wigner-splat

Constructively physical Gaussian representations for continuous-variable quantum-state tomography.

wigner-splat is a reproducible research prototype for fitting homodyne measurements without making a truncated Fock density matrix the primary model. It began from a Gaussian-splatting view of the Radon projections of a Wigner function, and now also contains constructively physical $\rho = BB^\dagger$ models with closed-form homodyne likelihoods and analytic gradients.

Research status. This is preliminary research software, not a claim of a universally superior tomography method. The repository preserves negative results, scope limits, and the exact experiments behind every headline below. The complete chronological research narrative is available in Japanese.

Highlights

• Real measured data. On the public propagating-light GKP homodyne dataset (Konno et al., Science 2024), a 92-parameter physical model ties the test-selected full-rank MLE frontier (255 parameters) at confidence-interval resolution (experiment 18).
• A recorded out-of-family win. On a synthetic full-rank thermal-noise target, the loss-channel-composed rank-2 model fitted blind reached fidelity 0.923 against 0.898 for a converged full-rank MLE with ~2.6×10⁵ parameters (experiment 19) — and the target is provably outside the model family: experiment 20 shows analytically that no efficiency and no finite rank can represent it exactly, while best approximations stop ≈2–3×10⁻³ short at the largest scored cutoff (n = 12).
• An honest scaling story. One mode: MLE is faster. Two modes: statistical tie at ~1/7.4 the compute. Three modes: the signed-splat score is not physical fidelity (non-PSD) and the repository says so.
• Falsification-first records. The formal gates and later experiments pre-declare their protocols and falsification conditions before running (the earliest experiments were exploratory and are recorded as such); negative results, scoring corrections, and superseded logs are all kept in the research log.

Visual demos: signed splats as a medium

The repository's signed (negative-weight) splats also work as a visual medium on ordinary 3D Gaussian-splatting scenes — negative splats subtract light instead of occluding, which enables effects that are difficult to express with ordinary non-negative-alpha 3DGS compositing.

Three 12-second animated-camera videos (players below; the archived files live in experiments/22_signed_splat_demo/media/):

Eraser — an invisible sphere honestly erases the object, revealing what was actually scanned behind it (file):

cc0-cactus-eraser-orbit.mp4

Dark flashlight — a beam of darkness: a cone filled with low-alpha negative splats (file):

cc0-cactus-dark-flashlight-orbit.mp4

Annihilation — the object meets its negative copy and the overlap cancels to nothing (file):

cc0-cactus-annihilation-orbit.mp4

See experiments/22_signed_splat_demo/ for code, provenance (CC0 cactus scene), and prior-art notes (negative Gaussians appear in NegGS, arXiv:2405.18163, for reconstruction quality; here they are used expressively).

What is being explored

Homodyne tomography observes one-dimensional quadrature marginals at local oscillator phases. The project investigates two related but distinct representations:

1. Signed anisotropic Gaussian mixtures ("splats"). A closed-form, differentiable Radon forward model fits Wigner-function marginals. Signed weights can express Wigner negativity, and gradient-driven birth, splitting, and pruning adapt the mixture.
2. Physical Gaussian-ket mixtures. Finite mixtures of displaced and squeezed product kets are fitted by per-sample likelihood. Their $BB^\dagger$ construction guarantees a positive-semidefinite density operator; loss and finite-rank extensions model mixed states.

The 3D Gaussian-splatting analogy is useful because a camera view corresponds to a homodyne phase, splatting corresponds to a phase-space Gaussian primitive, and rendering corresponds to a Radon projection. It is an origin and an experimental representation—not a claim that computer-vision 3DGS itself solves quantum tomography.

Evidence at a glance

Synthetic multimode tests

The signed-splat track was tested against iterative $R\rho R$ maximum likelihood estimation (MLE) on simulated cat states at matched shot budgets.

Setting	Observed result	Scope limit
1 mode	Splat has slightly higher score, but MLE is about 2x faster.	No computational advantage at this scale.
2 modes	Fidelity is statistically indistinguishable across 20 paired seeds; splat uses about 1/7.4 of the measured compute.	Requires full cross-mode covariance; separable splats fail.
3 modes	A signed-splat run reached a higher Wigner-overlap score in about 15 s while the 512-dimensional MLE run did not converge within 900 s.	This score is not state fidelity when the reconstruction is non-PSD. It is not a physical-tomography win.

The physical $BB^\dagger$ track resolves the PSD issue by construction. Its early synthetic high-fidelity results were in-family existence results, and a fair Fock-ket comparison showed that its main demonstrated advantage is compactness and speed, not a general fidelity advantage. A held-out full-rank gate (experiment 19) later recorded blind held-out performance above a converged full-rank MLE: on a thermal-noise lossy cat that no finite-rank ket mixture contains, the loss-channel-composed rank-2 model reached fidelity 0.923 against the MLE's 0.898 with roughly 110 real parameters — while the pure-detection ket mixtures landed almost exactly on their rank-capacity ceilings. A non-inclusion analysis (experiment 20) then settled the family boundary: for no efficiency does the target admit a finite-rank pre-image (proven analytically across the whole efficiency range, with a validated numerical scan as corroboration), so the target is strictly outside the winning family and the exp19 record is one instance of blind performance on a genuinely out-of-family target. The boundary is thin, however: a direct best-approximation study found the family approaches the target to ≈2–3 × 10⁻³ in 1 − F at the largest scored cutoff (n = 12; one-mode best-found values, still increasing slowly with cutoff — upper bounds on its true distance), so exp19's larger blind gap is a fit- and data-budget effect, not the family boundary. A robustness sweep (experiment 21) then repeated the blind comparison across three data seeds and a 4x range of noise strength: the pre-declared verdict holds on all five configurations (representative lossy fidelity 0.893–0.949 vs MLE 0.815–0.936), with no exp16-style basin collapse in any of the fifteen fits. The MLE opponent inherits exp19's 900-second budget and met its convergence criterion on only two of the five configurations, so exp21 demonstrates robustness against an equal-budget full-rank MLE; the margins over the three unconverged baselines are not guaranteed to survive further MLE optimization. Still one target class and exploratory; universal claims remain unwarranted. See the research log and prior-art survey for the evidence and comparisons.

Public homodyne data: GKP states

On the public propagating-light GKP homodyne dataset from Konno et al. (Science 2024; Dryad DOI), a pure physical model initially lost clearly to full-rank MLE. Adding a physical loss channel and rank-two squeezed-ket model improved held-out likelihood and, in a matched-degree-of-freedom control, outperformed a rank-one model of comparable capacity.

A rank-saturation study (experiment 18) then walked the remaining gap down: the rank curve saturates at R = 4–5, warm starts rule out under-optimization, and matched-degree-of-freedom controls at two frontier points attribute each gain to rank rather than parameter count. At rank 4 (92 real parameters) the physical model ties the empirical MLE frontier at confidence-interval resolution on both reshuffles (conditional 95% CIs [−0.00002, +0.00020] and [−0.00017, +0.00003] nats per held-out sample against the test-selected frontier best at 255 parameters). Earlier rounds' recorded losses (experiments 12–14) stand in the log. These analyses remain exploratory—the splits reuse the same observations, the MLE opponent is test-selected, and a tie at CI resolution is not preregistered confirmation.

The figures and their full protocols are in experiments/14_gkp_rank / experiments/18_gkp_saturation and the dated research-log entries (exp14, exp18).

Reproduce

The repository is intentionally lightweight. For the core synthetic experiment:

pip install numpy matplotlib pytest
python -m pytest tests/ -q
python experiments/01_cat_state/run.py

Experiment directories contain their own scripts, committed output logs, and where applicable the generated figures. The public GKP dataset is included with its source README and attribution under experiments/12_gkp_data/data/.

Repository map

wigner_splat/   forward models, fitters, physical Gaussian-ket models, and MLE baselines
experiments/    reproducible synthetic and public-data experiments
tests/          numerical and physical-consistency tests
docs/           research log, surveys, and reproducibility notes

Prior work and novelty boundary

This project builds on, rather than replaces, several lines of work:

• 3DGS-style differentiable Radon tomography: R2-Gaussian and X²-Gaussian
• Gaussian representations of Wigner negativity: Kenfack et al. (2004) and Tosca et al. (2025)
• Homodyne tomography and physical optimization: Strandberg (2022) and Gaikwad et al. (2025)

The narrow signed-splat research question is whether those two first lines can be combined as an inverse problem on measured homodyne data. The physical Gaussian-ket track has a different prior-art boundary and should not inherit that novelty claim. The prior-art survey records both boundaries and their remaining risks.

Citation and license

A short preprint summarizing the research record is published on Zenodo: W. Kawashima, Compact physical Gaussian-ket models for homodyne quantum-state tomography (2026), DOI: 10.5281/zenodo.21457048 (concept DOI — resolves to the latest version; source and build under docs/preprint/).

The theory companion to that preprint is published separately as a working paper: W. Kawashima, Zero-counting lower bounds and measured K–ε curves for approximate Gaussian rank (2026), DOI: 10.5281/zenodo.21698699 (concept DOI; source, PDF, and submission metadata under docs/kepsilon-note/). It states the certified lower-bound theorems for a restricted dictionary and the measured K–ε curves; the GKP robust-zero census in it is numerically supported, not computer-certified.

If this software or its research record is useful, please cite it using CITATION.cff. The exact archived v0.1.0 release is DOI: 10.5281/zenodo.21387212. The concept DOI always resolves to the latest archived version and is not the citation for this fixed release. The code is released under the MIT License.

Small support for compute and agent time will be welcome once a sponsorship link is configured; citations and careful technical feedback are already valuable.