Quantum Imaging with Undetected Photons

• QIUP is a nonlinear interferometric imaging method where signal interference encodes object information while the idler photons remain undetected.
• It exploits induced coherence between separate SPDC sources to transfer phase and amplitude details from the object into the detected beam.
• The technique underpins diverse applications including mid-infrared microscopy, terahertz sensing, and ancilla-assisted tomography, with performance driven by spatial correlations.

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to=arxiv_search.search 大发彩票快三 平台总代理 code: {"^^^^^^^^3query3 OR id:(&&&^^^^3all:\3all:\3^^^^&&&) OR id:(&&&^^^^3all:\3 OR ti:\3^^^^&&&) OR id:(&&&^^^^3all:\33^^^^&&&) OR id:(&&&^^^^3all:\3^^^^4&&&) OR id:(&&&^^^^3all:\3^^^^5&&&)","max_results":^^^^3all:\3query3

Quantum imaging with undetected photons (QIUP) is a nonlinear-interferometric imaging paradigm in which the photons that interrogate the object are not measured, while the image is reconstructed from partner photons that never interact with the object. In its canonical realization, two coherently pumped spontaneous parametric down-conversion (SPDC) sources are arranged so that the alternatives “pair created in source ^^^^3all:\3^^^^” and “pair created in source ^^^^3 OR ti:\3^^^^” are indistinguishable; the object is placed only in the undetected idler path, and its transmission or phase is transferred to the interference term of the detected signal beam (&&&^^^^3query3^^^^&&&, &&&^^^^3all:\3^^^^&&&).

^^^^3all:\3^^^^. Origins and induced-coherence basis

The experimental point of departure is the two-crystal induced-coherence arrangement of Lemos and co-workers, in which a 53^^^^3 OR ti:\3^^^^ nm pump illuminates two identical ppKTP crystals, the signal photons are detected around 8^^^^3all:\3query3^^^^ or 8^^^^3 OR ti:\3query3^^^^ nm on an EMCCD, and the idler photons probe the object at ^^^^3all:\3^^^^55^^^^3query3^^^^ or ^^^^3all:\3sort_by3all:\3^^^^5 nm. In that demonstration, the method imaged an intensity mask, a silicon phase object opaque at 8^^^^3all:\3query3^^^^ nm but transparent at ^^^^3all:\3^^^^55^^^^3query3^^^^ nm, and a fused-silica phase object that was nearly invisible at 8^^^^3 OR ti:\3query3^^^^ nm but visible when placed in the idler path (&&&^^^^3query3^^^^&&&). The defining physical statement is that interference of the detected photons is controlled by whether the undetected partner photons preserve or destroy which-source information.

The first detailed theory established that the object information is present only in the interference term of the detected intensity and not in the individual intensities of the interfering beams. In the geometry analyzed there, the magnification is

PRESERVED_PLACEHOLDER_^^^^3query3^^^^

so it depends on both the average detected wavelength and the average illuminating wavelength (&&&^^^^3all:\3^^^^&&&). This formulation also made explicit that QIUP is an instance of induced coherence without induced emission: the image is carried by first-order interference in the signal arm because the idler alternatives are rendered indistinguishable.

A persistent conceptual distinction from ghost imaging follows from that structure. In standard QIUP, object information is read out from single-photon interference in the detected beam, rather than from coincidence detection between two detected arms. Later near-field treatments emphasized that the method is closer in spirit to ghost imaging than to conventional imaging because image formation is governed by a two-photon correlation function, but still differs from standard ghost imaging in using single-photon interference without coincidence detection or post-selection (Viswanathan et al., 2021).

^^^^3 OR ti:\3^^^^. Image-formation laws and geometrical regimes

In the far-field, momentum-correlation geometry, the biphoton state from one source is written as

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^

with joint transverse-momentum probability

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^

For two weakly pumped crystals, the detected-camera intensity at position $r_{q_d}$ takes the form

$$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$

so the image is a weighted average over the conditional momentum distribution of the undetected photon given the detected one. For a purely absorptive object, the visibility becomes

$V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$

which makes the effective point-spread function the conditional distribution $p(q_u|q_d)$ itself (&&&^^^^3 OR ti:\3^^^^&&&).

A complementary near-field formulation places both object and camera in image planes of the source. There the key quantity is the joint position probability density

$$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$

and the image for a purely absorptive object is extracted from two interferometer phases as

$$G(\boldsymbol{\rho}_{c}) \propto \int d\boldsymbol{\rho}_{o}\, P\!\left(\frac{\boldsymbol{\rho}_{c}}{M_{S}}, \frac{\boldsymbol{\rho}_{o}}{M_{I}} \right) |T(\boldsymbol{\rho}_{o})|.$$

In this regime QIUP becomes a linear convolution-like imaging system whose kernel is the twin-photon position-correlation function; for standard nondegenerate SPDC, the magnification is

$M=\frac{M_S}{M_I}.$

This is one of the sharpest distinctions from far-field QIUP, where wavelength ratios appear explicitly in the magnification law (Viswanathan et al., 2021).

In the idealized limit of maximal position correlation,

PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^

the near-field image reduces to point-to-point mapping with

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^

and for standard nondegenerate SPDC the paper states PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^ (Viswanathan et al., 2021). This suggests that QIUP should be regarded less as a single fixed protocol than as a family of interferometric image-transfer schemes whose operative spatial variable is selected by geometry: momentum in the far field, position in the near field.

3. Spatial correlations, magnification, and resolution

The most developed far-field resolution theory showed that the decisive quantity is the transverse momentum correlation between detected and undetected photons. For a Gaussian pump, the conditional probability obeys

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^

so a larger pump waist PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4 yields tighter momentum anti-correlation and better resolution. In that geometry the camera-plane blur width is

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5

while the object-space blur is

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6

The reported experiment found that weaker momentum correlation makes a ^^^^3 OR ti:\3sort_by3query3^^^^ PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7 test feature visibly blurrier, that all measured PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8 values differed by less than two standard deviations from theory, and that at fixed correlation strength a setup with PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9 nm resolved finer features than one with PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3query3^^^^ nm even though the detected wavelength was slightly longer (&&&^^^^3 OR ti:\3^^^^&&&).

In the near-field, position-correlation regime, the principal resolution law is different. Under Gaussian pump and Gaussianized phase matching, the effective blur kernel has width

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^

and the minimum resolvable distance derived from a dip-to-peak criterion PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^ is

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^

Here the detected and undetected wavelengths enter symmetrically through PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4, and the correlation-imposed limit cannot be improved by conventional optical techniques in the detected arm (Viswanathan et al., 2021).

A later experimental study of position-correlation QIUP varied crystal length PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5 mm and pump waist PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6, and found that the visibility spread, rather than the amplitude-image spread, is the correct measure of image sharpness once finite-pump effects become important. In the large-pump limit, the magnification-adjusted visibility spread approaches

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7

confirming the earlier crystal-length scaling, but strong pump focusing broadens PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8 and can drive the biphoton state toward a separability condition at which the visibility becomes spatially constant and object information is lost (Basset et al., 2023).

A significant controversy concerns which wavelength sets the ultimate far-field resolution. The focused far-field resolution theory above is explicit that, in its regime, the object-space resolution is governed by the undetected wavelength rather than the detected one (&&&^^^^3 OR ti:\3^^^^&&&). A later beyond-paraxial treatment argued instead that, in the standard free-space far-field geometry, the ultimate diffraction-limited minimum resolvable distance is bounded by the longer wavelength of the pair,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9

more precisely with prefactors $r_{q_d}$^^^^3query3^^^^, $r_{q_d}$^^^^3all:\3^^^^, and $r_{q_d}$^^^^3 OR ti:\3^^^^ multiplying $r_{q_d}$3 in strongly nondegenerate and degenerate limits. In that account, the shorter-wavelength scaling found in earlier work is regime-specific rather than fundamental (Vega et al., 2022). The coexistence of these results has become a central interpretive issue in the QIUP resolution literature.

4. Experimental regimes and application domains

QIUP has diversified from near-IR proof-of-principle interferometry into microscopy, phase imaging, terahertz sensing, holography, and tomography.

Modality	Undetected / detected band	Reported capability
Mid-infrared microscopy (Kviatkovsky et al., 2021)	$r_{q_d}$4 / near $r_{q_d}$5	$r_{q_d}$6 resolution; unstained mouse heart tissue
Single-shot phase quadrature (Haase et al., 2022)	$r_{q_d}$7 / $r_{q_d}$8	phase and visibility from one camera exposure
Terahertz QIUP (&&&^^^^3all:\3query3^^^^&&&)	$r_{q_d}$9 / $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$^^^^3query3^^^^	amplitude and phase imaging; $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$^^^^3all:\3^^^^ resolution
QUOPT tomography (Gemmell et al., 9 Jan 2025)	$$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$^^^^3 OR ti:\3^^^^ / $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$3	3D projection tomography; $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$4 estimated resolution
Deep-learning QHUP (&&&^^^^3all:\3 OR ti:\3^^^^&&&)	QHUP signal detection near $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$5	one-shot hologram inversion; resolution improved from $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$6 to $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$7

Position-correlation microscopy provided the first experimental realization of near-field QIUP, with a Michelson-type nonlinear interferometer around a ppKTP crystal, $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$8 idler probing, and measured field of view $$I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},$$9 and resolution $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$^^^^3query3^^^^ at $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$^^^^3all:\3^^^^. The same work reported ultralow mid-IR illumination of about ^^^^3all:\3^^^^5 pW on the sample and demonstrated morphological imaging of mouse heart tissue on the cellular level (Kviatkovsky et al., 2021).

Single-shot phase-quadrature QIUP replaced sequential phase stepping by a polarization-optics scheme that produces four simultaneous outputs with phase offsets $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$^^^^3 OR ti:\3^^^^, allowing phase and visibility to be reconstructed from one camera frame. In calibration, the retrieved phase versus induced delay had $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$3, with mean visibility $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$4 and absolute standard deviation $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$5; the method was demonstrated on a static phase mask opaque to the detected photons, drying isopropanol film, and stretched adhesive tape (Haase et al., 2022).

Terahertz QIUP extended the spectral separation to more than two orders of magnitude in frequency by probing the sample at ^^^^3all:\3^^^^.5 THz and detecting only visible photons at 66^^^^3 OR ti:\3^^^^.^^^^3 OR ti:\3^^^^ nm on an uncooled sCMOS camera. The experiment reported spatial resolution $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$6, field of view $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$7, per-pixel interference visibility below $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$8, and a raw signal-to-noise ratio of at most ^^^^3all:\3^^^^:7^^^^3query3query3^^^^ due to thermal and parasitic backgrounds; a Fourier-domain “quantum distillation” procedure isolated the desired interference term and enabled amplitude-sensitive, phase-sensitive, and spectrally selective imaging (&&&^^^^3all:\3query3^^^^&&&).

The tomographic extension QUOPT combined QIUP with optical projection tomography: a ^^^^3all:\3^^^^55^^^^3query3^^^^ nm idler probed the sample, an 8^^^^3all:\3query3^^^^ nm signal was recorded on a silicon camera, fringe visibility was extracted from phase scans via FFT, and ^^^^3all:\3id:(Lemos et al., 2014) OR id:(Lahiri et al., 2015) OR id:(Fuenzalida et al., 2020) OR id:(Viswanathan et al., 2021) OR id:(Viswanathan et al., 2021) OR id:(Kviatkovsky et al., 2021) OR id:(Vega et al., 2022) OR id:(Haase et al., 2022) OR id:(Basset et al., 2023) OR id:(Gemmell et al., 9 Jan 2025)3query3^^^^ projections over ^^^^3all:\3id:(Lemos et al., 2014) OR id:(Lahiri et al., 2015) OR id:(Fuenzalida et al., 2020) OR id:(Viswanathan et al., 2021) OR id:(Viswanathan et al., 2021) OR id:(Kviatkovsky et al., 2021) OR id:(Vega et al., 2022) OR id:(Haase et al., 2022) OR id:(Basset et al., 2023) OR id:(Gemmell et al., 9 Jan 2025)3query3^^^^° were reconstructed by inverse Radon transform into a three-dimensional infrared absorption volume. The demonstrated object was a twisted-wire figurine about 5 mm high, and the spatial resolution was estimated as $V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,$9 (Gemmell et al., 9 Jan 2025).

Two further extensions altered the measurement architecture itself. Interaction-free single-pixel QIUP embedded an Elitzur–Vaidman-type Michelson interferometer in the idler arm and combined it with Hadamard-mask single-pixel imaging, using ^^^^3all:\3^^^^55^^^^3query3^^^^ nm undetected idlers and 8^^^^3all:\3query3^^^^ nm detected signals; in the binary-object regime, the relevant object information can be obtained from events in which the returning idler avoided the object arm (Yang et al., 2022). A distinct four-photon protocol based on “phase-subtractive interference by path identity” used two independent sources and two undetected photons, yielding a coincidence interferogram that is independent of spatially uniform pump, signal, and idler phases while retaining spatially dependent object phases acquired by the undetected photons (&&&^^^^3all:\3all:\3^^^^&&&).

5. Information-theoretic reformulations and generalized tomography

QIUP has repeatedly been recast in explicitly quantum-informational language. One early reinterpretation described the original Lemos protocol as a form of ancilla-assisted process tomography in which the idler is the system passing through the unknown object channel $p(q_u|q_d)$^^^^3query3^^^^, the signal is the ancilla, and the detected-photon measurement performs only a partial tomography of the object. In that model, the original setup measures only the combination $p(q_u|q_d)$^^^^3all:\3^^^^; a simple phase shifter before the final signal beamsplitter upgrades the protocol to full tomography of the two-parameter object model. The same analysis also argued that entanglement is not strictly necessary for image formation: for an extended Werner-state probe, the image visibility becomes $p(q_u|q_d)$^^^^3 OR ti:\3^^^^, so sufficiently strong correlations can suffice even when the probe state is separable (&&&4^^^^3query3^^^^&&&).

The induced-coherence paradigm has also been generalized from object imaging to tomography of the undetected subsystem itself. In “Quantum state tomography of undetected photons,” an unknown idler polarization qubit

$p(q_u|q_d)$3

was reconstructed from two signal-fringe measurements, with reported mean fidelities $p(q_u|q_d)$4 and $p(q_u|q_d)$5, and minimum fidelity $p(q_u|q_d)$6 (&&&4^^^^3all:\3^^^^&&&). This makes explicit that QIUP is not confined to spatial transmission maps; it is a broader ancilla-assisted measurement strategy for inaccessible quantum degrees of freedom.

A more recent conceptual development reframed QIUP in wave-particle-duality language. In the ideal case, the “imaging duality ellipse” is

$p(q_u|q_d)$7

where $p(q_u|q_d)$8 is visibility, $p(q_u|q_d)$9 is path predictability, and $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$^^^^3query3^^^^ is the object transmittance. With source decoherence $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$^^^^3all:\3^^^^ and idler misalignment $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$^^^^3 OR ti:\3^^^^, the relation becomes

$$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$3

Within that framework the object can be reconstructed from duality observables alone, and the ellipse form survives calibration against uniform imperfections (&&&^^^^3all:\33^^^^&&&).

Polarization-sensitive generalizations have begun to extend QIUP from scalar transmission to matrix-valued sample characterization. A theoretical Jones-matrix protocol uses selected idler polarization inputs, two special HWP settings, and signal-fringe visibilities and phase offsets to reconstruct a structured Jones matrix

$$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$4

thereby reducing the number of unknowns relative to a fully unconstrained complex $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$5 fit (&&&^^^^3all:\3^^^^5&&&). By contrast, a separate theoretical study on polarization distinguishability concluded that if the two SPDC alternatives are polarization-distinguishable at generation, then post-generation optics and a Mach–Zehnder interferometer cannot restore the lost interference. That result sharpens a basic QIUP design rule: downstream optics reveal coherence already present between the generation alternatives; they do not create it after source-side distinguishability has destroyed it (&&&^^^^3all:\3^^^^4&&&).

6. Limitations, controversies, and current directions

QIUP remains constrained by three interlocking resources: indistinguishability of the generation alternatives, usable transverse correlations in the biphoton state, and sufficiently stable interferometric readout. In practice, finite crystal size limits the pump waist and hence the achievable momentum correlation in far-field geometries; in one reported regime the diffraction-limited uncertainty from ^^^^3all:\3^^^^-inch optics was about $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$6, whereas the uncertainty due to imperfect momentum correlation was about $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$7 for $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$8, making correlation rather than lens diffraction the dominant limit (&&&^^^^3 OR ti:\3^^^^&&&). In the near field, the correlation-imposed blur cannot be undone by improving conventional optics in the detected arm, because the image has already been convolved with the twin-photon position kernel before detection (Viswanathan et al., 2021).

Operational bottlenecks differ by modality. Single-shot phase-quadrature QIUP still required calibration of branch-dependent throughput, numerical spot registration, Gaussian filtering, and phase unwrapping for large phase excursions, with 5^^^^3query3query3^^^^ ms integration time and stability only over several minutes (Haase et al., 2022). THz QIUP achieved visible-only detection at extreme nondegeneracy, but with per-pixel visibility below $$P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},$$9, ^^^^3all:\3query3query3query3^^^^ images averaged per delay step, and strong parasitic backgrounds imposed by room-temperature thermal photons and nonlinear side processes (&&&^^^^3all:\3query3^^^^&&&). QUOPT tomography demonstrated 3D infrared reconstruction, but retained phase scanning and a proof-of-concept resolution of about $$G(\boldsymbol{\rho}_{c}) \propto \int d\boldsymbol{\rho}_{o}\, P\!\left(\frac{\boldsymbol{\rho}_{c}}{M_{S}}, \frac{\boldsymbol{\rho}_{o}}{M_{I}} \right) |T(\boldsymbol{\rho}_{o})|.$$^^^^3query3^^^^ (Gemmell et al., 9 Jan 2025). Deep-learning-enhanced QHUP removed much of the multi-shot reconstruction burden, but explicitly acknowledged performance degradation when training and testing distributions are mismatched, especially between SLM-based virtual objects and real samples (&&&^^^^3all:\3 OR ti:\3^^^^&&&).

The most visible theoretical controversy remains the far-field wavelength question. One line of work, validated experimentally in a specific momentum-correlation regime, concludes that the camera-plane blur is set by the detected wavelength while the object-space resolution is governed by the undetected illumination wavelength (&&&^^^^3 OR ti:\3^^^^&&&). Another, beyond-paraxial analysis argues that the ultimate free-space limit is instead set by the longer wavelength of the pair, regardless of which photon illuminates the object (Vega et al., 2022). A plausible implication is that QIUP resolution must be discussed at two levels: regime-specific performance laws for a given geometry and source model, and the ultimate nonparaxial diffraction bound.

The current research trajectory is correspondingly bifurcated. One branch pushes spectral reach and modality: mid-IR microscopy, terahertz phase imaging, dynamic phase sensing, holography, and 3D tomography (Kviatkovsky et al., 2021, &&&^^^^3all:\3query3^^^^&&&, Haase et al., 2022, Gemmell et al., 9 Jan 2025). The other branch refines the formal structure of the protocol: ancilla-assisted tomography, duality-based reconstruction, polarization and Jones-matrix readout, and noise-resistant multi-photon interference (&&&4^^^^3query3^^^^&&&, &&&^^^^3all:\33^^^^&&&, &&&^^^^3all:\3^^^^5&&&, &&&^^^^3all:\3all:\3^^^^&&&). At the device level, recurrent proposals include increasing the pump waist through larger transverse crystal dimensions, decreasing $$G(\boldsymbol{\rho}_{c}) \propto \int d\boldsymbol{\rho}_{o}\, P\!\left(\frac{\boldsymbol{\rho}_{c}}{M_{S}}, \frac{\boldsymbol{\rho}_{o}}{M_{I}} \right) |T(\boldsymbol{\rho}_{o})|.$$^^^^3all:\3^^^^ in the undetected arm, using shorter undetected wavelengths, redesigning the geometry so that position rather than momentum correlations govern imaging, and extending nonlinear materials and interferometer architectures into still more difficult spectral bands, including explicit future concepts involving x-ray-scale illumination with optical detection (&&&^^^^3 OR ti:\3^^^^&&&).