High-dimensional quantum process tomography with undetected photons

Scientific Impact Assessment

1. Core Contribution

This paper proposes an interferometric method for high-dimensional quantum process tomography (QPT) that requires no measurement on the photon undergoing the unknown operation. The central advance over the authors' own prior work (Ref. [15], which handled only unitary transformations) is the extension to *arbitrary linear (non-unitary) operations*. The key technical maneuver is a quantum-field-theoretic treatment: since a non-unitary transformation does not preserve the bosonic commutation relations, the authors introduce an auxiliary vacuum field (Eqs. 5–7) whose admixture matrix  restores a legitimate field operator. This is the standard trick used to model lossy/dissipative photonic channels (e.g., beam-splitter loss models), applied here within the Zou-Wang-Mandel path-identity interferometer. The reconstruction then proceeds by applying a known two-mode rotation Ûb(q,r) in one signal arm and reading off the magnitudes and phases of matrix elements from single-photon interference visibilities and fringe phase shifts, requiring N(N−1)/2 unitary settings.

2. Methodological Rigor

As a theoretical proposal, the derivation is clean and self-consistent. The vacuum-field construction correctly recovers the unitary case as a limit, and the mapping from interference-pattern visibility/phase to matrix elements is derived transparently (Eqs. 17–22). The 4×4 non-unitary example (a lossy Hadamard) provides a concrete, checkable numerical demonstration with all 16 elements reconstructed in the appendix. However, the treatment is entirely idealized: there is no analysis of experimental imperfections (phase instability, source brightness mismatch, imperfect mode alignment/path identity, detector noise), no error propagation from finite-visibility measurements, and no discussion of how loss in Û affects reconstruction. Crucially, there is no experimental demonstration — the paper is a proof-of-principle proposal with a numerical illustration only. The scaling argument (N(N−1)/2 settings) is stated but not deeply analyzed against alternative QPT resource costs.

3. Potential Impact

The primary appeal is the ability to characterize operations at wavelengths where single-photon detectors are poor or unavailable — notably the mid- and far-infrared — because only one photon of the pair (the signal, which can be at a convenient wavelength) is ever detected. This "undetected photon" paradigm has already proven impactful in imaging and spectroscopy, and extending it to full process characterization (including non-unitary/lossy channels, which are ubiquitous in real devices) is genuinely useful for benchmarking quantum networks and components in otherwise inaccessible spectral regions. That said, the impact is bounded by the paper's status as a proposal: real influence hinges on experimental realization, which the authors do not provide. The technique also inherits the practical demands of path-identity interferometry (mutual coherence, precise idler alignment).

4. Timeliness & Relevance

QPT, high-dimensional (qudit) encoding, and undetected-photon techniques are all active, growing areas. The path-identity / quantum-imaging-with-undetected-photons field (anchored by the Zeilinger group RMP, Ref. [14]) is presently expanding into new functionalities, and process tomography is a natural and topical extension. Handling non-unitary operations is directly relevant to real-world quantum hardware, where loss and decoherence dominate. The work is well-timed.

5. Strengths & Limitations

*Strengths:* Conceptually elegant; correct and pedagogically clear field-theoretic handling of non-unitarity in path-identity interferometers; addresses a real capability gap (detector-free characterization in difficult spectral bands); concrete worked example; the claim that the framework extends in principle to non-photonic systems is plausible given its field-theoretic basis.

*Limitations:* (i) Incremental relative to Ref. [15] — the architecture and reconstruction strategy are largely inherited, with the non-unitary generalization being the main new content. (ii) No experiment; feasibility under realistic noise is unaddressed. (iii) The vacuum-field method for modeling non-unitary channels is well-established, so the theoretical novelty lies mainly in its transplant into this interferometric context. (iv) No comparison of resource cost/precision against standard or ancilla-assisted QPT beyond counting unitary settings. (v) The reconstruction assumes clean two-mode rotations are experimentally available; imperfections there are not modeled.

Additional observations: Reproducibility of the theory is high — all equations and the numerical example are fully specified, and a competent quantum-optics group could reproduce the derivation and simulate it directly. The translational potential is real but latent: it becomes significant only upon experimental demonstration and integration into IR quantum photonics. The work is likely to be cited as an enabling theoretical step within the undetected-photon subcommunity and by high-dimensional QPT researchers, but is unlikely on its own to redirect the broader field absent an experimental follow-up.

Overall, this is a solid, well-executed theoretical extension that fills a meaningful conceptual gap (non-unitary characterization without detecting the transformed photon), with clear but as-yet-unrealized practical value.