Quantum Vortex Tomography

• Quantum vortex tomography is a set of techniques that reconstruct and analyze quantized vortices in diverse quantum systems, highlighting phase singularities and topological features.
• Methods include optical Wigner function reconstruction, STM-based Fano factor mapping, and quantum circuit analysis to extract vortex signatures.
• These approaches enhance the understanding of quantum coherence and topological defects, enabling precise diagnostics in quantum fluids and vortex-bound Majorana modes.

Quantum vortex tomography denotes the class of methodologies aimed at reconstructing and characterizing the structure, quantum state, and spatial distribution of vortices in quantum systems. These approaches are pivotal in the contexts of topological superconductivity, quantum fluids, and photonic non-classical states, where quantized vorticity underlies nontrivial topological and phase-coherence phenomena. Quantum vortex tomography leverages both direct quantum-state reconstruction—such as Wigner function tomography for photonic vortices—and efficient feature extraction from quantum data, including spatially resolved Fano-factor shot noise for vortex-bound Majorana modes and quantum circuit pipelines for flow vorticity encoded in qubit registers.

1. Theoretical Foundations of Quantum Vortex States

Vortices in quantum systems are generally associated with topological defects or localized current structures characterized by quantized phase winding. In photonic systems, a quantum vortex can be constructed by preparing two orthogonal bosonic modes (denoted $x$ and $y$) in squeezed–coherent states, then coupling them via an SU(2) unitary transformation using a beam splitter (BS) or dual-channel directional coupler (DCDC). The vortex structure is explicitly embedded by repeated application ($m$ times) of a creation-operator superposition $\eta_x\hat a_x'^\dagger+i\eta_y\hat a_y'^\dagger$ on the coupled vacuum, producing the normalized vortex state (Bandyopadhyay et al., 2010):

$$|\Psi_{\mathrm{vortex}}\rangle = \mathcal N\,\bigl(\eta_x\,\hat a_x'^\dagger+i\,\eta_y\,\hat a_y'^\dagger\bigr)^m\,D_x(\alpha_x)S_x(\xi_x)\otimes D_y(\alpha_y)S_y(\xi_y)\,|0,0\rangle$$

Topological superconductors host vortex-bound Majorana zero modes (MZMs), modeled in two-dimensional (2D) electron systems using the Bogoliubov–de Gennes (BdG) Hamiltonian with a spatially varying order parameter $\Delta(\mathbf{r})$ vanishing at core locations, exemplified in the Fu–Kane model (Mei et al., 2023):

$$H_{\mathrm{BdG}} = \begin{bmatrix} v_F(\mathbf{p}\cdot\boldsymbol{\sigma}) - \mu & \Delta(\mathbf{r}) \ \Delta^*(\mathbf{r}) & -\sigma_y [v_F(\mathbf{p}\cdot\boldsymbol{\sigma})^* - \mu] \sigma_y \end{bmatrix}$$

The vortex-induced in-gap bound states exhibit nontrivial electron–hole symmetry, which is a central diagnostic feature of MZM physics.

2. Quantum-State Tomography of Optical Vortices

Full quantum-state tomography of photonic vortex states proceeds through homodyne detection along rotated quadratures for each mode, scanning the local oscillator phases $\theta_x$ and $\theta_y$. The protocol entails measuring joint histograms $P(q_x,q_y;\theta_x,\theta_y)$ for quadrature eigenvalues over a grid of phase settings (Bandyopadhyay et al., 2010).

Reconstruction of the two-mode Wigner quasiprobability distribution employs the inverse Radon transform:

$$W(\alpha,\beta) = \frac{1}{4\pi^2} \int_0^\pi d\theta_x \int_0^\pi d\theta_y \int_{-\infty}^{\infty} dq_x dq_y \, P(q_x,q_y;\theta_x,\theta_y) K_{\mathrm{kernel}}(q_x,q_y,\theta_x,\theta_y;\alpha,\beta)$$

with kernel

$$K_{\mathrm{kernel}} = \exp[iq_x\,\mathrm{Im}(e^{i\theta_x}\alpha^*)+iq_y\,\mathrm{Im}(e^{i\theta_y}\beta^*)]$$

The resulting Wigner function for an elliptical quantum optical vortex assumes the closed form:

$$W(x,y,p_x,p_y) = K \exp[-(X_x^2+X_y^2+P_x^2+P_y^2)]\,L_m^{(-1/2)}\left(\frac{1}{2}[(\eta_xX_x+\eta_yX_y)^2 +(\eta_xP_x+\eta_yP_y)^2]\right)$$

where $L_m^{(-1/2)}$ is the associated Laguerre polynomial, and $X_i$, $P_i$ are appropriately displaced and scaled phase space coordinates (Bandyopadhyay et al., 2010).

Analysis of $W$ reveals vortex core position, phase singularities, topological charge $m$ (from the number of interference petals), and ellipticity $\eta_x/\eta_y$. These features are robust to generalizations involving higher-order vortex states and multimode couplings.

3. Spatially Resolved Tomography in Topological Superconductors

In superconducting vortex lattices, spatially resolved quantum vortex tomography is achieved via scanning tunneling microscopy (STM) shot-noise measurements. Here, the key observables are the spatially mapped current noise $S(x,y;V)$ and average current $I(x,y;V)$ at high bias $eV \gg \Gamma_j$ (where $\Gamma_j$ is the site-dependent tunneling broadening at point $(x,y)$), allowing extraction of the Fano factor profile,

$F(x,y) = \frac{S(x,y;V)}{2e\,|I(x,y;V)|}$

Majorana zero modes yield a robust plateau $F(x,y) \to 1$ within distances $r \lesssim \xi$ of vortex cores, reflecting exact particle–hole symmetry. In contrast, trivial Caroli–de Gennes–Matricon (CdGM) or Yu–Shiba–Rusinov (YSR) bound states display $F(x,y)$ oscillations spanning $[1,2]$ without a stable plateau. The salient theoretical result is that $F(j) \approx 1 + [\delta_{\mathrm{ph}}(j)]^2$, where $\delta_{\mathrm{ph}}$ parameterizes the local electron–hole imbalance (Mei et al., 2023). Importantly, this topological Fano factor signature is robust to MZM hybridization and suppresses false positives from other bound states.

4. Quantum Circuit Tomography of Vorticity in Quantum Data

Quantum vortex tomography applied to quantum data derived from PDE solutions (e.g., Navier–Stokes flows) operates on amplitude-encoded quantum states:

$$|\psi_f\rangle = \sum_{i,j} \psi_{ij} |i\rangle_x \otimes |j\rangle_y$$

where $\psi_{ij}$ is the classical vorticity at grid point $(i,j)$. Vortex-detection employs quantum circuits implementing feature pooling as follows (Williams et al., 30 Jun 2025):

• Sliding-window extraction: Shift $S(n_f,d)$ and permutation $P(n_f,l_1)$ bring spatial subblocks into focus.
• Contour-shaped pooling: Permutation $P(n_w,l_2)$ selects a circular pixel ring on a windowed patch, parameterized by an inverse radius $\beta$.
• Quantum Fourier Transform (QFT) on the contour register extracts rotational invariants; a strong response at low-$k$ modes signals azimuthal symmetry characteristic of vortices.

Parallelization is achieved by using an ancilla register in superposition, permitting the coherent extraction of windowed Fourier features over all positions in a single pass. Measurements with projector $\Pi_{\mathrm{low-freq}}$ distinguish vortex-present from vortex-free contours by the power in low-frequency modes. Sequential (scan-based) and parallel (global spectrum) strategies support both localization and classification of vortex structures. Hyperparameters $(\alpha, \beta, \gamma)$—step size, contour size, detection threshold—are optimized via classical Bayesian search to minimize detection error or maximize classification accuracy, routinely achieving F1 scores $\approx90\%$ on simulated data (Williams et al., 30 Jun 2025).

5. Experimental and Computational Considerations

Achieving robust quantum vortex tomography demands strict control of physical and computational parameters:

• For optical tomography, squeezing levels $r_i\gtrsim1$ ($\sim$9 dB) and homodyne detection efficiency $\eta\gtrsim85\%$ are required to resolve negative Wigner features and preserve vortex structure (Bandyopadhyay et al., 2010).
• Beam splitter or coupler asymmetry must be maintained with high precision to preserve vortex ellipticity.
• In STM-based Fano-factor tomography, instrument resolution ($\sim$0.1$\xi$, $<$nm scale) and integration times ($\sim$1 s per pixel) are necessary to discriminate plateaus versus oscillations of $F(x,y)$ within statistical error $\pm0.05$ (Mei et al., 2023).
• Quantum-circuit-based tomography leverages shallow ($O(n^2)$ gate depth) quantum circuits relying on shift, permutation, QFT, and projectors. Coherent parallelization reduces the necessity for $O(W)$ repeated experimental runs to $O(1)$, up to ancillary overhead (Williams et al., 30 Jun 2025).

6. Key Discriminants and Extensions

The distinguishing signature of genuine quantum vortices—whether Majorana zero modes, photonic vortices, or hydrodynamic flow regions—is a robust tomographic signal invariant under parameter variations:

• Fano-factor plateau $F=1$ for MZM cores, surviving hybridization and tunnel matrix fluctuations (Mei et al., 2023).
• Elliptical vortex phase singularity and Laguerre polynomial node patterns in the Wigner function for optical states (Bandyopadhyay et al., 2010).
• Fourier-mode peaks at $k=1,2$ for rotationally symmetric contours in quantum data encoding vorticity (Williams et al., 30 Jun 2025).

Extensions to higher-order vortex states and multi-vortex entangled states in both photonics and synthetic quantum data are algorithmically straightforward, requiring only additional homodyne channels or expandability in register and window constructions.

Table: Comparison of Quantum Vortex Tomography Modalities

Physical System	Tomographic Observable	Vortex Signature
Photonic optical vortex	Two-mode Wigner function $W$	Phase winding, $m$-petal structure
Topological superconductor vortex	Fano factor $F(x,y)$ via STM noise	$F=1$ plateau near core
Quantum data (PDE encoded)	Ancilla density spectrum/QFT modes	Low-$k$ peak, windowed contour power

7. Significance and Outlook

Quantum vortex tomography provides a powerful suite of diagnostic and characterization tools tailored to the diverse manifestations of quantized vorticity in quantum matter and quantum data representations. These protocols bypass the need for full quantum-state tomography where possible, leveraging topological and symmetry-protected observables obtainable with reasonable experimental and computational resources. Persistent research directions include scaling to larger system sizes, extending methodologies to nonequilibrium and strongly correlated vortex ensembles, and integrating quantum vortex tomography into feedback-driven control of topological quantum devices and quantum-fluid dynamics (Bandyopadhyay et al., 2010, Mei et al., 2023, Williams et al., 30 Jun 2025).