Abstract
A recognizable quantum image need not determine the entanglement of the bipartite state that produced it. This paper studies that distinction in a fully specified synthetic spatial-biphoton model whose pump amplitude follows a Taichi Diagram morphology. The morphology is engineered preparation data, not a spontaneous image of entanglement and not historical evidence for quantum mechanics. The analysis separates pump morphology, the signal–idler tensor partition, reference-relative phase, and the complete complex joint kernel. For a normalized pure kernel, an integrated rectangle-defect identity equals one minus the reduced-state purity; in coordinate form it is the linear-entropy or I-concurrence invariant. A displayed diagonal nevertheless fails to determine entanglement, as exact separable and maximally entangled controls show.
The source-admitted finite model fixes cell-center coordinates, quadrature, exact Taichi Diagram masks, eye locations, Gaussian convention, phase maps, normalization, interventions, scan grids, and resolution grids. At the declared 18 × 18 grid per arm, the baseline has purity 0.441336, effective Schmidt number 2.265849, entropy 1.473719 nats, eye-region capacity 3.614 percent, and leading-two Schmidt weight 78.730 percent. Host continuation lowers the Schmidt number to 2.108193, whereas support excision raises it to 2.358247. A deterministic random midpoint phase leaves the complete joint intensity unchanged to numerical precision but changes the Schmidt number to 108.558946 and reduces fidelity with the zero-phase state to 0.000857. Complete joint-pixel dephasing preserves the image and both eyes while producing a separable state. Optimal Schmidt truncation requires 10 modes for 95 percent weight and 78 modes for 99 percent, yet an equal-weight two-mode projector already exceeds its separable bound; after the top-two local filter its target fidelity is 0.889270.
The paper gives a leakage-aware partial-transpose test, a two-copy purity test, explicit prospective mode-sorter and Hong–Ou–Mandel or local Z/X/Y analyzer maps, and exact one-sided binomial design calculations. A deterministic 80-block mock TRAIN/CALIBRATION/HOLDOUT execution is included, but the block-resolved experimental data and code described by the motivating article were not obtained or analyzed. The central result is both positive and negative: compact calibrated measurements can witness coherence in the full synthetic kernel, while the visible Taichi Diagram eye regions neither determine the leading Schmidt subspace nor constitute an entanglement witness.
Many modes to reconstruct, few modes to witness
Two numbers from the same model pull in opposite directions, and the paper keeps them apart:
- Reconstruction is expensive. Optimal Schmidt truncation needs 10 modes for 95 percent of the weight, and 78 for 99 percent.
- Witnessing is cheap. An equal-weight two-mode projector already exceeds its separable bound.
Recovering the state is a much harder task than certifying that it is coherent. The blind protocol is built on the second fact, not the first.
What the model is, and is not
The morphology is engineered preparation data. The block-resolved data and code of the motivating 2023 biphoton experiment were not obtained or analyzed, and the mock TRAIN/CALIBRATION/HOLDOUT run demonstrates software ordering rather than experimental certification.
Where it sits in the release
RIC-P2 applies and stress-tests the architecture of RIC-P1. Its synthetic model, scripts and outputs are frozen in RIC-D1.
Download paper (Zenodo) — CC-BY-4.0.