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pyBoLaNO: A Python symbolic package for normal ordering involving bosonic ladder operators

T0 review · 1 major / 8 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper presents pyBoLaNO, a SymPy-based package that normal-orders any bosonic ladder-operator polynomial through Blasiak's explicit formulae.

desk verdict Useful, honest software paper that needs one missing implementation detail documented before acceptance. read the letter →

arxiv 2501.01603 v2 pith:O5KZFEGO submitted 2025-01-03 quant-ph math-phmath.MPphysics.comp-ph

Hendry M. Lim , Donny Dwiputra , M. Shoufie Ukhtary , Ahmad R. T. Nugraha This is my paper · ORCID
classification quant-phmath-phmath.MPphysics.comp-ph
keywords bosonicladderoperatorsnormalorderingWickBlasiak'sformulaegeneralizedStirlingnumbersLindbladmasterequationopenquantumsystemsSymPysymboliccomputation
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

pyBoLaNO is a Python package built on SymPy that normal-orders any polynomial in bosonic creation and annihilation operators. Its central claim is that Blasiak's closed-form formulae, expressed through generalized Stirling numbers, make this faster and less error-prone than SymPy's built-in recursive flatten-and-swap routine; benchmark results show about an order-of-magnitude speedup on random monomials, with the gap widening as operator count and mode count grow. Because normal-ordered expressions turn coherent-state expectation values into ordinary c-number functions, the package also derives normal-ordered commutators and Lindblad-master-equation expectation-value evolutions directly, reproducing published equations for several open quantum systems. If the claim holds, researchers can automate a tedious, error-prone algebraic step in quantum optics and open quantum systems.

What carries the argument

Blasiak's formulae. For a monomial written as alternating blocks $\hat{b}^{\dagger r_1}\hat{b}^{s_1}\cdots \hat{b}^{\dagger r_M}\hat{b}^{s_M}$ with running excess $d_l = \sum_{m=1}^l (r_m-s_m)$, the generalized Stirling number $S_{r,s}(k)$ in Eq. (20) supplies the coefficient in Eq. (21), giving $N(\hat{X})$ as a sum over $k$ of $S_{r,s}(k)\,\hat{b}^{\dagger k}\hat{b}^k$ multiplied by an outer creation or annihilation power depending on the sign of the total excess. This explicit formula is the engine; the package's normal_ordering routine splits polynomials into monomials by subsystem, applies the formula, and then sorts creation and annihilation factors by subscript.

What would settle it

Run pyBoLaNO's normal_ordering on $\hat{b}\hat{b}^\dagger$: the correct normal-ordered equivalent is $1 + \hat{b}^\dagger\hat{b}$. If the output is only $\hat{b}^\dagger\hat{b}$, or if the input factors are reordered before the formula is applied, the ordering assumption has failed. A broader check would compare against hand-computed normal ordering for monomials such as $\hat{b}_1\hat{b}^\dagger_2\hat{b}_2\hat{b}^\dagger_1$.

Watch

Extended reading notes

Core claim

The paper's central claim is that normal ordering of a bosonic ladder-operator polynomial need not be built by recursively swapping adjacent operators. Instead, each monomial is classified by the running excess of creation over annihilation operators, and its normal-ordered form is written directly as a finite sum of generalized Stirling numbers times $\hat{b}^{\dagger k}\hat{b}^k$, with a possible leftover power of $\hat{b}^\dagger$ or $\hat{b}$. The same engine then gives the normal-ordered commutator of two polynomials and the normal-ordered evolution equation for an expectation value under the Lindblad master equation. The package is validated against known equations from the literature, including a quantum Rayleigh oscillator, a bipartite quantum battery, a PT-symmetric trimer, and a pair of nonreciprocal driven-dissipative resonators.

Load-bearing premise

The entire algorithm silently depends on SymPy preserving the order in which the custom ladder-operator objects are multiplied; if SymPy ever treats them as commuting and reorders them, every normal-ordering result would be wrong.

Editorial extensions

If this is right

  • Normal-ordered expectation values in the coherent-state basis reduce to c-number functions, so any expression the package normal-orders can be turned into a phase-space integrand automatically.
  • The benchmark claim implies that for random monomials with ten ladder operators and two subsystems, normal ordering is about ten times faster than SymPy's built-in routine, and the ratio grows with the number of operators and subsystems.
  • For polynomial inputs, the package's multiprocessing splits work over summands, so large sums gain near-linear speedup on multi-core machines.
  • Commutator normal ordering is computed as $AB-BA$ passed through the same routine, so the speedup carries over to commutator evaluations.
  • The Lindblad expectation-value routine reproduces the published evolution equations for the examples shown, which is the main validation evidence for the package.

Reading between the lines

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

  • Because the cost of Blasiak's formula grows with the powers $s_l$ and $r_l$ and the number of blocks $M$, the observed speedup should shrink for monomials that are already nearly normal-ordered or extremely high-power; a trivial pre-check for already-normal-ordered input would capture that regime, though it would add overhead elsewhere.
  • The same generalized-Stirling-number approach may carry over to other single-parameter algebras whose normal ordering is governed by combinatorial numbers, such as deformed or $q$-oscillator algebras, though the paper does not claim this.
  • Using the package as a symbolic back end, one could automatically generate closed systems of moment equations for nonlinear open quantum systems by differentiating a chosen set of normal-ordered observables and closing with a truncation rule; this is not done in the paper.
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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

1 major / 8 minor

Summary. This manuscript presents pyBoLaNO, a SymPy-based Python package for normal-ordering polynomials in bosonic ladder operators using Blasiak's generalized Stirling-number formula. The package provides normal_ordering/NO for polynomials, NO_commutator for commutators, and LME_expval_evo for expectation-value evolution in the Lindblad master-equation framework, with support for multiple mode indices and multiprocessing. The paper describes the workflow, gives usage examples, validates the package by reproducing five published evolution equations, and benchmarks runtime against SymPy's normal_ordered_form.

Significance. The package addresses a real need in quantum optics and open quantum systems: normal ordering is ubiquitous, and SymPy's built-in routine becomes slow for large monomials. The explicit-formula approach is sound in principle, and the manuscript has concrete strengths: open-source code with Jupyter notebooks, reproduction of several published results from different groups, and a parameter-free algorithm. The validation includes a co-authored result (Ref. [7]), but this is a reproduction of an already published equation and does not raise a circularity concern. The main weakness is that the paper's central correctness claim is not fully supported by the text as written; the most important missing piece is the mechanism that prevents SymPy from auto-sorting the ladder-operator factors.

major comments (1)
  1. [§3.2] The correctness of every normal-ordering result depends on SymPy preserving the left-to-right order of the custom ladder-operator factors, but the paper never states how this is guaranteed. Section 3.2 says only that BosonicAnnihilationOp and BosonicCreationOp "inherit from sympy.Expr without any modifications to methods handling interactions between SymPy objects"; it does not state that the classes set is_commutative=False (or otherwise disable Mul's auto-sorting of factors). In SymPy, an Expr subclass that does not declare noncommutativity is treated as commutative, so the input b*bd*b would be silently rewritten as b**2*bd before _NO_Blasiak is called, yielding 2*b + bd*b**2 instead of the correct b + bd*b**2. The correct outputs in Section 4 imply that the actual code does enforce noncommutativity in some way, but this load-bearing detail is undocumented and no regression test is shown. Please state the mechanism explicitly, include the relevant class definition or point to the exact lines in the repository, and add a unit test such as assert NO(b*bd*b) == b + bd*b**2 that would fail if the noncommutativity mechanism were removed.
minor comments (8)
  1. [§4.3.1] In the energy-conservation example, the LaTeX output is correct, but the "Output render" line repeats the previous result d⟨b⟩/dt = −iω0⟨b⟩ instead of showing d⟨b†b⟩/dt = 0; please fix the rendered display so that the validation of Eq. (28) is actually visible.
  2. [§4.2] In the multipartite commutator example, the "Input render" shows A = b†_1 b_2, while the code defines A = bd_1*bd_2 (i.e., b†_1 b†_2); the printed input is inconsistent with both the code and the output, so the render should be corrected.
  3. [§4.3.3, Eq. (33)] There is a typo in "γ1 = γ, ˆO1 = ˆO1 = ˆbc"; the second equality should read ˆO1 = ˆP1 = ˆbc.
  4. [§2.1, Eq. (16)] The coherent-state label β is said to be in Z; it should be in C, as also implied by the appearance of β* later in the same section.
  5. [§3.6 and §4] The multiprocessing configuration variable is introduced as mp_config in Section 3.6, but the setup snippet in Section 4 uses bl.mpconfig (without underscore); please unify the name so that the printed API is usable as written.
  6. [§3.6] The heading contains a typo: "Mutiprocessing" should be "Multiprocessing".
  7. [§4.3.5, Eq. (39)] The dissipator rates Γe^{±iϕ} are complex, while Eq. (22) states γj ≥ 0 for Lindblad dissipators; please clarify that this example is an effective or generalized master equation, or state how the package's output remains meaningful outside the strict Lindblad form.
  8. [§6] The conclusion's phrase "error-free normal ordering" is an overclaim relative to the evidence presented; the paper demonstrates agreement with several published results, which is better described as validation rather than a guarantee of error-freeness. I suggest softening the wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the package's normal-ordering engine is an independent implementation of Blasiak's published formulae, validated against external literature results rather than fitted to them.

full rationale

The derivation chain in pyBoLaNO is self-contained with respect to its central claim. The core algorithm implements Blasiak's formulae (Eqs. 18-21), which are cited from the external mathematical literature (Refs. [21,22]) and are not derived from the paper's own outputs. The normal-ordering results follow from explicit combinatorial sums over generalized Stirling numbers; there are no fitted parameters, no data-dependent calibration, and no ansatz whose validity is assumed from the authors' prior work. The package's validation examples in Section 4 reproduce published equations from the literature: the harmonic oscillator (standard result), the Rayleigh oscillator (Ref. [5]), the bipartite quantum battery (Ref. [7]), the PT-symmetric trimer (Ref. [11]), and the nonreciprocal resonators (Ref. [12]). One of these references, Ref. [7], includes a co-author (M. S. Ukhtary), but the comparison is a reproduction of an already-published equation, not a target the package was fitted to, so this does not constitute load-bearing self-citation. The paper also benchmarks against SymPy's built-in normal_ordered_form as an external baseline, and the performance advantage is an empirical measurement, not a tautology. The only concern noted by a skeptical reader is an undocumented implementation detail about how SymPy handles the noncommutativity of the custom operator classes (Section 3.2 states the classes 'inherit from sympy.Expr without any modifications' to interaction methods, and Section 3.3 step 6 explicitly describes a post-processing sorting step to handle noncommuting Mul factors). That is a potential correctness or robustness issue, not a circularity issue: even if the ordering assumption failed, the failure would be an implementation bug, not a case of the derivation reducing to its own inputs. Consequently, no circular step meets the evidentiary bar required by the review rules, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fit to data, and the package introduces no new physical entities. It relies on standard bosonic commutation relations, Blasiak's formula from the literature, trace identities, and SymPy's noncommutative product handling.

assumptions (5)
  • domain assumption Canonical bosonic commutation relations [b, b†]=1 and cross-mode commutation [b_j, b†_k]=0 for j≠k
    Standard bosonic algebra assumed throughout, stated in Eqs. (5) and (10).
  • standard math Blasiak's formula (Eqs. 19-21) correctly computes the normal-ordered form of any single-mode ladder monomial
    The package's core algorithm is this formula; the paper cites Refs. [21,22] but does not re-prove it.
  • standard math Trace identities for Lindblad expectation values (Eqs. 25-26) are correct
    Derived in the text from the cyclic trace property; standard and independently verifiable.
  • domain assumption SymPy correctly handles noncommutative products of the custom operator classes
    The implementation depends on SymPy preserving operator order; stated only implicitly in Section 3.2.
  • domain assumption Multipartite monomials can be split by subscript, normal-ordered per mode, and multiplied without additional combinatorial corrections
    Follows from cross-mode commutation, but not rigorously proven in the paper; used in workflow steps 3-6 of Section 3.3.

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Cite this review

Pith. "Pith review of pyBoLaNO: A Python symbolic package for normal ordering involving bosonic ladder operators." pith.science (2026). https://pith.science/paper/O5KZFEGO

@misc{pith2026250101603,
  author       = {Pith},
  title        = {Pith review of: pyBoLaNO: A Python symbolic package for normal ordering involving bosonic ladder operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5KZFEGO}},
  note         = {Machine review of arXiv:2501.01603}
}
read the original abstract

We present pyBoLaNO, a Python symbolic package based on SymPy to quickly normal-order (Wick-order) any polynomial in bosonic ladder operators. By extension, this package offers the normal ordering of commutators of any two polynomials in bosonic ladder operators and the evaluation of the normal-ordered expectation value evolution in the Lindblad master equation framework for open quantum systems. The package also supports multipartite descriptions and multiprocessing. We describe the package's workflow, show examples of use, and discuss its computational performance. All codes and examples are available on our GitHub repository.

Figures

Figures reproduced from arXiv: 2501.01603 by the authors.

Figure 1
Figure 1. Core functionalities of pyBoLaNO . The main functions are shown in blue, while the core utility functions are colored black. Meanwhile, the purple-colored mp_config is a Python variable. Red connectors show symbolic workflows, while green connectors show visual workflows. Dashed rectangles indicate functionalities that are not accessible by the user. describe the open system dynamics. Each dissipator is defined by t… view at source ↗
Figure 2
Figure 2. Benchmarks of pyBoLaNO’s normal ordering algorithm against sympy.physics.operatorordering.normal ordererd form involving random ladder operator monomials. Benchmark (a): execution time of the normal ordering of 1000 monomials containing 10 ladder operators of 2 subsystems. Benchmark (b): Average of the ratio between the average execution time of the normal ordering of 1000 monomials by SymPy and pyBoLaNO, for varyin… view at source ↗

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Reference graph

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