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arxiv: 1907.09277 · v1 · pith:AANRU7ENnew · submitted 2019-07-22 · 🧮 math-ph · math.MP· math.PR· quant-ph

Classical Noises Emerging from Quantum Environments

Stephane Attal , Julien Deschamps , Clement Pellegrini This is my paper

Pith reviewed 2026-05-24 18:07 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PRquant-ph
keywords quantum open systemsclassical noisecomplex obtuse random variablesrepeated quantum interactionsrandom walks on unitary groupdiffusive and Poisson limits3-tensor diagonalization
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The pith

Certain unitary interactions make quantum environments produce classical noise, tied to complex obtuse random variables whose 3-tensor diagonalizes to set the dynamics.

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

This paper seeks to characterize the unitary interactions in quantum open systems where the bath produces dynamics equivalent to classical noise, specifically random walks on the small system's unitary group. It establishes that such interactions correspond precisely to complex obtuse random variables, which carry a 3-tensor with symmetries permitting diagonalization in an orthonormal basis. That diagonalization then governs the entire random walk and fixes which directions yield diffusive versus Poisson behavior in the continuous-time limit. A reader would care because the result supplies an explicit bridge from fully quantum baths to effective classical stochastic processes, showing when quantum coupling reduces to ordinary randomness.

Core claim

The characterization of those unitary interactions which make the environment acting as if it were a classical noise is intimately related to the notion of complex obtuse random variables; these particular random variables have an associated 3-tensor whose symmetries make it diagonalizable in some orthonormal basis, and this diagonalisation entirely describes the behavior of the random walk and characterizes the directions along which the limit process is of diffusive or of Poisson nature.

What carries the argument

The 3-tensor associated with complex obtuse random variables, which is diagonalizable in an orthonormal basis owing to its symmetries and thereby determines the random-walk paths and the diffusive or Poisson character of their continuous limits.

If this is right

  • The dynamics generated by the environment reduce exactly to random walks on the unitary group of the small system.
  • The continuous-time scaling limits are diffusive along some directions and Poisson along others, with the assignment fixed by the eigenvalues and eigenvectors of the diagonalized 3-tensor.
  • All interactions admitting a classical-noise description in the repeated-interaction setting are thereby classified by the algebraic properties of the associated obtuse random variable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tensor machinery might be used to engineer discrete interactions that reproduce any prescribed classical noise statistics in the limit.
  • The classification supplies a criterion for deciding, from the interaction Hamiltonian alone, whether a given quantum bath will appear classical or retain quantum features at long times.
  • Extensions could test whether the same 3-tensor structure appears in continuous-time quantum stochastic differential equations outside the repeated-interaction model.

Load-bearing premise

The discrete-time repeated quantum interactions model is assumed to capture all relevant actions of the quantum environment that can produce classical-noise dynamics, with the complex obtuse random variable framework exhausting the cases that admit a 3-tensor diagonalization description.

What would settle it

A concrete unitary interaction that produces classical-noise random-walk statistics yet whose driving random variable is not complex obtuse, or whose 3-tensor fails to be diagonalizable in an orthonormal basis, or whose observed continuous limit fails to match the diffusive/Poisson assignment predicted by that diagonalization.

read the original abstract

In the framework of quantum open systems, that is, simple quantum systems coupled to quantum baths, our aim is to characterize those actions of the quantum environment which give rise to dynamics dictated by classical noises. First, we consider the discrete time scheme, through the model of repeated quantum interactions. We explore those unitary interactions which make the environment acting as if it were a classical noise, that is, dynamics which ought to random walks on the unitary group of the small system. We show that this characterization is intimately related to the notion of complex obtuse random variables. These particular random variables have an associated 3-tensor whose symmetries make it diagonalizable in some orthonormal basis. We show how this diagonalisation entirely describes the behavior of the random walk associated to the action of the environment. In particular this 3-tensor and its diagonalization characterize the behavior of the continuous-time limits; they characterize the directions along which the limit process is of diffusive or of Poisson nature.

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

0 major / 3 minor

Summary. The manuscript characterizes those unitary interactions in the discrete-time repeated quantum interactions model that cause the quantum environment to produce dynamics equivalent to classical noise, i.e., random walks on the unitary group of the small system. The central claim is that this occurs precisely when the interactions correspond to complex obtuse random variables; these variables possess a 3-tensor whose symmetries permit diagonalization in an orthonormal basis, and the resulting diagonalization fully determines the random-walk behavior and classifies the continuous-time limits into diffusive or Poissonian directions along specific axes.

Significance. If the characterization holds, the work supplies a concrete mathematical criterion, based on the diagonalizable 3-tensor of complex obtuse random variables, for identifying when quantum open-system dynamics reduce to classical noise processes. This supplies a precise link between the repeated-interaction model and the classification of limit processes (diffusive versus jump) that may prove useful in quantum stochastic calculus and the study of classical emergence from quantum baths. The paper employs the standard discrete-time framework and frames the result as a characterization rather than an exhaustive physical claim.

minor comments (3)
  1. [Introduction] The abstract asserts that the 3-tensor diagonalization 'entirely describes' the random walk and 'characterizes' the limit directions, but the introduction should state the precise theorem (including any hypotheses on the unitary or the random-variable moments) that makes this rigorous.
  2. Notation for the 3-tensor (its components, symmetries, and the orthonormal basis in which it is diagonal) should be introduced with an explicit definition or displayed equation before it is used to classify diffusive versus Poissonian directions.
  3. The manuscript would benefit from at least one fully worked low-dimensional example (e.g., a 2- or 3-dimensional unitary interaction) that computes the 3-tensor, performs the diagonalization, and exhibits the resulting random-walk increments and continuous-time limit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments appear in the provided report, so we have no individual points requiring rebuttal or clarification at this stage. We remain ready to incorporate any minor editorial adjustments the editor may request.

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained characterization theorem

full rationale

The paper's central claim is a mathematical characterization: unitary interactions in the repeated quantum interaction model yield classical-noise dynamics precisely when they correspond to complex obtuse random variables whose 3-tensor admits orthonormal diagonalization, which in turn classifies the diffusive/Poisson directions in the continuous limit. This chain is presented as a derived equivalence (interaction → obtuse RV → tensor symmetries → random-walk behavior → limit classification) without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. The discrete-time model is invoked as the standard framework rather than an ansatz smuggled in; no equations or steps in the abstract equate a claimed prediction to its own inputs by construction. The result is scoped as a characterization within the model, remaining independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard repeated-interaction model of quantum open systems and on the mathematical definition of complex obtuse random variables; no free parameters are visible in the abstract, and the only invented entity is the complex obtuse random variable itself, introduced to encode the interaction symmetries.

axioms (1)
  • domain assumption Repeated quantum interactions model captures the discrete-time action of the quantum environment
    Abstract opens by placing the problem inside this standard framework of quantum open systems.
invented entities (1)
  • complex obtuse random variables no independent evidence
    purpose: To encode the unitary interactions that produce classical-noise dynamics and to supply the 3-tensor whose diagonalization classifies the limit process
    Abstract states that the characterization is intimately related to this notion and that the associated 3-tensor describes the random walk and continuous limits.

pith-pipeline@v0.9.0 · 5697 in / 1431 out tokens · 41789 ms · 2026-05-24T18:07:27.831025+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Attal, J

    S. Attal, J. Deschamps, C. Pellegrini, ”Complex Obtuse Random Wa lks and Their Continuous-Time Limits”, Probability Theory and Related Fields , to appear

    work page

  2. [2]

    Attal, J

    S. Attal, J. Deschamps, C. Pellegrini, ”Entanglement of Bipartite Quantum Systems driven by Repeated Interactions”, Journal of Statistical Physics , 154 (2014), no. 3, 819-837

    work page 2014

  3. [3]

    Attal, ”Approximating the Fock space with the toy Fock space ”, S´ eminaire de Probabilit´ esXXXVI, Springer L.N.M

    S. Attal, ”Approximating the Fock space with the toy Fock space ”, S´ eminaire de Probabilit´ esXXXVI, Springer L.N.M. 1801 (2003) , p. 477-497

    work page 2003

  4. [4]

    Repeated Quantum Interactions and Unita ry Random Walks

    S. Attal, A. Dhahri, “Repeated Quantum Interactions and Unita ry Random Walks”, Journal of Theoretical Probability , 23, p. 345-361, 2010

    work page 2010

  5. [5]

    Equations de structure pour des martinga les vectorielles

    S. Attal, M. Emery, “Equations de structure pour des martinga les vectorielles”, S´ eminaire de Probabilit´ es, XXVIII, p. 256–278, Lecture Notes in Math., 1583, Springer, Berlin, 1994

    work page 1994

  6. [6]

    From repeated to continuous quantum in teractions

    S. Attal, Y. Pautrat, “From repeated to continuous quantum in teractions”, Annales Henri Poincar´ e. A Journal of Theoretical and Mathematical Physics, 7, 2006, p. 59– 104

    work page 2006

  7. [7]

    From (n+1)-level atom chains to n-dimensional noises

    S. Attal, Y. Pautrat, “From (n+1)-level atom chains to n-dimensional noises”, Ann. Inst. H. Poincar´ e Probab. Statist.41 (2005), no. 3, p. 391–407

    work page 2005

  8. [8]

    Thermal relaxation of a QED cavity

    L. Bruneau, C.-A. Pillet, “Thermal relaxation of a QED cavity”, J. Stat. Phys. 134 (2009), no. 5-6, p. 1071–1095. 32

    work page 2009

  9. [9]

    Scattering induced cur rent in a tight-binding band

    L. Bruneau, S. De Bi` evre, C.-A. Pillet, “Scattering induced cur rent in a tight-binding band” , J. Math. Phys. 52 (2011), no. 2, 022109, 19 pp

    work page 2011

  10. [10]

    On some classes of b ipartite unitary operators

    J. Deschamps, I. Nechita and C. Pellegrini, “On some classes of b ipartite unitary operators” J. of Physics A: Mathematical and Theoretical , vol 49 , Number 33 (2016)

    work page 2016

  11. [11]

    Quantum jumps of light recording the birt h and death of a photon in a cavity

    S. Haroche, S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Del´ eglise, U. Busk-Hoff, M. Brune and J-M. Raimond, “Quantum jumps of light recording the birt h and death of a photon in a cavity”, Nature 446, 297 (2007)

    work page 2007

  12. [12]

    Real- time quantum feedback prepares and stabilizes photon number sta tes

    S. Haroche, C. Sayrin, I. Dotsenko, XX. Zhou, B. Peaudecer f, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M.Brune and J-M. Raimond , “Real- time quantum feedback prepares and stabilizes photon number sta tes”, Nature, 477, 73 (2011)

    work page 2011

  13. [13]

    On Simultaneous Reduction of Families of Matrices to Triangular or Diagonal Form by Unitary Congruence

    Y. P. Hong, R. A. Horn, “On Simultaneous Reduction of Families of Matrices to Triangular or Diagonal Form by Unitary Congruence”, Linear and Multilinear Algebra, 17 (1985), p.271-288

    work page 1985

  14. [14]

    Hudson, K.R

    R.L. Hudson, K.R. Parthsarathy, ”Quantum Stochastic Evolut ions and ...”, Commu- nications in Mathematical Physics

    work page

  15. [15]

    Existence, uniqueness and approximation of a sto chastic Schr¨ odinger equation: the diffusive case

    C. Pellegrini, “Existence, uniqueness and approximation of a sto chastic Schr¨ odinger equation: the diffusive case”, Ann. Probab. 36 (2008), no. 6, p. 2332–2353

    work page 2008

  16. [16]

    Existence, Uniqueness and Approximation of the j ump-type Stochastic Schr¨ odinger Equation for two-level systems

    C. Pellegrini, “Existence, Uniqueness and Approximation of the j ump-type Stochastic Schr¨ odinger Equation for two-level systems”, Stochastic Process and their Applica- tions, 2010 vol 120 No 9, pp. 1722-1747

    work page 2010

  17. [17]

    Markov Chain Approximations of Jump-Diffusion Sto chastic Master Equations

    C. Pellegrini, “Markov Chain Approximations of Jump-Diffusion Sto chastic Master Equations”, Annales de l’institut Henri Poincar´ e: Probabilit´ es et Statistiques, 2010, vol 46, pp. 924-948. St´ ephane ATTAL Universit´ e de Lyon Universit´ e de Lyon 1, C.N.R.S. Institut Camille Jordan 21 av Claude Bernard 69622 Villeubanne cedex, France Julien DESCHAMPS U...

    work page 2010