arxiv: 2604.06466 · v1 · submitted 2026-04-07 · 🪐 quant-ph

One-to-one correspondence between Hierarchical Equations of Motion and Pseudomodes for Open Quantum System Dynamics

Kai M\"uller , Walter T. Strunz This is my paper

Pith reviewed 2026-05-10 18:35 UTC · model grok-4.3

classification 🪐 quant-ph

keywords open quantum systemshierarchical equations of motionpseudomodesnon-Markovian dynamicsbath correlation functionLindblad formquantum dissipation

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The pith

Every exponential bath correlation function arises from a Lindblad-damped pseudomode model and maps exactly onto the HEOM hierarchy via a linear transformation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that any physical bath correlation function expressible as a finite sum of exponential terms can be realized by a concrete model of N interacting pseudomodes whose damping follows Lindblad form. It further shows that the time evolution of the combined system-plus-pseudomode state can be mapped onto the HEOM hierarchy by a non-unitary linear transformation that works in both directions. The proofs supply explicit formulas for both the pseudomode Lindbladian and the mapping itself. This equivalence supplies fresh derivations of the stochastic pure-state hierarchy and its nearly-unitary variant. Readers interested in non-Markovian open-system simulations gain a concrete bridge between two widely used numerical approaches.

Core claim

We prove that every physical bath correlation function that can be written as a sum of N exponential terms can be obtained from a physical model with N interacting pseudomodes which are damped in Lindblad form. For every such bath correlation function there exists a non-unitary, linear transformation which mirrors the evolution of the system-pseudomode state onto the HEOM hierarchy, and vice versa. The proofs are constructive and we give explicit expressions for the mirror transformation as well as for the pseudomode Lindbladian corresponding to a given exponential bath correlation function.

What carries the argument

The non-unitary linear mirror transformation that equates the density-matrix evolution of the system plus Lindblad-damped pseudomodes with the HEOM hierarchy, constructed directly from the exponential coefficients of the bath correlation function.

Load-bearing premise

Any bath correlation function of physical interest can be represented exactly as a finite sum of exponential terms.

What would settle it

A concrete calculation for an exponential bath correlation function in which the transformed pseudomode state fails to satisfy the corresponding HEOM equations at any time after the initial condition.

Figures

Figures reproduced from arXiv: 2604.06466 by Kai M\"uller, Walter T. Strunz.

read the original abstract

We unite two of the most widely used approaches for strongly damped, non-Markovian open quantum dynamics, the Hierarchical Equations of Motion (HEOM) and the pseudomode method by proving two statements: First, every physical bath correlation function (BCF) that can be written as a sum of $N$ exponential terms can be obtained from a physical model with $N$ interacting pseudomodes which are damped in Lindblad form. Second, for every such BCF there exists a non-unitary, linear transformation which mirrors the evolution of the system-pseudomode state onto the HEOM hierarchy, and vice versa. Our proofs are constructive and we give explicit expressions for the mirror transformation as well as for the pseudomode Lindbladian corresponding to a given exponential BCF. This approach also gives insight and provides elegant derivations of the corresponding Hierarchy of stochastic Pure States (HOPS) method and its nearly-unitary version, nuHOPS. Our result opens several avenues for further optimization of non-Markovian open quantum system dynamics methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit, constructive mapping between HEOM and a Lindblad pseudomode model for any bath correlation function that is exactly a finite sum of exponentials.

read the letter

The main takeaway is that for bath correlation functions written as sums of N exponentials, there is now a direct, invertible link to a system of N interacting pseudomodes whose damping is fully specified by Lindblad operators, plus a non-unitary linear transformation that carries the pseudomode state onto the HEOM hierarchy and back. The proofs are constructive, so the Lindbladian and the mirror map are given in closed form from the exponential coefficients and rates. This also produces clean derivations of the HOPS and nuHOPS methods as by-products. That is the concrete advance over prior work that treated the two approaches separately. The paper is upfront that the result applies only when the bath correlation function is already in exponential form, which is the usual practical starting point after a fit. Within that scope the mapping appears free of positivity violations or non-invertibility problems, and the stress-test found no load-bearing gaps in the stated derivations. A minor practical note is that real correlation functions are rarely exact finite sums, so users will still need to decide how many terms to keep and how to fit them; the equivalence itself does not remove that step. The citation pattern is standard and appropriate, drawing on the established HEOM and pseudomode literature without circularity. This is useful reading for anyone who already runs HEOM or pseudomode codes for non-Markovian problems in quantum chemistry or information. The explicit constructions could help with hybrid implementations or with moving between the two formalisms for numerical efficiency. It is mathematically grounded enough to warrant a serious referee, even if the final impact will depend on how widely the mapping gets adopted in code. I would send it for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a one-to-one correspondence between the Hierarchical Equations of Motion (HEOM) and the pseudomode method for open quantum system dynamics. For any physical bath correlation function (BCF) expressible as a finite sum of N exponential terms, it constructs an equivalent physical model of N interacting pseudomodes damped via Lindblad operators. It further supplies an explicit non-unitary linear transformation that maps the evolution of the system-pseudomode state onto the HEOM hierarchy (and vice versa). The proofs are constructive, providing closed-form expressions for the Lindbladian and the mirror map; the work also yields derivations of the Hierarchy of stochastic Pure States (HOPS) and its nearly-unitary variant (nuHOPS).

Significance. If the claimed equivalence holds, the result unifies two standard numerical frameworks for strongly non-Markovian open-system dynamics and supplies a rigorous route to hybrid or optimized implementations. The constructive, parameter-free character of the proofs (derived from standard open-system axioms without fitted quantities or ad-hoc entities) is a clear strength, as is the explicit invertibility of the linear map. This should facilitate cross-validation between HEOM and pseudomode codes and clarify the status of stochastic unravelings such as HOPS.

minor comments (2)

[Abstract / §3] The abstract states that the pseudomode model remains physical for all parameter regimes, but the main text should include a short explicit check (e.g., after Eq. (X) defining the Lindblad operators) that the resulting rates and interaction matrices preserve complete positivity for arbitrary exponential coefficients.
[§4] Notation for the auxiliary density operators in the HEOM hierarchy and the pseudomode state vector should be aligned more clearly (perhaps in a dedicated comparison table) to make the linear map immediately usable by readers implementing the correspondence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the main results, and recommendation for minor revision. The report correctly identifies the constructive nature of the proofs and the potential for unifying HEOM and pseudomode approaches. Below we respond to the referee summary as the primary point raised.

read point-by-point responses

Referee: The manuscript proves a one-to-one correspondence between the Hierarchical Equations of Motion (HEOM) and the pseudomode method for open quantum system dynamics. For any physical bath correlation function (BCF) expressible as a finite sum of N exponential terms, it constructs an equivalent physical model of N interacting pseudomodes damped via Lindblad operators. It further supplies an explicit non-unitary linear transformation that maps the evolution of the system-pseudomode state onto the HEOM hierarchy (and vice versa). The proofs are constructive, providing closed-form expressions for the Lindbladian and the mirror map; the work also yields derivations of the Hierarchy of stochastic Pure States (HOPS) and its nearly-unitary variant (nuHOPS).

Authors: We appreciate the referee's concise and accurate encapsulation of our contributions. The one-to-one correspondence is indeed established via the explicit construction of the Lindbladian for the interacting pseudomodes and the invertible non-unitary linear mirror map, both derived directly from the exponential form of the BCF without additional assumptions. The derivations of HOPS and nuHOPS follow naturally from the same framework as stochastic unravelings of the pseudomode dynamics. revision: no

Circularity Check

0 steps flagged

No significant circularity in the equivalence proof

full rationale

The paper presents constructive proofs deriving explicit Lindblad operators for interacting pseudomodes and a non-unitary linear map between the pseudomode state and HEOM auxiliary operators, directly from the assumption that the bath correlation function is a finite sum of exponentials and standard open-system Lindblad dynamics. No derivations reduce to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations; the equivalence follows from first-principles mappings without circular reduction. The result is self-contained against external benchmarks such as standard quantum master equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work is a proof of equivalence that rests on standard domain assumptions of open quantum systems rather than new fitted parameters or postulated entities.

axioms (2)

domain assumption Bath correlation functions of physical interest can be represented as finite sums of exponential terms
This is the prerequisite that allows the pseudomode construction to match any given BCF exactly.
domain assumption Pseudomodes damped by Lindblad operators constitute a valid physical model for the environment
Invoked when the paper states that the exponential BCF is obtained from a physical model with N interacting pseudomodes.

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discussion (0)

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One-to-one correspondence between Hierarchical Equations of Motion and Pseudomodes for Open Quantum System Dynamics

like uniTEMPO (uniform time evolving matrix product operator) [14] can be understood in this manner. Here we focus specifically on the relation between the pseudomode approach, and the hierarchical methods (HEOM, HOPS, nuHOPS), and we touch chain mapping approaches later. While they all embed the system into an effective environment, the construction and ...

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PSEUDOMODE INTERACTION PICTURE As mentioned in the main text the pseudomode method uses an Ansatz-Lindbladian describing damped, coupled harmonic os- cillators to approximate the bath correlation function (BCF) of the original Gaussian environment. The system and pseudomode dynamics is given by the Lindbladian in Eq. (6) as ˙ρ=−i[H pm,ρ] + ∑ k LkρL† k − 1...

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,zN(t))T ∈C N for anN-dimensional, complex OU-vector

GENERAL RESULTS ON MULTIV ARIATE ORNSTEIN-UHLENBECK PROCESSES In the following we present an analysis of (c-number) multivariate Ornstein-Uhlenbeck (OU) processesz(t):= (z1(t),z 2(t), . . . ,zN(t))T ∈C N for anN-dimensional, complex OU-vector. An insertion of the identity in terms of coherent states in Eq. (S6) makes it clear that the pseudomode BCF is id...

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band gap model

PROOF OF PSEUDOMODE REPRESENTABILITY We start from a bath correlation function (BCF) that can be expressed as a sum of exponential terms αexp(τ) = N ∑ j=1 G je−λ jτ ,G j,λ j ∈C,forτ≥0,(S30) withλ j =γ j +iω j and positive real partsγ j >0, and we setα exp(τ) =α exp(−τ) ∗ forτ≤0, as required from the fundamental relation Eq. (2). Note that this impliesα ex...

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empty", incoming modes of the second input port. As quantum state in this enlarged Hilbert space we consider|Ψ(t)⟩ ⊗ |vac⟩ diss, where|vac⟩ diss is the dissipon vacuum (the

DISSIPON TRANSFORMATION Now assume that we have constructed the pseudomode model – as explained above – corresponding to a physical bath correlation function of the usual exponential form (S30). Thus, the parameters of that pseudomode model Eq. (S43), including V,λ i are known. In the following we explicitly derive the HEOM and HOPS equations from the dis...

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