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arxiv: 2505.13067 · v2 · submitted 2025-05-19 · 🪐 quant-ph

Verifying Quantum Memory in the Dynamics of Spin Boson Models

Charlotte B\"acker , Valentin Link , Walter T. Strunz This is my paper

Pith reviewed 2026-05-22 14:46 UTC · model grok-4.3

classification 🪐 quant-ph
keywords spin boson modelquantum memorynon-Markovian dynamicsprocess tensordynamical mapopen quantum systemsmatrix product operator
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The pith

Local quantum memory criteria applied to process tensors show that spin-boson model dynamics require quantum memory in the environment at low temperatures.

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

The paper examines memory effects in the non-Markovian dynamics of spin-boson models by applying local criteria that indicate when the reduced system evolution necessarily involves quantum memory from the environment. It computes dynamical maps and process tensors exactly using a matrix product operator influence functional method across wide parameter ranges. With single-intervention process tensors, the criteria generally predict quantum memory at low temperatures. When only the dynamical map is available, detection still works for resonant environments at short evolution times. The work also confirms quantum memory in the stationary regime by using process tensors initialized with the correlated steady state of system and environment.

Core claim

The authors show that local quantum memory criteria, drawn from different literature definitions, applied to single-intervention process tensors generally indicate the presence of quantum memory in the dynamics of spin-boson and two-spin-boson models at low temperatures. When restricted to the dynamical map alone, the same criteria detect quantum memory for resonant environments at short times. In the stationary regime, process tensors prepared with the correlated system-environment steady state as initial condition confirm quantum memory. These results follow from numerically exact computations that cover broad parameter regimes without uncontrolled approximations.

What carries the argument

Local quantum memory criteria applied to single-intervention process tensors and dynamical maps, obtained via matrix product operator influence functionals.

Load-bearing premise

The matrix product operator influence functional method yields numerically exact dynamical maps and process tensors without approximations that would change the outcomes of the memory criteria.

What would settle it

An exact computation of the single-intervention process tensor for a spin-boson model at low temperature in which the local quantum memory criteria return no indication of memory would falsify the general prediction.

Figures

Figures reproduced from arXiv: 2505.13067 by Charlotte B\"acker, Valentin Link, Walter T. Strunz.

Figure 1
Figure 1. Figure 1: FIG. 1. Upper: Influence matrix (time step [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Upper: A single-intervention process tensor has clas [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Extraction of the two dynamical maps [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Deformation of the Bloch sphere under the dynamics [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Quantum memory criteria for the spin-boson model [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Deformation of the Bloch sphere under the dynamics [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Quantum memory criterion based on the process [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Convergence of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

We investigate the nature of memory effects in the non-Markovian dynamics of spin boson models. Local quantum memory criteria can be used to indicate that the reduced dynamics of an open system necessarily requires a quantum memory in its environment. We apply two such criteria, derived from different definitions put forward in the literature, to spin boson and two-spin boson models. For the computation of dynamical maps and process tensors, we employ a numerically exact method for non-Markovian open system dynamics based on matrix product operator influence functionals, that can be applied across broad parameter regimes. We find that, with access to single-intervention process tensors, one can generally predict quantum memory in the dynamics at low temperatures. Given instead only the dynamical map, we are still able to detect quantum memory in the case of resonant environments at short evolution times. Moreover, we confirm quantum memory in the stationary dynamical regime using process tensors with the correlated steady state of system and environment as initial condition.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript applies two local quantum memory criteria to the reduced dynamics of spin-boson and two-spin-boson models. Dynamical maps and single-intervention process tensors are obtained via a matrix product operator influence functional method asserted to be numerically exact over broad parameter regimes. The central claims are that process tensors generally detect quantum memory at low temperatures, dynamical maps suffice to detect it for resonant baths at short times, and process tensors initialized from the correlated steady state confirm memory in the stationary regime.

Significance. If the numerical results hold, the work supplies concrete, falsifiable evidence that standard quantum-memory witnesses can be applied to a canonical open-system model using quantities accessible in experiment or simulation. The use of an established numerically exact technique across wide ranges of temperature and coupling is a methodological strength that could guide future studies of non-Markovianity in quantum thermodynamics and information processing.

major comments (2)
  1. [Numerical method and results sections] The central claims rest on the numerical exactness of the MPO influence functional for the memory criteria. No convergence data (bond dimension, singular-value cutoff, or truncation error) are shown for the low-temperature or resonant regimes where bath correlations decay slowly and coherent oscillations must be faithfully reproduced. Because the criteria detect small deviations from Markovianity, uncontrolled truncation errors could alter the reported outcomes.
  2. [Stationary dynamical regime subsection] The stationary-regime claim uses process tensors initialized with the correlated steady state. It is unclear how this initial condition is obtained numerically and whether the same MPO truncation that affects the dynamics also affects the steady-state preparation; any inconsistency would undermine the confirmation of quantum memory in that regime.
minor comments (2)
  1. [Introduction and criteria definitions] Notation for the two memory criteria should be unified or clearly cross-referenced when both are applied to the same data sets.
  2. [All figures] Figure captions would benefit from explicit statements of the bond dimension and cutoff values used for each plotted curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and positive assessment of the significance of our work. We address each major comment below and will revise the manuscript accordingly to improve clarity and substantiate the numerical claims.

read point-by-point responses

  1. Referee: The central claims rest on the numerical exactness of the MPO influence functional for the memory criteria. No convergence data (bond dimension, singular-value cutoff, or truncation error) are shown for the low-temperature or resonant regimes where bath correlations decay slowly and coherent oscillations must be faithfully reproduced. Because the criteria detect small deviations from Markovianity, uncontrolled truncation errors could alter the reported outcomes.

    Authors: We agree that explicit convergence data are necessary to support the claim of numerical exactness, especially in the challenging low-temperature and resonant regimes. In the revised manuscript we will add supplementary figures showing the bond-dimension dependence of the memory criteria (both process-tensor and dynamical-map witnesses) together with the associated truncation errors for representative parameter sets in these regimes. These data will confirm that the reported deviations from Markovianity remain stable under further increases in bond dimension. revision: yes

  2. Referee: The stationary-regime claim uses process tensors initialized with the correlated steady state. It is unclear how this initial condition is obtained numerically and whether the same MPO truncation that affects the dynamics also affects the steady-state preparation; any inconsistency would undermine the confirmation of quantum memory in that regime.

    Authors: We thank the referee for highlighting this ambiguity. The correlated steady state is prepared by propagating the joint system-environment state under the full Hamiltonian using the same MPO influence-functional representation until the reduced system state converges; the resulting MPO is then used directly as the initial condition for the process-tensor calculation. In the revision we will insert a dedicated paragraph describing this procedure, including the truncation schedule applied during steady-state preparation and a brief convergence check demonstrating consistency with the subsequent dynamical evolution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct numerical application of literature criteria

full rationale

The paper computes dynamical maps and single-intervention process tensors for standard spin-boson Hamiltonians using the established matrix product operator influence functional method, then applies two quantum-memory criteria taken from the existing literature. No parameter is fitted to a data subset and then relabeled as a prediction, no quantity is defined in terms of itself, and no load-bearing step reduces by construction to a self-citation or prior ansatz of the same authors. The numerical results are presented as direct simulations across parameter regimes; the detection of memory signatures follows from the computed objects rather than from any redefinition or statistical forcing internal to the paper. This is a self-contained numerical verification against external benchmarks and therefore carries no circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard spin-boson Hamiltonian and the validity of the matrix product operator method for computing exact reduced dynamics; no new free parameters are introduced beyond conventional model parameters such as temperature and coupling strength, which are varied rather than fitted to the memory criteria.

free parameters (2)
  • temperature
    Temperature is varied across regimes to test low-temperature predictions; treated as an input parameter rather than fitted to memory outcomes.
  • system-environment coupling strength
    Coupling is a standard tunable parameter in the model and is scanned to explore different dynamical regimes.
axioms (2)
  • domain assumption The spin-boson model and its two-spin extension faithfully represent the open quantum system under study.
    Invoked throughout as the physical setup for applying the memory criteria.
  • domain assumption The matrix product operator influence functional technique yields numerically exact dynamical maps and process tensors in the studied parameter ranges.
    Basis for all reported computations and comparisons between criteria.

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discussion (0)

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    We apply two such criteria, derived from different definitions put forward in the literature, to spin boson and two-spin boson models. For the computation of dynamical maps and process tensors, we employ a numerically exact method... based on matrix product operator influence functionals

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exact Floquet dynamics of strongly damped driven quantum systems

    quant-ph 2025-11 unverdicted novelty 7.0

    A periodic matrix product operator representation of the influence functional yields a numerically exact Floquet propagator for non-Markovian dynamics in strongly damped driven quantum systems.

  2. Uniform process tensor approach for the calculation of multi-time correlation functions of non-Markovian open systems

    quant-ph 2026-03 unverdicted novelty 6.0

    A time-translation-invariant MPO process tensor from uniTEMPO gives direct Fourier-space access to multi-time correlations, reducing the cost of multi-dimensional spectra for strongly coupled non-Markovian reservoirs.

  3. Revealing the quantum nature of memory in non-Markovian dynamics on IBM Quantum

    quant-ph 2025-10 unverdicted novelty 5.0

    IBM quantum hardware verifies quantum memory in non-Markovian single-qubit dynamics via collision-model circuits, with a toy example for two-qubit cases.

Reference graph

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