Quantum Skyrmions in Mixed States of Light and their Nested Topology

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.

• The paper introduces a novel framework that encodes skyrmionic topology directly in the density matrix of mixed quantum states of light for robustness against decoherence.
• It employs minimal eigenmode decomposition and entangled biphoton states to establish nested topological structures resilient to noise.
• The study proposes an integrated photonic platform for scalable quantum sensing, linking programmable phase transitions to enhanced metrology.

Quantum Skyrmions in Mixed States of Light and their Nested Topology

Introduction and Theoretical Framework

This work introduces an approach for encoding topological quantum numbers associated with skyrmions directly within the density matrices of mixed quantum states of light, extending the scope of skyrmion realization fundamentally beyond prior demonstrations limited to pure or fully coherent states. Skyrmions, which are topological quasiparticles, classically and quantum mechanically, have been extensively studied in the context of magnetism, particle physics, and photonics. Here, the focus is set on optical and quantum skyrmions whose topology is intrinsically encoded in entanglement between a pseudospin (such as polarization) and a one-dimensional mode space (spatial or synthetic), with robust persistence in the presence of decoherence or environmental mixing.

A key insight is the formal identification of a topological texture directly on the density matrix—treating the coherence terms as a Stokes-like vector field, denoted as the coherence Stokes texture $\mathbf{s}(x,x')$, which supports winding numbers quantized as the skyrmion charge. This unifies the treatment of classical partial coherence and quantum mixed states.

Figure 1: (a) Skyrmion encoded in 1D partial coherence; (b) minimal eigenvectors required for a target skyrmion number; (c-d) density matrix and associated skyrmion texture.

Analysis reveals that the density matrix, in principle, admits a spectral decomposition; however, only a minimal number $d = |Q|+1$ of eigenmodes are necessary to reconstruct a skyrmion of charge $Q$, yielding compressed, low-rank encodings. Construction proceeds via formulating an auxiliary (not necessarily physical) Hermitian matrix with the desired texture, followed by its PSD spectral truncation—ensuring the resulting density matrix encodes the designed skyrmion.

Entangled and Nested Skyrmions in Mixed States

The formalism is generalized from single photons to entangled biphoton states. Skyrmions can be encoded using maximally entangled states over polarization and mode, producing both local (single-particle) and non-local (hybridized degrees of freedom) skyrmion topologies. Crucially, the skyrmion structure is not confined to the joint state but persists in reduced density matrices corresponding to all combinations of pseudospin and external degrees of freedom—whether traced locally or non-locally.

Figure 2: (a) Topological charge in an entangled two-photon system; (b) local skyrmions in reduced subsystems; (c) non-local (hybrid) skyrmions in mixed subspaces.

This phenomenon, identified as "nested topology," is characterized by the coexistence of multiple, interdependent topological charges in the various reduced subsystems extracted from the global entangled state. In contrast to earlier optical skyrmion realizations, topology in this setting is not exclusive to the full quantum state and survives partitioning, providing significant implications for decoherence-resilient protocols.

Robustness to Noise and Topological Transition

A critical result is the quantification and demonstration of robustness of both local and non-local skyrmions under noise channels: specifically, under Gaussian phase randomization and additive mixture with random high-rank noise (Wishart) matrices. The local skyrmion number $Q$ remains invariant under arbitrarily strong local dephasing, while the non-local skyrmion retains quantization up to a critical noise threshold—beyond which entanglement-based topology is destroyed but local topology persists. The topological transition is shown to depend parametrically on the number of encoding modes.

Figure 3: (a) Skyrmion number vs. dephasing strength for reduced subspaces; (b) scaling of non-local skyrmion robustness with mode number; (c) invariance of skyrmion charge under additive Wishart noise; (d) texture transition from Néel- to Bubble-type.

Notably, for large mode numbers, the non-local skyrmion also acquires noise-immunity due to a transition from a Bloch-type to Bubble-type skyrmion texture—demonstrating that the underlying topology can be multiplexed and made resilient by engineering the encoding structure.

Nested Topology in Multiphoton Mixed States

The construction is generalized to multiphoton mixed states, specifically via mixtures of bipartite Bell skyrmion states embedded within mixtures over larger Hilbert spaces. Even after tracing out $N-2$ photons from a $N$-photon system, the reduced two-photon state exhibits nested topology in all plausible local and hybrid subspaces. This is substantiated explicitly through numerical generation and evaluation of the skyrmion charge in various reduced statistics.

Figure 4: Multi-photon generalization—nested topology in Bell and mixed states, as evidenced by local, non-local, and hybrid reduced textures.

These results highlight the prospects for use of nested topological charges in robust many-body encoding, where the destruction of coherence or loss of subsystem access does not eradicate the topological signature of the state.

Experimental Implementation and Sensing Applications

A practical experimental path is proposed via integrated photonic platforms. The architecture leverages path and polarization entanglement, robust modal engineering (via MZI meshes), and programmable basis transformations to prepare and measure the constructed quantum skyrmion states in realistic chip-scale devices.

Figure 5: (a) Integrated photonic setup for skyrmion generation and analysis; (b) measured skyrmion number as a function of modulated phase, demonstrating sharp topological transitions usable for sensing.

The chip-based system enables high-fidelity realization of the modal and polarization superpositions required, with scalable extension to higher values of $|Q|$ and more complex entangled states. Importantly, the abrupt transition of topological charge as a function of programmable phase (Fig. 4b) links the framework directly to phase-enhanced sensing and metrology, highlighting one clear practical application.

Conclusion

This work establishes that skyrmionic topology can be encoded, manipulated, and made robust in quantum mixed states via direct structuring of the density matrix, with a unifying framework connecting classical coherence and quantum information. Encoding in lower-dimensional, low-rank representations reduces physical resource usage and measurement complexity, yielding scalable architectures for topological quantum information. The concept of nested topology is shown to transcend simple bipartite entanglement, underlying a distributed, persistent hierarchy of topological invariants observable in both local and non-local subsystems—even amid strong decoherence.

These advancements suggest practical avenues for topological encoding in quantum photonics and general quantum many-body systems (including atoms and ions), provide robust topological signatures for metrology, and motivate further study of density-matrix based realization of other quasiparticle topologies with complex nesting and resilience properties.