Photon Mode Erasure: Nonlocal Quantum Effects

• Photon mode erasure is the controlled removal of accessible modal information from photonic states, enabling nonlocal interference and nonclassical state transformations.
• Techniques include local photon subtraction, erasure of distinguishing labels (e.g., polarization, OAM), and engineered detection that restore indistinguishability.
• Applications span quantum imaging, error correction protocols, and nonlocal measurements by converting modality information into measurable correlations.

Searching arXiv for the core and related papers on photon mode erasure. Photon mode erasure denotes a family of quantum-optical phenomena in which information tied to a photonic mode is removed, rendered inaccessible, or re-routed so that interference, nonlocal state transformation, error mitigation, or number resolution becomes possible. Across the literature, the phrase does not refer to a single mechanism. In one line of work, a local photon annihilation event applied to part of a distributed nonclassical mode subtracts a photon from the entire mode while leaving the mode profile unchanged, the “quantum vampire” effect (Fedorov et al., 2014). In other work, “erasure” means removing mode labels such as polarization, orbital angular momentum, phase structure, source identity, color, or path reality, thereby restoring interference between alternatives that would otherwise remain distinguishable (Heuer et al., 2015, Nape et al., 2017, Yang et al., 2024, Qu et al., 2019). In yet another usage, an erasure is the loss of an entire optical mode to vacuum, motivating dedicated erasure-correcting codes and erasure checks in optical and circuit-QED architectures (Lassen et al., 2010, Graaf et al., 2024). The unifying principle is that experimentally accessible behavior depends not only on the photon number or trajectory in a classical sense, but on whether the relevant mode information remains available in principle.

1. Distributed-mode annihilation and the “quantum vampire” effect

A canonical formulation of photon mode erasure begins with a nonclassical state prepared in a single optical mode $\hat a$, with all orthogonal modes in vacuum, and then distributed by a beam splitter between remote modes $\hat a_1$ and $\hat a_2$ according to

$\hat a=\mu \hat a_1+\lambda \hat a_2.$

Using the orthogonal mode

$$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$

the key operator identity is

$$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$

because the state occupies only mode $\hat a$ and the orthogonal mode $\hat a_\perp$ is vacuum (Fedorov et al., 2014). A local application of the annihilation operator to only one portion of the distributed state is therefore equivalent, up to the scalar factor $\mu^\ast$, to annihilating a photon in the global occupied mode.

For Fock input states this gives

$$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$

so one photon is removed from the entire delocalized excitation while the lower-Fock-state mode is preserved (Fedorov et al., 2014). The paper explicitly distinguishes this from ordinary linear absorption, which is described by a Lindblad-type evolution,

$\hat a_1$0

and does create an ordinary shadow (Fedorov et al., 2014). Photon subtraction conditioned on a detector click approximates the annihilation operator; ordinary attenuation does not.

The experiment realizing this effect used heralded one-photon or two-photon Fock states produced by type-II parametric down-conversion in a PPKTP crystal pumped by a 390 nm, frequency-doubled Ti:sapphire laser. A $\hat a_1$1 waveplate set $\hat a_1$2 in a polarization decomposition, photon subtraction on one local component was implemented with a strongly imbalanced partially polarizing beam splitter with around $\hat a_1$3 transmission for one polarization, and balanced homodyne detection with tomography reconstructed the output state (Fedorov et al., 2014). With no subtraction click, the reconstructed state remained the prepared one-photon or two-photon Fock state. With a subtraction click, the reconstruction showed

$\hat a_1$4

with the same spatial, temporal, and polarization mode structure; detection efficiency was about $\hat a_1$5, and a small residual population in the two-photon case was attributed to dark counts of $\hat a_1$6 (Fedorov et al., 2014).

This usage of photon mode erasure is specifically about global photon-number reduction without deformation of the occupied mode. A common misconception is to identify it with absorption or collapse localized to one beam region. The cited experiment emphasizes the opposite point: the photon is “stolen” without leaving a shadow, and the action is nonlocal but probabilistic, so it cannot be used for superluminal signaling (Fedorov et al., 2014).

2. Quantum erasure as removal of distinguishability between photonic alternatives

A broader and historically dominant meaning of photon mode erasure concerns the removal of information that distinguishes two or more photonic alternatives. In “Phase Selective Quantum Eraser,” two synchronously pumped parametric down converters are coupled by induced coherence. When the idler modes are made indistinguishable, signal photons from the two crystals interfere; when source information leaks into the idler sector, interference disappears (Heuer et al., 2015). The experiment extends standard induced-coherence arrangements by adding an idler Mach-Zehnder interferometer, so the erasure is controlled continuously by an idler phase $\hat a_1$7. Under the assumptions $\hat a_1$8 and balanced beam splitters, the coincidence visibility becomes

$\hat a_1$9

with $\hat a_2$0 for $\hat a_2$1 and $\hat a_2$2 for odd multiples of $\hat a_2$3; experimentally, coincidence visibilities reached about $\hat a_2$4 in the erasure condition and dropped to about $\hat a_2$5 in the distinguishable condition (Heuer et al., 2015).

In Hong-Ou-Mandel settings, the same logic appears as erasure of hidden spectral-temporal labels. “Erasing Quantum Distinguishability via Single-Mode Filtering” shows that single-mode filtering with a time gate of duration $\hat a_2$6 and a spectral filter of bandwidth $\hat a_2$7 can erase modal information when

$\hat a_2$8

The number operator of the filtered field is written as

$\hat a_2$9

with Schmidt eigenvalues $\hat a=\mu \hat a_1+\lambda \hat a_2.$0 determined by the filter kernel, and the single-mode regime corresponds to $\hat a=\mu \hat a_1+\lambda \hat a_2.$1 and $\hat a=\mu \hat a_1+\lambda \hat a_2.$2 (Patel et al., 2012). In heralded HOM interference using two independent SFWM sources, the measured visibility improved from $\hat a=\mu \hat a_1+\lambda \hat a_2.$3 in a multimode regime to $\hat a=\mu \hat a_1+\lambda \hat a_2.$4 in a single-mode regime, matching theory values of $\hat a=\mu \hat a_1+\lambda \hat a_2.$5 and $\hat a=\mu \hat a_1+\lambda \hat a_2.$6 with no fitting parameter (Patel et al., 2012).

These works establish a technically precise distinction between photons being different and differences being experimentally accessible. This suggests that “mode erasure” is often less about changing the photons than about engineering a measurement or interaction so that the mode degree of freedom ceases to carry usable which-alternative information.

3. Structured-light and orbital-angular-momentum realizations

Mode erasure has been generalized from polarization tags to structured photonic degrees of freedom, especially orbital angular momentum and transverse phase structure. In “Erasing the orbital angular momentum information of a photon,” opposite OAM states $\hat a=\mu \hat a_1+\lambda \hat a_2.$7 and $\hat a=\mu \hat a_1+\lambda \hat a_2.$8 play the role of abstract paths, while the partner photon’s polarization marks or erases which-OAM information (Nape et al., 2017). After a $\hat a=\mu \hat a_1+\lambda \hat a_2.$9-plate and postselection, the hybrid entangled state takes the form

$$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$0

and projection of photon $$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$1 onto diagonal or anti-diagonal polarization yields coherent superpositions $$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$2 on photon $$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$3 (Nape et al., 2017). The joint detection probability is

$$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$4

so the fringe visibility is

$$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$5

This realizes complementarity in OAM space, with the paper explicitly stating $$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$6 (Nape et al., 2017).

“Quantum erasure based on phase structure” pushes the same idea to the wavefront itself in a Mach-Zehnder interferometer with a first-order spiral phase plate in one arm (Yang et al., 2024). The spiral phase plate converts the Gaussian mode $$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$7 into an OAM mode, and the output states become superpositions such as

$$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$8

Because the flat equiphase surface and the helical equiphase surface are distinguishable, interference is absent. A shifted output spiral phase plate acts as a mode converter, mixing the OAM basis states so that both paths contribute to the same measurable Gaussian component. The recovered detection probability is

$$\hat a_\perp=\lambda^\ast \hat a_1-\mu^\ast \hat a_2,$$9

and the measured fringe visibility after erasure was

$$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$0

The paper states that the delayed erasure choice satisfies space-like separation (Yang et al., 2024).

A further generalization appears in “Engineering quantum states from a spatially structured quantum eraser,” where vector-vortex modes make polarization distinguishability vary across the transverse plane (Schiano et al., 2023). The radial and $$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$1 modes are

$$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$2

For temporally indistinguishable photons, the coincidence probability depends on transverse position, and the reported visibilities include

$$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$3

with several other projection pairs yielding zero visibility (Schiano et al., 2023). Here the erasure is local in the transverse mode profile: some regions exhibit bunching, some antibunching, and some no interference.

These structured-light variants show that “path” in a quantum eraser need not be a literal spatial slit. Any orthogonal mode basis that stores distinguishing information can serve the same role, including OAM, equiphase structure, and spatially varying polarization (Nape et al., 2017, Yang et al., 2024, Schiano et al., 2023).

4. Nonlocal and remote forms of mode erasure

Several experiments and proposals use erasure to produce nonlocal behavior in ways not reducible to ordinary local interference. “Quantum observable’s reality erasure with spacelike-separated operations” employs polarization-entangled photons sent to laboratories separated by more than $$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$4 (Araújo et al., 2024). The source state is

$$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$5

and the analysis uses the Bilobran-Angelo irreality measure

$$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$6

When Alice measures in $$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$7, Bob’s path is real, with $$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$8 and the auxiliary degrees of freedom $$\hat a_1\psi_{\hat a}=(\mu^\ast \hat a+\lambda \hat a_\perp)\psi_{\hat a}=\mu^\ast \hat a\psi_{\hat a},$$9 also real. When Alice measures in $\hat a$0, Bob’s path becomes indefinite with

$\hat a$1

and the same value applies to $\hat a$2 and $\hat a$3 after the beam displacers; at maximal entanglement, $\hat a$4 (Araújo et al., 2024). The operations are arranged to be spacelike separated and no classical communication occurs.

In “Photonic realization of erasure-based nonlocal measurements,” local path meters and ancillas are used to access a nonlocal parity observable of two polarization qubits (Pan et al., 2019). After local CNOT-like interactions, a Hadamard transform on one meter and postselection erase local which-branch information, leaving

$\hat a$5

so the ancilla outcome $\hat a$6 encodes even parity and $\hat a$7 odd parity (Pan et al., 2019). The point of erasure here is not restoration of a familiar fringe but conversion of nonlocal information into a locally accessible meter record upon successful postselection.

A different nonlocal usage appears in “Nonlocal Quantum Erasure of Phase Objects,” which uses Franson-type fourth-order interference rather than first-order interference (Gao et al., 2019). Two polarization Sagnac interferometers receive entangled photons in the state

$\hat a$8

and the Franson phase is

$\hat a$9

A phase object inserted in one arm destroys or alters coincidence interference locally in the image; an identical phase object inserted in the corresponding arm of the distant interferometer restores the phase condition and erases the image in the overlap region (Gao et al., 2019). The measured image quality was quantified by

$\hat a_\perp$0

The proposal “Counterfactual quantum erasure: spooky action without entanglement” radicalizes the remote-control aspect by removing entanglement from the resource list (Salih, 2016). Alice marks which path in a Michelson interferometer by a switchable polarization rotator so that $\hat a_\perp$1 and $\hat a_\perp$2 become equally likely. Bob then uses a chained quantum Zeno effect module: if he blocks the channel, the polarization of the arm heading toward him is remotely flipped, restoring indistinguishability and hence destructive interference at $\hat a_\perp$3 (Salih, 2016). The visibility is

$\hat a_\perp$4

with reported example values of about $\hat a_\perp$5 for $\hat a_\perp$6 and about $\hat a_\perp$7 for $\hat a_\perp$8 (Salih, 2016). The successful events are counterfactual in the sense stated by the paper: if Alice detects the photon at $\hat a_\perp$9 or $\mu^\ast$0, that photon could not have traveled to Bob.

A common misunderstanding across these nonlocal cases is to interpret erasure as superluminal causal influence. The cited works frame the effects instead as correlations, postselected nonunitary operations, or entanglement-enabled mode transformations that do not permit faster-than-light signaling (Fedorov et al., 2014, Araújo et al., 2024, Salih, 2016).

5. Mode erasure in detection, sensing, and interference across unconventional mode labels

Mode erasure can also be engineered at the detector or sensing architecture level. “Color Erasure Detectors Enable Chromatic Interferometry” constructs detectors that cannot distinguish between photons of very different wavelengths (Qu et al., 2019). The physical implementation uses upconversion single-photon detectors based on periodically poled lithium niobate waveguides pumped at $\mu^\ast$1, with signal wavelengths approximately $\mu^\ast$2 and $\mu^\ast$3 satisfying

$\mu^\ast$4

The interaction Hamiltonian is

$\mu^\ast$5

and in the large-pump limit it reduces to an effective color-space rotation (Qu et al., 2019). After postselection on a particular output color, the detector states corresponding to different input colors have near-unit overlap, so the coincidence probability contains the interference term

$\mu^\ast$6

The paper reports interference visibility about $\mu^\ast$7, close to the ideal $\mu^\ast$8, and demonstrates the effect with coherent lasers, thermal-like light, and a free-space geometry (Qu et al., 2019).

In imaging with undetected photons, “Coupling undetected sensing modes by quantum erasure” uses a polarization-state quantum eraser to interpolate continuously between induced-coherence IUP and nonlinear-interferometry IUP (Gemmell et al., 2023). The first-pass signal polarization is rotated by an internal QWP with angle $\mu^\ast$9, and an external HWP with angle $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$0 either preserves or erases distinguishability between two generation alternatives. The limiting cases are explicitly identified: $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$1 gives an NI-IUP-like situation, $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$2 makes the alternatives fully distinguishable and interference vanishes, and the output HWP plus PBS can erase the distinguishability to yield an IC-IUP-like operation (Gemmell et al., 2023). The detected signal count rate $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$3 contains incoherent and interference terms, and visibility is defined as

$$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$4

The setup uses a $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$5 pump, a PPLN crystal, signal photons at $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$6, and idler photons at $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$7 (Gemmell et al., 2023).

The same conceptual theme appears outside interference imaging. “Photon number resolution without optical mode multiplication” removes the need to create distinguishable optical modes at all (Vetlugin et al., 2022). Instead of spatial or temporal multiplexing, a distributed detector array coherently absorbs a single standing-wave mode. For $$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$8 sublayers in the counterpropagating geometry, the required per-layer response for perfect total absorption is

$$\hat a_1|N\rangle_{\hat a}\propto |N-1\rangle_{\hat a},$$9

The paper gives a striking example: a 10-layer distributed detector can resolve two-photon states with about $\hat a_1$00 probability and three-photon states with about $\hat a_1$01 probability, even though individual sublayers may absorb only around $\hat a_1$02 each in a traveling-wave picture (Vetlugin et al., 2022). In this usage, the “erasure” consists in avoiding optical mode multiplication rather than erasing which-path information in the usual sense.

These detector-centered works make explicit that photon mode erasure may occur not only in propagation and state preparation but also in the measurement apparatus itself. A plausible implication is that the boundary between “source engineering” and “detector engineering” is less fundamental than descriptions based purely on input photonic states might suggest.

6. Loss, deterministic extraction, and error-correction notions of erasure

A distinct technical usage of erasure treats the disappearance of a whole mode as the error itself. In “Quantum optical coherence can survive photon losses: a continuous-variable quantum erasure correcting code,” an erasure channel is modeled as

$\hat a_1$03

so the mode is either transmitted intact or completely replaced by vacuum (Lassen et al., 2010). The four-mode codeword is built from a coherent input state, a vacuum input, two ancillary squeezed-vacuum modes, and three balanced beam splitters. Decoding uses balanced beam splitters, entangled homodyne measurements, and either deterministic feedforward,

$\hat a_1$04

or probabilistic postselection with the threshold condition

$\hat a_1$05

for discarding an event (Lassen et al., 2010). The paper reports a maximum deterministic fidelity of approximately $\hat a_1$06, surpassing the classical benchmark of $\hat a_1$07, and in the probabilistic protocol reports direct transmission at $\hat a_1$08 with about $\hat a_1$09 versus corrected output $\hat a_1$10 for the entangled four-mode code and $\hat a_1$11 for an entanglement-free version (Lassen et al., 2010).

A modern hardware-specific variant is “A mid-circuit erasure check on a dual-rail cavity qubit using the joint-photon number-splitting regime of circuit QED” (Graaf et al., 2024). There, the dual-rail qubit is encoded in the single-photon manifold

$\hat a_1$12

and the main erasure state is the joint vacuum $\hat a_1$13. Under a strong parametric beamsplitter drive, a single ancilla coupled statically to one cavity becomes sensitive to the total photon number in two cavities. For symmetric detuning $\hat a_1$14, the transition frequencies are

$\hat a_1$15

so an ancilla pulse can be selective for $\hat a_1$16 (Graaf et al., 2024). The reported performance includes a missed erasure fraction of $\hat a_1$17 in the abstract, and the mechanism-focused results report a false negative rate $\hat a_1$18 per erasure check, a total erasure rate $\hat a_1$19, and a Pauli error rate $\hat a_1$20, dominated by cavity errors (Graaf et al., 2024).

By contrast, “Extraction of a single photon from an optical pulse” uses “mode erasure” in the sense of removing exactly one photon from an original mode while preserving that photon in another mode (Rosenblum et al., 2015). The relevant operator is not the annihilation operator but

$\hat a_1$21

which removes one photon regardless of $\hat a_1$22 (Rosenblum et al., 2015). Using single-photon Raman interaction with a single $\hat a_1$23 atom near a nanofibre-coupled whispering-gallery-mode microresonator, the first photon in an input pulse is reflected into another mode and transfers the atom from $\hat a_1$24 to $\hat a_1$25, after which subsequent photons are transmitted. The measured extraction efficiency for low input photon number was about $\hat a_1$26, compared to about $\hat a_1$27 for an ideal extractor with the same linear loss; the current limitation was mostly linear loss, and the paper argues that efficiencies close to unity should be attainable with realistic improvements (Rosenblum et al., 2015).

These examples show that “erasure” in quantum optics can mean at least three non-equivalent things: removal of distinguishing information, replacement of a mode by vacuum, or deliberate subtraction/rerouting of occupancy from one mode into another. The term therefore requires contextual reading.

7. Conceptual synthesis, scope, and recurrent points of confusion

Across the cited literature, photon mode erasure is best understood as a controlled removal of accessible modal information rather than as a single protocol. The erased quantity may be photon number in a global delocalized mode (Fedorov et al., 2014), source identity (Heuer et al., 2015), spectral-temporal multimode structure (Patel et al., 2012), OAM or wavefront labels (Nape et al., 2017, Yang et al., 2024), local branch information in a meter (Pan et al., 2019), color labels at detection (Qu et al., 2019), or the occupation of an entire transmission mode in an erasure channel (Lassen et al., 2010).

Several controversies or misconceptions recur. One is the equation of erasure with mere ignorance. The color-erasure work explicitly rejects that view: simply discarding frequency information in data processing is not enough; the detector must be engineered so that it cannot retain which-color information in the first place (Qu et al., 2019). Another is the assumption that all erasure experiments are delayed-choice restatements of wave-particle duality. Some papers do fit that pattern, especially the OAM and phase-structure erasers (Nape et al., 2017, Yang et al., 2024), but others concern nonlocal measurements (Pan et al., 2019), distributed-mode annihilation (Fedorov et al., 2014), or error correction against full-mode loss (Lassen et al., 2010), where the operative concept is not wave-particle duality in the narrow sense.

A further point of confusion is the relation between erasure and collapse. The “quantum vampire” paper explicitly frames its effect as nonlocal action at a distance without local state collapse by either party (Fedorov et al., 2014). The spacelike-separated reality-erasure experiment similarly frames its result in terms of correlations with Bob’s observable reality, not superluminal signaling or classical causation (Araújo et al., 2024). This suggests that the most careful way to speak about photon mode erasure is operational: one should specify which degree of freedom carries the distinguishing information, what physical transformation or measurement removes access to that information, and what observable consequence follows.

Taken together, these works place photon mode erasure at the intersection of nonclassical mode structure, engineered indistinguishability, quantum measurement theory, and loss-tolerant information processing. The topic spans from foundational studies of nonlocality and complementarity to practical protocols for imaging with undetected photons, distributed detection, deterministic photon extraction, and erasure-aware fault tolerance (Gemmell et al., 2023, Vetlugin et al., 2022, Rosenblum et al., 2015, Graaf et al., 2024).