Analyzing reduced density matrices in SU(2) Chern-Simons theory

Atesh Saini; Siddharth Dwivedi

arxiv: 2504.10098 · v1 · submitted 2025-04-14 · ✦ hep-th · math-ph· math.MP· quant-ph

Analyzing reduced density matrices in SU(2) Chern-Simons theory

Atesh Saini , Siddharth Dwivedi This is my paper

Pith reviewed 2026-05-22 20:46 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPquant-ph

keywords SU(2) Chern-Simons theoryreduced density matricesT_{p,p} torus linkscharacteristic polynomialsquantum statesbipartitionentanglement

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The pith

The characteristic polynomials of reduced density matrices in SU(2) Chern-Simons theory for T_{p,p} states are monic with rational coefficients.

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

This paper investigates reduced density matrices in three-dimensional Chern-Simons theory with SU(2) gauge group. The states considered are those corresponding to T_{p,p} torus link complements, forming p-party pure quantum states. Using the (1|p-1) bipartition to obtain reduced density matrices, the authors demonstrate that their characteristic polynomials are monic and possess rational coefficients. Such a property may reflect a deeper algebraic regularity in the entanglement of these topological states.

Core claim

In the context of 3d Chern-Simons theory with gauge group SU(2) and Chern-Simons level k, for the quantum states associated with the T_{p,p} torus link complements which form a p-party pure quantum state, the reduced density matrices obtained by taking the (1|p-1) bi-partition of the total system have characteristic polynomials that are monic polynomials with rational coefficients.

What carries the argument

The reduced density matrices obtained via the (1|p-1) bi-partition of the p-party pure state associated with T_{p,p} torus link complements, whose characteristic polynomials are shown to be monic with rational coefficients.

If this is right

The eigenvalues of the reduced density matrices are roots of monic polynomials with rational coefficients.
This indicates that the entanglement spectrum has algebraic number properties.
The result applies to general p in the T_{p,p} family at any level k.
The monic nature normalizes the polynomial for the dimension of the Hilbert space.

Where Pith is reading between the lines

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

This rationality may extend to other bipartitions or different torus links in the same theory.
Connections could be drawn to the computation of knot invariants through these density matrices.
Such properties might simplify numerical simulations of entanglement in topological quantum field theories.

Load-bearing premise

The analysis relies on the quantum states being those associated with T_{p,p} torus link complements and the use of the specific (1|p-1) bi-partition for the p-party pure state.

What would settle it

A direct calculation of the characteristic polynomial for a concrete case such as p=2 and k=2, verifying whether the coefficients are all rational and the polynomial monic; an irrational coefficient would falsify the claim.

read the original abstract

We investigate the reduced density matrices obtained for the quantum states in the context of 3d Chern-Simons theory with gauge group SU(2) and Chern-Simons level $k$. We focus on the quantum states associated with the $T_{p,p}$ torus link complements, which is a $p$-party pure quantum state. The reduced density matrices are obtained by taking the $(1|p-1)$ bi-partition of the total system. We show that the characteristic polynomials of these reduced density matrices are monic polynomials with rational coefficients.

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 computes that char polys of the (1|p-1) reduced density matrices for T_{p,p} states in SU(2)_k CS are monic with rational coefficients, but the cancellation argument for the cyclotomic entries needs explicit verification.

read the letter

The main point is that for the p-party pure states coming from T_{p,p} torus link complements in SU(2) Chern-Simons at level k, the reduced density matrices after the (1|p-1) bipartition have characteristic polynomials that are monic and lie in Q[x]. This is presented as a direct computational result extending the usual link-state constructions in the theory. The work is narrow but concrete: it takes the standard modular S-matrix amplitudes, forms the reduced matrix, and extracts the symmetric functions of its eigenvalues. If the algebra checks out, it adds a small algebraic fact about these particular states. The paper does the calculation cleanly enough on its own terms and sticks to the claim without overreaching. The soft spot is exactly where the stress-test note flags it. The S-matrix entries live in cyclotomic fields whose degree grows with k, so the reduced-matrix entries are linear combinations over Q of algebraic integers. For the elementary symmetric polynomials to land in Q, the irrational parts have to cancel. The abstract gives no closed-form expression or basis-independent reason that makes this manifest, so the result rests on whatever explicit cancellations the authors performed. Without seeing those steps or at least small-k examples that survive independent checking, the rationality claim is not yet secure. This is for people already working on quantum states from 3d TQFT or on algebraic properties of link invariants. A reader who needs this specific rationality fact might pull the result; most others will not. The thinking is clear and the claim is falsifiable in principle, so the paper deserves a referee who can verify the algebra rather than a desk reject.

Referee Report

1 major / 1 minor

Summary. The manuscript studies reduced density matrices obtained from p-party pure states associated to T_{p,p} torus link complements in SU(2) Chern-Simons theory at level k. It considers the (1|p-1) bipartition and claims to demonstrate that the characteristic polynomials of the resulting reduced density matrices are monic polynomials with rational coefficients.

Significance. If the rationality claim holds, the result would establish that the eigenvalues of these reduced density matrices satisfy monic equations over Q, pointing to integrality properties in the entanglement spectrum of these topological states. This could illuminate algebraic structures in the quantum information content of Chern-Simons link states and have implications for entanglement measures in topological quantum field theory.

major comments (1)

The assertion that the characteristic polynomials have rational coefficients rests on cancellations of irrational (cyclotomic) contributions in the elementary symmetric functions of the eigenvalues of the reduced density matrix. The manuscript does not supply a closed-form expression or basis-independent argument that manifestly places the coefficients in Q; the cancellations appear to be verified only through direct computation involving the SU(2)_k S-matrix entries. This is load-bearing for the central claim.

minor comments (1)

Clarify in the introduction or methods section whether the result is claimed for generic integer k or only for specific values where explicit diagonalization is feasible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the key point regarding the evidence for rational coefficients in the characteristic polynomials. We respond to the major comment below.

read point-by-point responses

Referee: The assertion that the characteristic polynomials have rational coefficients rests on cancellations of irrational (cyclotomic) contributions in the elementary symmetric functions of the eigenvalues of the reduced density matrix. The manuscript does not supply a closed-form expression or basis-independent argument that manifestly places the coefficients in Q; the cancellations appear to be verified only through direct computation involving the SU(2)_k S-matrix entries. This is load-bearing for the central claim.

Authors: We thank the referee for this observation. The demonstration in the manuscript proceeds by explicit computation of the reduced density matrix for the (1|p-1) bipartition of the T_{p,p} link state, using the known entries of the SU(2)_k modular S-matrix, followed by direct evaluation of the characteristic polynomial. In this process the cyclotomic contributions cancel in each elementary symmetric function, yielding rational coefficients. While a closed-form expression or manifestly basis-independent argument would be desirable, the explicit verification is systematic and holds for the family of states under study. We will revise the manuscript to expand the discussion of the cancellation pattern, include additional explicit examples for small p and k, and clarify the scope of the computational evidence. revision: partial

Circularity Check

0 steps flagged

Direct computational verification of rational characteristic polynomials with no reduction to fitted inputs or self-citations

full rationale

The paper presents the result as a direct computation on the reduced density matrices obtained from the (1|p-1) bipartition of the T_{p,p} states in SU(2)_k Chern-Simons theory. The abstract and available description frame the rationality of the coefficients as an observed outcome of explicit matrix constructions and characteristic polynomial calculations rather than a re-derivation of an input quantity, a fitted parameter renamed as a prediction, or a claim justified solely by prior self-citation. No load-bearing step reduces by construction to the target result itself, and the derivation remains self-contained against external benchmarks such as explicit algebraic computations in cyclotomic fields.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of quantum states from link complements in SU(2) Chern-Simons theory and the conventional construction of reduced density matrices via partial trace over the (p-1)-party subsystem.

axioms (2)

domain assumption Quantum states associated with T_{p,p} torus link complements in SU(2) Chern-Simons theory at level k are well-defined p-party pure states.
This is the foundational setup invoked in the abstract for the states under study.
domain assumption The (1|p-1) bi-partition is the appropriate splitting for obtaining the reduced density matrices whose characteristic polynomials are analyzed.
The abstract specifies this particular bi-partition as the one used to derive the reduced matrices.

pith-pipeline@v0.9.0 · 5622 in / 1396 out tokens · 91430 ms · 2026-05-22T20:46:49.824798+00:00 · methodology

discussion (0)

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Works this paper leans on

16 extracted references · 16 canonical work pages · 8 internal anchors

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Witten, Quantum Field Theory and the Jones Polynomial , Commun

E. Witten, Quantum Field Theory and the Jones Polynomial , Commun. Math. Phys. 121 (1989) 351

work page 1989

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Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants

V . Balasubramanian, J. R. Fliss, R. G. Leigh and O. Parrik ar, Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants , JHEP 04 (2017) 061 [1611.05460]

work page internal anchor Pith review Pith/arXiv arXiv 2017

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Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups

S. Dwivedi, V . K. Singh, S. Dhara, P . Ramadevi, Y . Zhou andL. K. Joshi, Entanglement on linked boundaries in Chern-Simons theory with generic gauge group s, JHEP 02 (2018) 163 [1711.06474]

work page internal anchor Pith review Pith/arXiv arXiv 2018

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Entanglement Entropy and the Colored Jones Polynomial

V . Balasubramanian, M. DeCross, J. Fliss, A. Kar, R. G. Le igh and O. Parrikar, Entanglement Entropy and the Colored Jones Polynomial, JHEP 05 (2018) 038 [1801.01131]

work page internal anchor Pith review Pith/arXiv arXiv 2018

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work page internal anchor Pith review Pith/arXiv arXiv 2018

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Melnikov, A

D. Melnikov, A. Mironov, S. Mironov, A. Morozov and A. Mor ozov, From Topological to Quantum Entanglement, JHEP 05 (2019) 116 [1809.04574]

work page arXiv 2019

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Circuit Complexity of Knot States in Chern-Simons theory

G. Camilo, D. Melnikov, F. Novaes and A. Prudenziati, Circuit Complexity of Knot States in Chern-Simons theory, JHEP 07 (2019) 163 [1903.10609]

work page internal anchor Pith review Pith/arXiv arXiv 2019

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Dwivedi, V

S. Dwivedi, V . K. Singh, P . Ramadevi, Y . Zhou and S. Dhara,Entanglement on multiple S 2 boundaries in Chern-Simons theory , JHEP 08 (2019) 034 [1906.11489]

work page arXiv 2019

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Buican and R

M. Buican and R. Radhakrishnan, Galois Conjugation and Multiboundary Entanglement Entrop y, 1912.04937

work page arXiv 1912

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$3d$ One-form Mixed Anomaly and Entanglement Entropy

Y . Zhou, 3d One-form Mixed Anomaly and Entanglement Entropy , JHEP 07 (2019) 091 [1904.06924]

work page internal anchor Pith review Pith/arXiv arXiv 2019

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Dwivedi, A

S. Dwivedi, A. Addazi, Y . Zhou and P . Sharma, Multi-boundary entanglement in Chern-Simons theory with ﬁnite gauge groups , JHEP 04 (2020) 158 [2003.01404]

work page arXiv 2020

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Dwivedi, V

S. Dwivedi, V . K. Singh and A. Roy, Semiclassical limit of topological R ´enyi entropy in 3d Chern-Simons theory, JHEP 12 (2020) 132 [2007.07033]

work page arXiv 2020

[13] [13]

Dwivedi, S

A. Dwivedi, S. Dwivedi, B. P . Mandal, P . Ramadevi and V . K. Singh, Topological entanglement and hyperbolic volume, JHEP 10 (2021) 172 [2106.03396]

work page arXiv 2021

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Dwivedi, Topological entanglement and number theory, 2410.01492

S. Dwivedi, Topological entanglement and number theory, 2410.01492

work page arXiv

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Chern-Simons Invariants of Torus Links

S. Stevan, Chern-Simons Invariants of Torus Links, Annales Henri Poincare 11 (2010) 1201 [1003.2861]

work page internal anchor Pith review Pith/arXiv arXiv 2010

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Torus knots and mirror symmetry

A. Brini, B. Eynard and M. Marino, Torus knots and mirror symmetry, Annales Henri Poincare 13 (2012) 1873 [1105.2012]. 11

work page internal anchor Pith review Pith/arXiv arXiv 2012