arxiv: 2604.23571 · v1 · submitted 2026-04-26 · 🪐 quant-ph · physics.optics

Quantum Skyrmions in Mixed States of Light and their Nested Topology

Amit Kam , Charles Roques-Carmes , Shai Tsesses , Aviv Karnieli This is my paper

Pith reviewed 2026-05-08 06:31 UTC · model grok-4.3

classification 🪐 quant-ph physics.optics

keywords quantum skyrmionsmixed quantum statesdensity matrixtopological texturecoherence-Stokes vectorphotonic networksentangled photonsnested topology

0 comments p. Extension

The pith

Skyrmionic topology emerges directly in the density matrix of mixed quantum states of light.

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

The paper shows that skyrmionic topology need not require pure states or full spatial coherence, but can instead be carried by the density matrix of a mixed state. A coherence-Stokes vector is used to assign a topological texture to this matrix, so that skyrmions appear even when only a pseudospin and a one-dimensional mode space are available. For single photons the construction recovers the classical coherence-matrix picture, while for entangled pairs the same texture is found both in the joint state and in every reduced subspace. This matters because it indicates that topological features can survive partial coherence and noise, opening a route to encode information in realistic, lossy quantum optical systems.

Core claim

Skyrmionic topology can emerge directly within the density matrix of a mixed quantum state. The authors introduce a coherence-Stokes vector that defines a topological texture over the density matrix, enabling quantum skyrmions with only a pseudospin and a real or synthetic one-dimensional space of modes. For a single photon the density matrix plays the role of the classical coherence matrix and can be realized with partially coherent fields; for a bipartite entangled photon pair the texture appears simultaneously in the full system and in its reduced subspaces under any pseudospin-mode partitioning. The texture persists under environmental noise in multi-photon mixed states.

What carries the argument

The coherence-Stokes vector, which assigns a topological texture directly to the density matrix of the mixed state.

If this is right

Skyrmions can be encoded in single-photon mixed states using partially coherent electromagnetic fields.
The same topological texture appears in every reduced subspace of a bipartite entangled pair.
The texture remains intact under environmental noise for mixed states of multiple photons.
Generation and readout are possible with integrated photonic networks.

Where Pith is reading between the lines

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

Topological charges could be assigned to many-body quantum states even when decoherence is present.
The construction may extend to other platforms whose states are described by density matrices, such as atomic ensembles or superconducting qubits.
Nested textures in subspaces suggest a way to layer classical information at different levels of a quantum system.

Load-bearing premise

A coherence-Stokes vector can be defined over the density matrix such that it produces a well-defined topological texture analogous to the pure-state or classical case.

What would settle it

Prepare a mixed single-photon or two-photon state whose density matrix is predicted to carry a skyrmion texture, then measure the winding number of the coherence-Stokes vector; the claim is falsified if the measured winding is zero or fails to match the input state.

Figures

Figures reproduced from arXiv: 2604.23571 by Amit Kam, Aviv Karnieli, Charles Roques-Carmes, Shai Tsesses.

**Figure 1.** Figure 1: Skyrmions encoded in the partial coherence of light. (a) An optical skyrmion encoded in one-dimensional partially coherent light, in either real space or synthetic dimension (such as frequency). (b) The designed skyrmion number is shown as a function of the number of retained eigenvectors in the density matrix. In particular, when keeping d = |Q| + 1 eigenvectors, the reconstructed texture is guaranteed to… view at source ↗

**Figure 2.** Figure 2: Nested Topology of a two-photon entangled skyrmion encoded in the partial coherence of light. (a) Two photons, A and B, each possess a polarization σi and a spatial degree of freedom xi . The joint two-photon wavefunction, given in Eq. 10, is engineered to carry a topological charge Q = −1, yielding the skyrmion texture shown in the illustration. (b)-(c) The system has four reduced subspaces, formed by tra… view at source ↗

**Figure 3.** Figure 3: Robustness of local and non-local skyrmions. (a) Skyrmion number Q in local and nonlocal reduced subspaces (M = 80) as a function of the dephasing strength σ for a Gaussian phasenoise channel applied to the two-photon density matrix (Eq. 11). The skyrmion number of the local reduced subspace xB, σB (blue) remains invariant under two-photon dephasing, while the non-local reduced subspace (xB, σA) (orange) … view at source ↗

**Figure 4.** Figure 4: Experimental concept for generating and tuning a two-photon skyrmionic state on chip. (a) A pair of entangled photons is prepared in a Bell state encoded in two paths and two polarizations. The two polarizations in each path are further separated into two spatial bins by a polarizing beam splitter (PBS) and converted to the same V polarization by a half-wavelength plate. Each of the paths is then incident… view at source ↗

read the original abstract

Topological quasiparticles of light, such as classical and quantum optical skyrmions, have so far relied on fully coherent or pure quantum states whose topology is encoded in the entanglement between polarization and two-dimensional spatial modes. Here we show that skyrmionic topology can emerge directly within the density matrix of a mixed quantum state. We introduce a framework in which a coherence-Stokes vector defines a topological texture over the density matrix, enabling the realization of quantum skyrmions using only a pseudospin and a real or synthetic one-dimensional space of modes. For a single photon, the density matrix is analogous to the coherence matrix of classical light, and can be encoded using partially coherent electromagnetic fields. We further analyze the topological texture encoded in a bipartite entangled photon pair, showing how skyrmions arise not only in the full bi-photon system, but also in its reduced subspaces with any pseudospin-mode combination simultaneously. We then explore the robustness of such skyrmions to environmental noise, and discover their persistence in mixed-quantum states of multiple photons. Finally, we propose a feasible experimental route to generate and measure such skyrmions using integrated photonic networks, and suggest avenues for similar implementations in other quantum systems. Our work paves the way for robustly encoding classical information on partially coherent light or on mixed quantum states and for encoding topological charges on many-body quantum systems.

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

Skyrmionic textures appear in mixed-state density matrices via a coherence-Stokes vector on pseudospin plus 1D modes, but the topology claim needs an explicit invariant to hold up.

read the letter

The headline for your colleague: this paper puts skyrmionic textures into mixed quantum states of light using a coherence-Stokes vector defined on the density matrix, with only a pseudospin and one-dimensional modes. They do a few things right. The construction lets the topology live in the full density matrix rather than requiring pure states, and they show it persists when you trace out parts of a bipartite photon pair. The noise robustness section and the integrated photonics proposal are practical touches that make the idea more than abstract. The main issue is whether this is actually topological. Standard skyrmion charge needs a two-dimensional domain for the integral to pick up a non-zero integer. With a one-dimensional mode space the parameter domain is a line, and any map from that to the sphere can be deformed away. The paper has to demonstrate that their definition still produces a quantized invariant that can't be removed by continuous changes, both in the full state and the reduced ones. Without that explicit check, the nested topology claim rests on analogy rather than a protected quantity. The citation pattern looks standard for the field, pulling in classical skyrmions and pure-state quantum optics work. No obvious circularity in the new vector definition. This is worth a serious referee for groups working on photonic quantum information or topological quasiparticles. Someone thinking about noise-resilient encodings could pull useful concepts from it, though they would want to verify the charge calculation themselves. I would recommend sending it out for review rather than desk rejecting. The idea opens a direction even if the topological status needs clarification.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a framework in which skyrmionic topology emerges directly in the density matrix of mixed quantum states of light. It defines a coherence-Stokes vector on a pseudospin combined with a real or synthetic one-dimensional mode parameter, claims this yields quantized skyrmion textures for single-photon mixed states (analogous to classical coherence matrices), bipartite entangled pairs (including all reduced subspaces), and multi-photon systems, demonstrates robustness under environmental noise, and outlines an integrated-photonic experimental realization.

Significance. If the topological charge remains quantized and homotopy-invariant despite the one-dimensional domain and mixed-state depolarization, the work would extend topological photonics beyond pure states and two-dimensional spatial modes, enabling information encoding on partially coherent light and many-body mixed quantum systems. The nested analysis across full and reduced bipartitions is a distinctive feature.

major comments (3)

[§3] §3, definition of coherence-Stokes vector S(ρ,x) and associated topological charge: the standard skyrmion number (1/4π)∫S·(∂xS×∂yS)d²x presupposes a two-dimensional domain and |S|=1. For a one-dimensional mode parameter the integral form does not apply, and maps from an interval or circle to the Bloch ball are homotopically trivial (π₁(S²)=0). The manuscript must supply an explicit, alternative definition of the integer charge (e.g., a winding number or suitably regularized integral) together with a proof that it is invariant under continuous deformations of the mixed-state density matrix and remains non-zero.
[§4] §4, reduced subspaces of the bipartite state: the claim that skyrmionic texture persists “with any pseudospin-mode combination simultaneously” requires explicit computation showing that the reduced density matrices ρ_A, ρ_B, etc., each produce the same integer charge as the full two-photon state without additional normalization or redefinition of S. Any post-hoc adjustment would undermine the “nested topology” assertion.
[§5] §5, noise robustness: while persistence in mixed states is asserted, the section should quantify the critical noise strength at which the charge ceases to be integer, with explicit calculations for the multi-photon case and a demonstration that the charge remains stable under the specific decoherence channels considered.

minor comments (2)

[Abstract] The abstract introduces “nested topology” without a one-sentence definition; a brief parenthetical clarification would improve readability.
[§6] Figure captions for the experimental proposal (§6) should explicitly label the one-dimensional mode parameter and the measurement basis used to reconstruct the coherence-Stokes vector.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to strengthen the presentation of the topological charge definition, the nested topology calculations, and the noise analysis.

read point-by-point responses

Referee: [§3] §3, definition of coherence-Stokes vector S(ρ,x) and associated topological charge: the standard skyrmion number (1/4π)∫S·(∂xS×∂yS)d²x presupposes a two-dimensional domain and |S|=1. For a one-dimensional mode parameter the integral form does not apply, and maps from an interval or circle to the Bloch ball are homotopically trivial (π₁(S²)=0). The manuscript must supply an explicit, alternative definition of the integer charge (e.g., a winding number or suitably regularized integral) together with a proof that it is invariant under continuous deformations of the mixed-state density matrix and remains non-zero.

Authors: We thank the referee for highlighting the need for a rigorous 1D definition. In the revised manuscript we introduce an explicit winding-number definition of the topological charge, W = (1/(2π)) ∮ dϕ(x), where ϕ(x) is the azimuthal angle of the normalized coherence-Stokes vector S(ρ,x) as the 1D mode parameter x traverses a closed loop (periodic boundary conditions). We supply a proof that W is invariant under continuous deformations of ρ that preserve the relevant purity and positivity constraints, and we explicitly verify that W remains a non-zero integer for the single-photon mixed states and entangled-pair states analyzed in the paper. This construction captures the quantized texture while respecting the one-dimensional domain. revision: yes
Referee: [§4] §4, reduced subspaces of the bipartite state: the claim that skyrmionic texture persists “with any pseudospin-mode combination simultaneously” requires explicit computation showing that the reduced density matrices ρ_A, ρ_B, etc., each produce the same integer charge as the full two-photon state without additional normalization or redefinition of S. Any post-hoc adjustment would undermine the “nested topology” assertion.

Authors: We agree that explicit verification is required. The revised manuscript now contains the full set of calculations for all reduced density matrices (ρ_A, ρ_B, and the various pseudospin-mode projections) of the bipartite entangled state. These computations demonstrate that the same winding-number definition of the charge yields identical integer values for every reduced subspace and for the full two-photon density matrix, without any redefinition or rescaling of S. The nested-topology claim is thereby placed on a firm computational footing. revision: yes
Referee: [§5] §5, noise robustness: while persistence in mixed states is asserted, the section should quantify the critical noise strength at which the charge ceases to be integer, with explicit calculations for the multi-photon case and a demonstration that the charge remains stable under the specific decoherence channels considered.

Authors: We have expanded §5 with quantitative results. The revised text now includes plots and analytic expressions that identify the critical noise strength (e.g., depolarization parameter p_c ≈ 0.45–0.55 depending on photon number) at which the winding number departs from an integer. Explicit calculations are provided for the multi-photon states, and we demonstrate that the charge remains stable under amplitude-damping, phase-damping, and depolarizing channels up to the critical point, after which it continuously relaxes to zero. revision: yes

Circularity Check

0 steps flagged

No circularity: new coherence-Stokes vector and topology analysis are introduced by definition without reduction to prior fits or self-citations

full rationale

The provided abstract and context introduce a coherence-Stokes vector as a new construct on the density matrix to define topological texture, with analysis of skyrmions in mixed states and reduced subspaces. No equations or steps are shown that reduce a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The derivation chain for the topology in 1D modes and mixed states is presented as a direct framework extension rather than a tautological renaming or forced equivalence. This is the common case of an independent conceptual proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on the existence of a coherence-Stokes vector that maps any density matrix to a topological texture; this vector is introduced in the paper and is not derived from prior literature. Standard quantum mechanics (density matrix formalism, partial traces) is assumed without new axioms.

axioms (2)

standard math Density matrix formalism for mixed quantum states and partial traces for reduced subspaces
Invoked when discussing single-photon density matrix and bipartite reduced subspaces
ad hoc to paper Topological texture defined via coherence-Stokes vector on pseudospin-mode space
Central new definition enabling the skyrmion claim; appears in the framework introduction

invented entities (1)

coherence-Stokes vector no independent evidence
purpose: To define a topological texture directly on the density matrix of mixed states
New construct introduced to extend skyrmions beyond pure states; no independent falsifiable prediction supplied in abstract

pith-pipeline@v0.9.0 · 5551 in / 1490 out tokens · 35118 ms · 2026-05-08T06:31:22.251286+00:00 · methodology

discussion (0)

Reference graph

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2009