability guarantees exact GDM simulation is stronger than the demonstrated benchmarks, because the required boundary behavior and sampling convergence are not established generally. The displayed estimator for K=Var(N)/<N> omits a bosonic normal-ordering correction equal to average[tilde n]/average[tilde n+x]. The assertion that K vanishes at mean-field level contradicts its displayed variance-to-mean definition, which gives K=1 for coherent bosons without spin excitations. The printed SU(2) operator correspondences omit scalar terms required by the normalized kernel, although this alone does not invalidate the final SU(2) stochastic equations.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Comments
18 pages, 11 figures

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The positive-P representation for bosons allows scalable exact simulations of many-body open quantum systems, provided dissipation is strong enough to stabilise stochastic trajectories. Designing analogous methods for spin systems, relevant to quantum technologies, remains challenging, as previous spin positive-P formulations can exhibit systematic errors even when simulations are numerically stable. Here, we develop a phase-space representation that maps spin-12\frac{1}{2}21​ degrees of freedom onto parameters of a local generalized density matrix (GDM), rather than SU(2) coherent states. In all stable regimes, we observe no deviations from exact solutions, restoring the positive-P property: stable solutions are exact up to stochastic sampling noise. The GDM representation combines seamlessly with bosonic positive-P, enabling highly scalable treatment of coupled spin-boson systems. We map simulation stability versus spin and boson dissipation in the Jaynes-Cummings model and apply the method to a Jaynes-Cummings-Hubbard quasi-Lieb lattice, showing how spin dissipation and driving can destroy or restore excitation incompressibility.
Positive-P simulations of dissipative spin systems can exhibit systematic errors even when trajectories appear numerically stable. This paper investigates whether replacing SU(2) coherent-state kernels with a nonsingular local kernel can avoid this problem while retaining compatibility with bosonic positive-P simulations.
The proposed generalized density matrix kernel is ΛS=(xzw1−x)\Lambda_S=\begin{pmatrix}x&z\\w&1-x\end{pmatrix}ΛS​=(xw​z1−x​), with three independent complex variables per spin. Operator correspondences yield stochastic differential equations for driven-dissipative Jaynes–Cummings–Hubbard models, using five complex variables per spin–boson site. A diffusion gauge redistributes noise between spin and boson variables without changing the formal ensemble means.
Single-site benchmarks against direct master-equation solutions use 1,000 trajectories per parameter set, generally through gt=100gt=100gt=100 and sometimes gt=1000gt=1000gt=1000. No statistically significant discrepancies are observed before noise instability. For gauge G=1G=1G=1, simulations broadly remain stable over the tested times when spin decay satisfies Γ↓>g\Gamma_\downarrow>gΓ↓​>g; choosing G=10G=10G=10 stabilizes some cases with stronger bosonic loss, while destabilizing others. A lattice demonstration treats 20 periodic unit cells containing 40 bosonic modes and 20 spins. Strong spin decay removes a previously reported excitation plateau versus bosonic drive; direct spin driving produces a low-bosonic-drive plateau and correlations absent from mean field.
Accessible times remain limited by stochastic noise amplification, and sustained stability requires sufficient dissipation. Accuracy in stable regimes is supported by benchmarks rather than a general guarantee. Pure interacting spin lattices and optimized, variable-dependent gauges are proposed extensions, not demonstrated applications.
Spin-boson models involving many interacting spins and bosons are ubiquitous in quantum simulation platforms. At the same time, characterizing the dynamics of these quantum systems represents a significant challenge. Here, we consider general spin-boson models where bosons are subject to Markovian dissipation (e.g., due to cavity loss). We present an exact hybrid quantum-classical stochastic approach where the solution of a classical stochastic equation—mimicking the bosonic modes—is input into a quantum stochastic equation for the spins. Furthermore, the spins are effectively decoupled for each stochastic realization, which nevertheless comes at the expense of sampling over unphysical states. In contrast with existing stochastic approaches based on the influence functional formalism, we place no restriction (factorizability or Gaussianity) on the initial state, or the spin-boson coupling (except that it be linear in the bosonic operator). Markovian dissipation, being at the heart of our approach, renders the stochastic equations Markovian even in the strong coupling regime. Furthermore, it ensures hermiticity (though not positivity) of the density matrix for each realization, thus improving the convergence of stochastic sampling. Interestingly, we find a condition on the classical simulability of the system based solely on the single atom cooperativity even in a many-body setting. We benchmark and showcase the utility of our approach in several examples, specifically in cases where a direct numerical computation is unfeasible.
This thesis presents our work on two separate subfields and has for this reason been divided into two parts: (I) the development and implementation of a numerical technique for non-equilibrium dynamics and a (II) detailed investigation of the dirty-boson problem. In part (I), we optimize and develop the quantum phase space method known as the positive-P representation (PPR) specifically for the simulations of the real time quench dynamics of quantum spin systems. The main benefit of this approach is that the dynamics of the density operator is mapped onto ∼N\sim N∼N stochastic variables (where NNN is the size of the system) that obey Langevin-type stochastic differential equations, thereby greatly reducing the complexity of the problem. The first publication presents our initial use of the PPR on spin systems by using a Schwinger mapping for spin operators on to Bosonic operators since the underlying basis comprises of Bosonic coherent states. We simulate the quench dynamics of the generalized transverse-field spin-1/2 XXZ model in 1d showing that simulations of up to 100 spins are possible, albeit for relatively short simulation times. In our second publication, we reformulate the PPR using SU(2) spin coherent states and further optimize simulation lifetimes by implementing an extrapolation scheme, to achieve several-fold improvements over the results of our first publication. We focus solely on the transverse Ising model and simulate its quench dynamics in 1d and 2d using up to 10410^4104 spins, while significantly extending simulation lifetimes. The second part of this thesis is a numerical study of the universality class of the Superfluid-Bose Glass transition of the dirty-boson system. Recently the longstanding exact result z=dz=dz=d, where zzz is the dynamic critical exponent and ddd is the dimensionality of the system, has been challenged by a series of numerical studies, suggesting the alternative scenario that zzz should be instead unconstrained. To address this controversy, we use large scale quantum Monte-Carlo simulations on two independent quantum models, and average over 5×104−1055\times10^4-10^55×104−105 disorder realizations to numerically determine the universality class of the dirty-boson transition, paying particular attention to the dynamic critical exponent, zzz and without any biasing assumptions.
We describe an efficient numerical method for simulating the dynamics and steady states of collective spin systems in the presence of dephasing and decay. The method is based on the Schwinger boson representation of spin operators and uses an extension of the truncated Wigner approximation to map the exact open system dynamics onto stochastic differential equations for the corresponding phase space distribution. This approach is most effective in the limit of very large spin quantum numbers, where exact numerical simulations and other approximation methods are no longer applicable. We benchmark this numerical technique for known superradiant decay and spin-squeezing processes and illustrate its application for the simulation of non-equilibrium phase transitions in dissipative spin lattice models.
We describe an efficient numerical method for simulating the dynamics and steady states of collective spin systems in the presence of dephasing and decay. The method is based on the Schwinger boson representation of spin operators and uses an extension of the truncated Wigner approximation to map the exact open system dynamics onto stochastic differential equations for the corresponding phase space distribution. This approach is most effective in the limit of very large spin quantum numbers, where exact numerical simulations and other approximation methods are no longer applicable. We benchmark this numerical technique for known superradiant decay and spin-squeezing processes and illustrate its application for the simulation of non-equilibrium phase transitions in dissipative spin lattice models.
The paradigmatic spin-boson model considers a spin degree of freedom interacting with an environment typically constituted by a continuum of bosonic modes. This ubiquitous model is of relevance in a number of physical systems where, in general, one has neither control over the bosonic modes, nor the ability to tune distinct interaction mechanisms. Despite this apparent lack of control, we present a suitable transformation that approximately maps the spin-boson dynamics into that of a tunable multiphoton Jaynes-Cummings model undergoing dissipation. Interestingly, the latter model describes the coherent interaction between a spin and a single bosonic mode via the simultaneous exchange of n bosons per spin excitation. Resorting to the so-called reaction coordinate method, we identify a relevant collective bosonic mode in the environment, which is then used to generate multiphoton interactions following the proposed theoretical framework. Moreover, we show that spin-boson models featuring structured environments can lead to non-Markovian multiphoton Jaynes-Cummings dynamics. We discuss the validity of the proposed method depending on the parameters and analyse its performance, which is supported by numerical simulations. In this manner, the spin-boson model serves as a good analogue quantum simulator for the inspection and realization of multiphoton Jaynes-Cummings models, as well as the interplay of non-Markovian effects and, thus, as a simulator of light-matter systems with tunable interaction mechanisms.
We reformulate the full quantum dynamics of spin systems using a phase-space representation based on SU(2) coherent states which generates an exact mapping of the dynamics of any spin system onto a set of stochastic differential equations. This representation is superior in practice to an earlier phase-space approach based on Schwinger bosons, with the numerical effort scaling only linearly with system size. By also implementing extrapolation techniques from quasiclassical equations to the full quantum limit, we are able to extend useful simulation times severalfold. This approach is applicable in any dimension including cases where frustration is present in the spin system. The method is demonstrated by simulating quenches in the transverse-field Ising model in one and two dimensions.