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REVIEW 5 major objections 5 minor 46 references

Quantum model reduction based on Oja's flow

T0 review · 5 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Oja's principal-component flow can reduce a Markovian quantum open system to its slowest dynamics without perturbative expansions; a constrained version preserves complete positivity and finds approximate decoherence-free subspaces.

desk verdict First algorithm is a clean, likely correct Oja-flow reduction for Lindbladians; the second algorithm's advertised 'guaranteed CPTP' is not proven as written. read the letter →

arxiv 2607.26669 v1 pith:XRPCFCCH submitted 2026-07-29 quant-ph

Miguel Casanova , Kentaro Ohki , Francesco Ticozzi This is my paper · ORCID
classification quant-ph PACS 03.65.Yz03.67.Pp
keywords quantummodelreductionOja'sflowadiabaticeliminationLindbladgeneratorslowmanifolddecoherence-freesubspacecompletepositivityprincipalcomponentanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops two numerical algorithms to approximate the slowest degrees of freedom of a Markovian quantum open system described by a Lindblad generator. The first applies Oja's continuous-time principal-component flow to the vectorized generator, converging to a projector onto the slow operator subspace and producing a reduced generator whose slow eigenvalues match those of the full generator exactly. The second restricts the flow to reductions of the form V = V*⊗V, which guarantees the reduced dynamics remain completely positive and trace preserving, and it identifies approximate decoherence-free subspaces as the maximizers of a real cost function. Together the methods form a non-perturbative alternative to adiabatic elimination, requiring no series expansions or matrix inversions, and the CP-preserving variant is shown on a central-spin model to find a 2-dimensional noise-protected code. A sympathetic reader would care because the methods offer a systematic, memory-efficient path to simulating large open quantum systems and to designing quantum codes.

What carries the argument

Oja's continuous-time principal-component flow, the matrix differential equation ϵV̇ = (1 - VV†)AV on the Stiefel manifold St(C^m, ℓ), whose stable steady states span the eigenspace of A with largest real parts. For Lindbladians the paper proves that a unitary SWAP conjugation makes A real, ensuring convergence of the flow; the reduced generator is then V∞† L̂ V∞. The CP-preserving variant arrives at a real cost function J(V) by substituting V = V*⊗V, and the Riemannian gradient of J defines the modified flow (32), which works on n×r matrices instead of n²×r², cutting memory by a square factor.

What would settle it

Apply the CP-Oja flow to a small Lindbladian whose center manifold is known analytically to have two operator blocks (m>1 in the structure of §8.1) and compare the eigenvalues of the reduced generator to the slow eigenvalues of the full generator; a mismatch would disprove the claim that the flow extracts the slowest degrees of freedom. Independently, for the unconstrained flow, check the exactness identity ||e^{Lt}P - V_inf e^{L_Vinf t} V_inf^dagger||_sop = 0 for a random initial state in the slow subspace; any deviation above machine precision would falsify the exact-simulation claim.

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Extended reading notes

Core claim

Given a quantum dynamical semigroup generator L with a spectral gap between the eigenvalues λ_{r²} and λ_{r²+1}, the paper claims that Oja's flow ϵV̇ = (1 - VV†)L̂V, run on the matrix representation L̂ of the Lindbladian, converges to a steady state V∞ whose columns span the r²-dimensional subspace of the slowest decaying operators. The reduced generator L̂_{V∞} = V∞† L̂ V∞ has the same r² slow eigenvalues, and the error between the full and reduced evolutions, measured as ||e^{Lt}P - V∞ e^{L_{V∞}t}V∞†||, is exactly zero for initial states in the slow subspace and decays as e^{Re(λ_{r²+1})t} otherwise. By imposing V = V*⊗V, the flow becomes a genuine gradient ascent of the real cost J(V) = ½

Load-bearing premise

The unconstrained reduction needs only a spectral gap in the generator, but the complete-positive-preserving flow assumes the slow manifold is a single Hilbert subspace of the form range(V)⊗range(V) — a condition the authors admit is rarely met in practice — so for generic Lindbladians the CP reduced model may not approximate the slow dynamics.

Editorial extensions

If this is right

  • If the generator has a spectral gap after the r²-th eigenvalue, the Oja-flow reduction reproduces the slow dynamics exactly for states in the slow subspace, with error shrinking at the fast-decay rate for all other initial states.
  • Model reduction is achieved without Jordan decomposition, dense inversions, or perturbative series; per iteration the cost is O(N_{L̂}^{nz} r² + (nr)²), and sparse generators stay sparse throughout.
  • The complete-positive-preserving flow returns a reduced generator that is itself a valid Lindbladian, so the reduced model could in principle be implemented on another open quantum system.
  • For time-dependent generators, piecewise-constant updates of the reduction matrix, with a transition matrix that renormalizes the state, keep the error of order Δt, dominated by leakage out of the slow subspace; updating only at input thresholds is enough.
  • When the system has an exact decoherence-free subspace, the CP-Oja cost J is maximized exactly by the DFS projector, and for weakly perturbed systems the flow numerically locates a slowly decohering approximate DFS.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the unconstrained reduction is exact on the slow subspace, one could certify a reduced simulation in practice by verifying that the initial state's projection onto the computed slow subspace is near identity; this is a guarantee that perturbative adiabatic elimination cannot match at finite order.
  • The CP-Oja flow's objective is real, so the same algorithm might extend to discrete-time CPTP maps by feeding the logarithm of the map through the flow, with the natural power method the authors cite as a discrete-time counterpart.
  • The structural limitation to single-block Hilbert subspaces suggests a concrete path: generalizing the flow to V = Σ_i V_i*⊗V_i, which the authors list as future work, would allow multi-block center manifolds and likely recover the full Oja-flow accuracy while retaining partial physical structure.
  • The central-spin numerics indicate that the quality of the CP-reduced model is not monotone in r: dimensions that do not match the invariant-subspace structure of the dissipative part (like r=6) introduce initialization-dependent errors, hinting that choosing r from the algebra of the noise operators matters as much as the spectral gap.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes two numerical algorithms for model reduction of finite-dimensional Markovian open quantum systems, both built on Oja's continuous-time principal component flow. The first algorithm vectorizes the Lindbladian, applies a swap-based similarity transformation to make the generator real, runs Oja's flow to obtain an isometry spanning the slow invariant subspace, and forms the reduced generator V†LV; it is extended to time-dependent generators via piecewise-constant updates. The second algorithm constrains the reduction to V=V*⊗V so that the reduced generator preserves Hermiticity and conditional complete positivity, and it is proposed as a way to identify approximate decoherence-free subspaces. Both methods are tested on a central-spin model with a dissipative bath.

Significance. If the convergence claims are accepted, the first algorithm is a genuinely useful numerical alternative to adiabatic elimination: it avoids perturbative expansions, works directly from the generator, preserves the slow eigenvalues via Eq. (12), and has favorable memory scaling for sparse generators. The realification in Prop. 1/Cor. 2 and the explicit complexity estimates are strong points. The second algorithm is the main weakness: its advertised CPTP guarantee is not established by the proof in Section 7 and is not automatically supplied by the flow in Section 8. The structural limitation admitted in Section 8.1—that the CP reduction is exact only for a single Hilbert-space slow block—is significant. Because the first algorithm is sound and the second may be repairable by a reformulation or additional hypotheses, the paper warrants a major revision rather than rejection.

major comments (5)
  1. The proof of trace preservation for \hat V' is incomplete at a load-bearing step. The proposition chooses |ψ1⟩ as an arbitrary right null vector of \hat L_V^†, i.e. \hat V^† \hat L^† \hat V|ψ1⟩=0, but the evaluation of \hat W' later uses the identity '|ψ*_1⟩=|ψ_1⟩=|ω⟩'. This identity is false: |ω_r⟩ is not generally in the kernel of \hat L_V^†, and |ψ1⟩ is not |ω_r⟩. The preceding assertion that \hat V|ψ1⟩ is a non-defective right eigenvector of \hat L^† does not follow from the defining equation unless range(\hat V) is invariant under \hat L^†, which is not established. Consequently the conclusion that |ω⟩ is an eigenvector of \hat W'—and hence that \hat L_{\hat V'} is trace preserving—is unsupported. In addition, the constructed \hat V' = \hat V\hat W\sqrt{(\hat V\hat W)^†(\hat S \hat V\hat W\hat S)^*} is not shown to have the product structure required by Proposition 8, so the CCP-pre
  2. The CP-Oja flow directly produces \hat V_∞ = V_∞^*⊗V_∞ and no trace-preserving correction from Section 7 is applied. For the resulting reduced generator \hat L_V = (V^*⊗V)^† \hat L (V^*⊗V), one has \hat L_V^†|ω_r⟩ = (V^*⊗V)^† \hat L^† (V^*⊗V)|ω_r⟩ = (1/√r)(V^*⊗V)^† \hat L^† vec(VV^†), which is not zero for a generic steady state. Thus the reduced generator is not shown to be trace preserving. The statement in Section 1 that the second method yields a reduced semigroup 'guaranteed to be completely positive and trace preserving (CPTP)' is therefore unsupported as written; at most Hermiticity and conditional complete positivity are ensured.
  3. The paper concedes that the CP-Oja reduction is exactly the intended slow dynamics only when the slow manifold is a single Hilbert-space block, and that 'the last situation is hard to obtain in practical situations.' This is a structural limitation, not a technical one: for a generic Lindbladian whose slow manifold is an operator subspace with multiple independent blocks, the CP-Oja reduced model is not guaranteed to approximate the slow manifold. The abstract's claim that the second algorithm provides a model reduction to 'the subspace associated to their slowest degrees of freedom' should be qualified accordingly; the algorithm is more accurately described as searching for the best approximate unitarily evolving subspace.
  4. Equation (21) is presented as an upper bound for the time-dependent reduction error, but the derivation is a first-order Taylor heuristic. Even under the stated assumptions (small Δt and P_i ρ=ρ), the O(Δt²) remainder is not uniform without additional assumptions on the commutator of L(t) with the projector, and the expression e^{L(t)Δt}P_i mixes the full generator with the projected initial state. Since this bound is used to justify the threshold-update strategy, it should be either proved rigorously or explicitly labeled as a heuristic.
  5. The global convergence statement for the first algorithm—the characterization of stable fixed points of the complex Oja flow—is imported from Ref. [33], a co-authored unpublished preprint. Because the main method's correctness depends on this external result, the paper should either include a self-contained proof or clearly state that the validity of the first algorithm is conditional on that preprint; the current text cites it without hedging.
minor comments (5)
  1. [Section 7, Prop. 6] Typo: '\hat V^† \hat L^† \hat V^†|ψ1⟩=0' should read '\hat V^† \hat L^† \hat V|ψ1⟩=0'.
  2. [Section 6, Eq. (16)] The Oja flow is written with \hat L_i \hat V_i^†; the last factor should be \hat V_i.
  3. [Section 6, before Eq. (19)] The text states tr(vec^{-1}(·)) = r⟨ω_r|·⟩; because |ω_r⟩ = (1/√r)|1⟩⟩, the correct factor is √r⟨ω_r|·⟩.
  4. [Sections 4–5] The reduced-dimension notation is inconsistent: Section 4 uses ℓ, Section 5 switches to r², and Section 8 uses r for the Hilbert-space dimension. Please introduce ℓ for the operator-subspace rank explicitly and keep it throughout.
  5. [Section 7, Lemma 7] The notation |x⟩=vec(x^†) makes the expression tr(\hat L x^*⊗x) opaque on first reading; an explicit index form or a short example would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the reduced model is obtained by eigenspace projection, with no fitted parameters renamed as predictions.

full rationale

The paper defines the reduced generator as Lhat_V∞ = V∞† Lhat V∞, with V∞ the Oja-flow steady state, and the matching of the reduced eigenvalues to the slow eigenvalues (Eq. 12) follows from the characterization of stable fixed points as the principal eigenspace. That characterization is imported from Ref. [33], a co-authored preprint, but it is a general parameter-free theorem about square matrices, not a fit to the data being predicted; under the review rules this counts as independent support and does not raise the circularity score. No parameter is fitted to the target dynamics, and the numerical central-spin benchmarks are checked against the full model, so the reported slow eigenvalues are predictions rather than constructions. Section 8.1 explicitly limits the CP-preserving variant's exactness to the case where the slow manifold is a single block on H, which is a scope limitation, not a circular step. Proposition 6's trace-preservation proof contains an unjustified identification (|ψ_1*> = |ψ_1> = |omega>), which is a correctness risk, but it is not an instance of the derivation reducing to its own input. Overall, no circular step can be exhibited, so the score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The derivation itself has no fitted constants; the algorithm's output is determined by Lhat and r. The main external inputs are the cited convergence theorem [33] and the structural assumptions on the spectrum and the slow manifold. Hand-chosen model parameters and thresholds are test inputs, not parameters of the method.

free parameters (4)
  • Reduced dimension r = 2, 6, 10, 22, 30, 32 (tested)
    Design parameter chosen by user; spectral gap must exist after the r-th eigenvalue. No principled selection rule is given.
  • Oja rate-control epsilon = not reported
    Controls convergence speed and numerical stability of the Oja flow; no values or tuning details are given in Section 9.
  • Central-spin model parameters = omega_s=1.01, A_x=0.12, A_z=0.03, omega_b=1.92, lambda_b=0.31, beta=0.1
    Hand-chosen to create a spectral gap after the fourth eigenvalue; they parameterize the testbed, not the derivation.
  • Time-dependent control envelope parameters = E_y=1.28, tau_y=5, tau_0=25; thresholds {0,0.3,0.6,0.9,1.2}
    Hand-chosen for the time-dependent test; algorithm behavior may depend on threshold choice.
assumptions (4)
  • domain assumption Spectral gap Delta_r = Re(lambda_r) - Re(lambda_{r+1}) > 0 exists after the chosen r.
    Stated in Section 4 ('In the remainder of this work we assume such a gap to be present') and Section 6; without it the slow/fast decomposition and Oja convergence target are undefined.
  • domain assumption Complex Oja flow converges to the dominant eigenspace of a general square matrix (Ref [33]).
    Used in Corollary 2 to guarantee convergence for Lindbladian matrices; the cited result is an unpublished preprint by co-author Ohki, so the paper does not supply a self-contained proof.
  • ad hoc to paper The slow manifold of a Lindbladian can be represented by a Hilbert subspace for the CP-preserving flow.
    The tensor-product constraint V=V*⊗V forces reduction onto range(VV†) in Hilbert space; Section 8.1 notes exactness only when the center manifold is a single subspace block.
  • standard math Zoutendijk-style convergence of Riemannian gradient ascent on the compact Stiefel manifold.
    Invoked for the convergence of the CP-Oja flow to local maxima of J(V) in Section 8.

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Cite this review

Pith. "Pith review of Quantum model reduction based on Oja's flow." pith.science (2026). https://pith.science/paper/XRPCFCCH

@misc{pith2026260726669,
  author       = {Pith},
  title        = {Pith review of: Quantum model reduction based on Oja's flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XRPCFCCH}},
  note         = {Machine review of arXiv:2607.26669}
}
read the original abstract

We propose a novel approach to numerically derive approximate reduced dynamical models for Markovian quantum open systems without perturbative iterations, projecting the evolution to the subspace associated to their slowest degrees of freedom. The two algorithms we develop are based on Oja's continuous-time principal component flow: the first returns the optimal reduction to the slowest decaying operator-subspace, and is extended to time-dependent dynamics, while the second one is designed to reduce the dynamics on a subspace of the system's Hilbert space, and thus preserve conditional complete positivity. The methods represent a non-perturbative alternative to well-established Adiabatic Elimination (AE) methods, and the second can be used to find noise-protected subspace codes for quantum information processing. Both are tested on a paradigmatic central spin model.

Figures

Figures reproduced from arXiv: 2607.26669 by the authors.

Figure 1
Figure 1. Comparison between the full model (solid blue line) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 3
Figure 3. Comparison between the full time-dependent model [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 2
Figure 2. Comparison between the full model (solid blue line) [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

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Works this paper leans on

46 extracted references · 2 linked inside Pith

  1. [33]

    Global convergence of Oja’s component flow for general square matrices and its applications.Preprint arXiv:2510.00801v2, October 2025

    Daiki Tsuzuki and Kentaro Ohki. Global convergence of Oja’s component flow for general square matrices and its applications.Preprint arXiv:2510.00801v2, October 2025

    arXiv 2025

  2. [1]

    Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21(6–7):467– 488, Jun 1982

    1982

  3. [2]

    Scully and M

    Marlan O. Scully and M. Suhail Zubairy.Quantum Optics. Cambridge University Press, 1997

    1997

  4. [3]

    Cambridge University Press, 2021

    Alejandro Lopez.Statistical Mechanics of Quantum ManyBody Systems. Cambridge University Press, 2021

    2021

  5. [4]

    Oxford University Press, Oxford, 1 edition, January 2007

    Heinz-Peter Breuer and Francesco Petruccione.The Theory of Open Quantum Systems. Oxford University Press, Oxford, 1 edition, January 2007

    2007

  6. [5]

    Huelga.Open Quantum Systems: An Introduction

    `Angel Rivas and Susana F. Huelga.Open Quantum Systems: An Introduction. SpringerBriefs in Physics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012

    2012

  7. [6]

    A practical introduction to tensor networks: Matrix product states and projected entangled pair states

    Rom´ an Or´ us. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349:117–158, 2014

    2014

  8. [7]

    Ignacio Cirac

    Frank Verstraete, Valentin Murg, and J. Ignacio Cirac. Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems.Advances in Physics, 57(2):143–224, 2008

    2008

Show all 46 references
  1. [8]

    Lecture Notes in Physics

    Robert Alicki and Karl Lendi.Quantum Dynamical Semigroups and Applications. Lecture Notes in Physics. Springer, Berlin, Germany, 1987 edition, September 1987

    1987

  2. [9]

    Model reduction for quantum systems: Discrete-time quantum walks and open markov dynamics.IEEE Transactions on Information Theory, 71(11), 2025

    Tommaso Grigoletto and Francesco Ticozzi. Model reduction for quantum systems: Discrete-time quantum walks and open markov dynamics.IEEE Transactions on Information Theory, 71(11), 2025. 14

    2025

  3. [10]

    Exact model reduction for continuous-time open quantum dynamics.Quantum, 9:1814, 2025

    Tommaso Grigoletto et al. Exact model reduction for continuous-time open quantum dynamics.Quantum, 9:1814, 2025

    2025

  4. [11]

    Hendra I. Nurdin. Structures and transformations for model reduction of linear quantum stochastic systems.IEEE Transactions on Automatic Control, 59(9):2413–2425, 2014

    2014

  5. [12]

    Huelga, and Martin B

    Dario Tamascelli, Andrea Smirne, Susana F. Huelga, and Martin B. Plenio. Non-perturbative treatment of non- markovian dynamics of open quantum systems.Physical Review Letters, 120:030402, 2018

    2018

  6. [13]

    Time evolution of a quantum system in contact with a nearly Gaussian- Markoffian noise bath.Journal of the Physical Society of Japan, 58(10):101–114, 1989

    Yoshitaka Tanimura and Ryogo Kubo. Time evolution of a quantum system in contact with a nearly Gaussian- Markoffian noise bath.Journal of the Physical Society of Japan, 58(10):101–114, 1989

    1989

  7. [14]

    J. J. Hope. Steady-state solutions of the master equation for atom-laser models.Phys. Rev. A, 64:053608, 2000

    2000

  8. [15]

    Towards generic adiabatic elimination for bipartite open quantum systems.Quantum Science and Technology, 2(4):044011, 2017

    R Azouit, F Chittaro, A Sarlette, and P Rouchon. Towards generic adiabatic elimination for bipartite open quantum systems.Quantum Science and Technology, 2(4):044011, 2017

    2017

  9. [16]

    Adiabatic elimination and subspace evolution of open quantum systems

    Daniel Finkelstein-Shapiro, David Viennot, Ibrahim Saideh, Thorsten Hansen, T˜ onu Pullerits, and Arne Keller. Adiabatic elimination and subspace evolution of open quantum systems. Physical Review A, 101(4):042102, 2020

    2020

  10. [17]

    Low-rank approximated Kalman-Bucy filters using Oja’s principal component flow for linear time-invariant systems.IEEE Control Systems Letters, 8:1583–1588, 2024

    Daiki Tsuzuki and Kentaro Ohki. Low-rank approximated Kalman-Bucy filters using Oja’s principal component flow for linear time-invariant systems.IEEE Control Systems Letters, 8:1583–1588, 2024

    2024

  11. [18]

    Quantum channels & operations: Guided tour, 2012

    Michael M Wolf. Quantum channels & operations: Guided tour, 2012. Lecture notes available athttps://citeseerx. ist.psu.edu/document?repid=rep1&type=pdf&doi= afa58291b0b8bd47504acb1ab8f553f0b37685cf

    2012

  12. [19]

    Complete positivity violation of the reduced dynamics in higher-order quantum adiabatic elimination

    Masaaki Tokieda, Cyril Elouard, Alain Sarlette, and Pierre Rouchon. Complete positivity violation of the reduced dynamics in higher-order quantum adiabatic elimination. Physical Review A, 109(6):062206, 2024

    2024

  13. [20]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang.Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 1 edition, June 2012

    2012

  14. [21]

    Number v.717 in Lecture Notes in Physics Ser

    Robert Alicki.Quantum Dynamical Semigroups and Applications. Number v.717 in Lecture Notes in Physics Ser. Springer Berlin / Heidelberg, Berlin, Heidelberg, 2nd ed edition, 2007

    2007

  15. [22]

    Clarendon Press, Oxford, 1998

    David Evans and Yasuyuki Kawahigashi.Quantum Symmetries on Operator Algebras. Clarendon Press, Oxford, 1998

    1998

  16. [23]

    A Review of Multiple- Time-Scale Dynamics: Fundamental Phenomena and Mathematical Methods

    Kristian Uldall Kristiansen. A Review of Multiple- Time-Scale Dynamics: Fundamental Phenomena and Mathematical Methods. In Bernhelm Booß-Bavnbek, Jens Hesselbjerg Christensen, Katherine Richardson, and Oriol Vall` es Codina, editors,Multiplicity of Time Scales in Complex Syste...

    2023

  17. [24]

    Explicit formulas for adiabatic elimination with fast unitary dynamics.arXiv:2404.01802v2, 2024

    Angela Riva, Alain Sarlette, and Pierre Rouchon. Explicit formulas for adiabatic elimination with fast unitary dynamics.arXiv:2404.01802v2, 2024

    2024 arXiv

  18. [25]

    E. M. Kessler. Generalized Schrieffer-Wolff formalism for dissipative systems.Physical Review A, 86(1):012126, 2012

    2012

  19. [26]

    Projection-based adiabatic elimination of bipartite open quantum systems.Physical Review A, 102(3):032212, 2020

    Ibrahim Saideh, Daniel Finkelstein-Shapiro, T˜ onu Pullerits, and Arne Keller. Projection-based adiabatic elimination of bipartite open quantum systems.Physical Review A, 102(3):032212, 2020

    2020

  20. [27]

    Lidar and Todd A

    Daniel A. Lidar and Todd A. Brun, editors.Quantum Error Correction. Cambridge University Press, 1 edition, 2013

    2013

  21. [28]

    D. A. Lidar, I. L. Chuang, and K. B. Whaley. Decoherence- Free Subspaces for Quantum Computation.Physical Review Letters, 81(12):2594–2597, September 1998

    1998

  22. [29]

    Quantum wall states for noise mitigation and eternal purity bounds.Quantum Science and Technology, 11(1):015002, March 2026

    Miguel Casanova and Francesco Ticozzi. Quantum wall states for noise mitigation and eternal purity bounds.Quantum Science and Technology, 11(1):015002, March 2026

    2026

  23. [30]

    Numerical method for finding decoherence-free subspaces and its applications.Physical Review A, 87(1):012338, January 2013

    Xiaoting Wang, Mark Byrd, and Kurt Jacobs. Numerical method for finding decoherence-free subspaces and its applications.Physical Review A, 87(1):012338, January 2013

    2013

  24. [31]

    Simplified neuron model as a principal component analyzer.Journal of Mathematical Biology, 15:267–273, 1982

    Erkki Oja. Simplified neuron model as a principal component analyzer.Journal of Mathematical Biology, 15:267–273, 1982

    1982

  25. [32]

    Helmke, and J.B

    Wei-Yong Yan, U. Helmke, and J.B. Moore. Global analysis of Oja’s flow for neural networks.IEEE Transactions on Neural Networks, 5(5):674–683, September 1994

    1994

  26. [34]

    Projection Robust Wasserstein Distance and Riemannian Optimization (supplementary material)

    Tianyi Lin, Chenyou Fan, Nhat Ho, Marco Cuturi, and Michael I Jordan. Projection Robust Wasserstein Distance and Riemannian Optimization (supplementary material). Conference on Neural Information Processing Systems, 2020

    2020

  27. [35]

    Horn and Charles R

    Roger A. Horn and Charles R. Johnson.Matrix Analysis. Cambridge University Press, 2 edition, 2012

    2012

  28. [36]

    Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems.SIAM Journal on Control and Optimization, 60(4):2514–2537, 2022

    Markus Tranninger, Richard Seeber, Martin Steinberger, Martin Horn, and Christian P¨ otzsche. Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems.SIAM Journal on Control and Optimization, 60(4):2514–2537, 2022

    2022

  29. [37]

    On duality between quantum maps and quantum states.Open systems & information dynamics, 11(1):3–42, 2004

    Karol ˙Zyczkowski and Ingemar Bengtsson. On duality between quantum maps and quantum states.Open systems & information dynamics, 11(1):3–42, 2004

    2004

  30. [38]

    M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac. Assessing non-Markovian dynamics.Physical Review Letters, 101(15):150402, 2008

    2008

  31. [39]

    SpringerBriefs in Electrical and Computer Engineering

    Hiroyuki Sato.Riemannian Optimization and Its Applications. SpringerBriefs in Electrical and Computer Engineering. Springer International Publishing, Cham, 2021

    2021

  32. [40]

    Analysis of quantum semigroups with GKS–Lindblad generators: II

    Bernhard Baumgartner and Heide Narnhofer. Analysis of quantum semigroups with GKS–Lindblad generators: II. General.Journal of Physics A: Mathematical and Theoretical, 41(39):395303, 2008

    2008

  33. [41]

    Adiabatic Markovian Dynamics.Physical Review Letters, 105(5):050503, 2010

    Ognyan Oreshkov and John Calsamiglia. Adiabatic Markovian Dynamics.Physical Review Letters, 105(5):050503, 2010

    2010

  34. [42]

    Finding quantum noiseless subsystems: A linear-algebraic approach

    Francesco Ticozzi and Augusto Ferrante. Finding quantum noiseless subsystems: A linear-algebraic approach. PhysCon07, 2007

    2007

  35. [43]

    Man-Duen Choi and David W. Kribs. Method to Find Quantum Noiseless Subsystems.Physical Review Letters, 96(5):050501, February 2006

    2006

  36. [44]

    Theory of Quantum Error Correction for General Noise

    Emanuel Knill, Raymond Laflamme, and Lorenza Viola. Theory of Quantum Error Correction for General Noise. Physical Review Letters, 84(11):2525–2528, March 2000

    2000

  37. [45]

    On Low-Rank Approximation of Kalman- Bucy Filters for Linear Time-Varying Systems.Proceedings of SICE Festival with Annual Conference (to appear), 2026

    Kentaro Ohki. On Low-Rank Approximation of Kalman- Bucy Filters for Linear Time-Varying Systems.Proceedings of SICE Festival with Annual Conference (to appear), 2026

    2026

  38. [46]

    Convergence analysis of natural power method and its applications to control

    Daiki Tsuzuki and Kentaro Ohki. Convergence analysis of natural power method and its applications to control. Preprint arXiv:2512.21469v1, December 2025. 15

    2025 arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.