---
title: 'Photon Mode Erasure: Nonlocal Quantum Effects'
url: https://www.emergentmind.com/topics/photon-mode-erasure
type: topic
---

# Photon Mode Erasure: Nonlocal Quantum Effects

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 [1408.1769]. 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 [1501.00817] [1702.06888] [2406.04358] [1905.01823]. 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 [1006.3941] [2406.14621]. 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 [1408.1769]. 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 [1408.1769]. The paper explicitly distinguishes this from ordinary linear absorption, which is described by a Lindblad-type evolution,
\[
\frac{\partial \hat\rho}{\partial z}\propto \hat a_1\hat\rho \hat a_1^\dagger-\frac{1}{2}\left(\hat\rho \hat a_1\hat a_1^\dagger+\hat a_1\hat a_1^\dagger \hat\rho\right),
\]
and does create an ordinary shadow [1408.1769]. 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 $\lambda/2$ waveplate set $\mu=\lambda=1/\sqrt{2}$ in a polarization decomposition, photon subtraction on one local component was implemented with a strongly imbalanced partially polarizing beam splitter with around $6\%$ transmission for one polarization, and balanced homodyne detection with tomography reconstructed the output state [1408.1769]. With no subtraction click, the reconstructed state remained the prepared one-photon or two-photon Fock state. With a subtraction click, the reconstruction showed
\[
|1\rangle_{\hat a}\to |0\rangle_{\hat a},\qquad |2\rangle_{\hat a}\to |1\rangle_{\hat a},
\]
with the same spatial, temporal, and polarization mode structure; detection efficiency was about $53\%$, and a small residual population in the two-photon case was attributed to dark counts of $\mathrm{SPCM}_3$ [1408.1769].

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 [1408.1769].

## 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 [1501.00817]. The experiment extends standard induced-coherence arrangements by adding an idler Mach-Zehnder interferometer, so the erasure is controlled continuously by an idler phase $\Delta\varphi_i$. Under the assumptions $C_1=C_2$ and balanced beam splitters, the coincidence visibility becomes
\[
V(\Delta\varphi_i)=\frac{|\sqrt{8}\cos(\Delta\varphi_i/2)|}{2-\cos(\Delta\varphi_i)},
\]
with $V=0$ for $\Delta\varphi_i=n2\pi$ and $V=1$ for odd multiples of $\pi/2$; experimentally, coincidence visibilities reached about $97\%$ in the erasure condition and dropped to about $40\%$ in the distinguishable condition [1501.00817].

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 $T$ and a spectral filter of bandwidth $B$ can erase modal information when
\[
BT<1.
\]
The number operator of the filtered field is written as
\[
\hat n=\sum_{j=0}^\infty \chi_j c_j^\dagger c_j,
\]
with Schmidt eigenvalues $\chi_j$ determined by the filter kernel, and the single-mode regime corresponds to $\chi_0\approx 1$ and $\chi_{j\neq 0}\ll 1$ [1205.4346]. In heralded HOM interference using two independent SFWM sources, the measured visibility improved from $19\pm 2\%$ in a multimode regime to $72\pm 7\%$ in a single-mode regime, matching theory values of $17\%$ and $72\%$ with no fitting parameter [1205.4346].

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 $\ket{+\ell}$ and $\ket{-\ell}$ play the role of abstract paths, while the partner photon’s polarization marks or erases which-OAM information [1702.06888]. After a $q$-plate and postselection, the hybrid entangled state takes the form
\[
\psi_{AB}=\frac{1}{\sqrt{2}}\left(\ket{H}_A\ket{\ell}_B+e^{i\delta}\ket{V}_A\ket{-\ell}_B\right),
\]
and projection of photon $A$ onto diagonal or anti-diagonal polarization yields coherent superpositions $\ket{\ell}_B\pm\ket{-\ell}_B$ on photon $B$ [1702.06888]. The joint detection probability is
\[
P(\alpha,\theta)\propto \frac{1}{2}\left(1+\sin(2\alpha)\cos(2\theta+\delta)\right),
\]
so the fringe visibility is
\[
V=|\sin(2\alpha)|.
\]
This realizes complementarity in OAM space, with the paper explicitly stating $V^2+D^2\le 1$ [1702.06888].

“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 [2406.04358]. The spiral phase plate converts the Gaussian mode $\ket{0}$ into an OAM mode, and the output states become superpositions such as
\[
|\psi_{P4}\rangle=\frac{(|0\rangle+e^{i(\phi+\pi)}|1\rangle)}{\sqrt{2}}.
\]
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
\[
|\langle 0|\hat S_{1/2}|\psi^+\rangle|^2=\frac{1-\cos\phi}{2},
\]
and the measured fringe visibility after erasure was
\[
84.35\%\pm 1.7\%.
\]
The paper states that the delayed erasure choice satisfies space-like separation [2406.04358].

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 [2306.13821]. The radial and $\pi$ modes are
\[
\mathbf e_{\mathrm{rad}}(\varphi)=\mathbf e_H\cos\varphi+\mathbf e_V\sin\varphi,\qquad
\mathbf e_{\pi}(\varphi)=\mathbf e_H\cos\varphi-\mathbf e_V\sin\varphi.
\]
For temporally indistinguishable photons, the coincidence probability depends on transverse position, and the reported visibilities include
\[
\mathcal V_{HH}=1,\qquad \mathcal V_{HV}=-1,\qquad
\mathcal V_{AH}=\cos(2\varphi_1),\qquad \mathcal V_{AV}=-\cos(2\varphi_1),
\]
with several other projection pairs yielding zero visibility [2306.13821]. 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 [1702.06888] [2406.04358] [2306.13821].

## 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 $45\,\mathrm m$ [2407.07185]. The source state is
\[
|\Psi_+\rangle_{AB}=c|00\rangle_{AB}+s|11\rangle_{AB},\qquad
c=\cos(\theta/2),\quad s=\sin(\theta/2),
\]
and the analysis uses the Bilobran-Angelo irreality measure
\[
\mathfrak I_X(\rho)=S(\Phi_X(\rho))-S(\rho).
\]
When Alice measures in $\mathscr C_z$, Bob’s path is real, with $\mathfrak I_b=0$ and the auxiliary degrees of freedom $d_1,d_2$ also real. When Alice measures in $\mathscr C_x$, Bob’s path becomes indefinite with
\[
\mathfrak I_b=-c^2\log_2 c^2-s^2\log_2 s^2,
\]
and the same value applies to $\mathfrak I_{d_1}$ and $\mathfrak I_{d_2}$ after the beam displacers; at maximal entanglement, $\mathfrak I=1$ [2407.07185]. 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 [1910.10936]. After local CNOT-like interactions, a Hadamard transform on one meter and postselection erase local which-branch information, leaving
\[
\psi=\left[(\alpha a_H b_H+\eta a_V b_V)b_1+(\beta a_H b_V+\gamma a_V b_H)b_0\right]\otimes a_d b_d,
\]
so the ancilla outcome $b_1$ encodes even parity and $b_0$ odd parity [1910.10936]. 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 [1904.11889]. Two polarization Sagnac interferometers receive entangled photons in the state
\[
\psi=\frac{1}{\sqrt{2}}\left(H_sH_i+e^{i\varphi}V_sV_i\right),
\]
and the Franson phase is
\[
\phi=\frac{\omega_p}{2c}(\Delta L_s+\Delta L_i)+\varphi.
\]
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 [1904.11889]. The measured image quality was quantified by
\[
\mathrm{SNR}=\frac{1}{\sigma}\left|\bar I_{\mathrm{in}}-\bar I_{\mathrm{out}}\right|.
\]

The proposal “Counterfactual quantum erasure: spooky action without entanglement” radicalizes the remote-control aspect by removing entanglement from the resource list [1606.03081]. Alice marks which path in a Michelson interferometer by a switchable polarization rotator so that $D_1$ and $D_2$ 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 $D_1$ [1606.03081]. The visibility is
\[
\mathrm{Visibility}=\frac{2Y[M]}{(X[M])^2+(Y[M])^2+1},
\]
with reported example values of about $89\%$ for $M=2,N=4$ and about $99\%$ for $M=2,N=14$ [1606.03081]. The successful events are counterfactual in the sense stated by the paper: if Alice detects the photon at $D_1$ or $D_2$, 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 [1408.1769] [2407.07185] [1606.03081].

## 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 [1905.01823]. The physical implementation uses upconversion single-photon detectors based on periodically poled lithium niobate waveguides pumped at $1950\,\mathrm{nm}$, with signal wavelengths approximately $1550\,\mathrm{nm}$ and $863\,\mathrm{nm}$ satisfying
\[
\nu_{\gamma_1}+\nu_{\gamma_3}=\nu_{\gamma_2}.
\]
The interaction Hamiltonian is
\[
H=i\chi\left(a_{\gamma_1}a_{\gamma_2}^\dagger a_{\gamma_3}-a_{\gamma_1}^\dagger a_{\gamma_2}a_{\gamma_3}^\dagger\right),
\]
and in the large-pump limit it reduces to an effective color-space rotation [1905.01823]. 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
\[
|D_{1A}D_{2B}+D_{1B}D_{2A}|^2.
\]
The paper reports interference visibility about $0.4$, close to the ideal $0.5$, and demonstrates the effect with coherent lasers, thermal-like light, and a free-space geometry [1905.01823].

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 [2311.13421]. The first-pass signal polarization is rotated by an internal QWP with angle $\theta_1$, and an external HWP with angle $\theta_2$ either preserves or erases distinguishability between two generation alternatives. The limiting cases are explicitly identified: $\theta_1=0^\circ$ gives an NI-IUP-like situation, $\theta_1=90^\circ$ 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 [2311.13421]. The detected signal count rate $N$ contains incoherent and interference terms, and visibility is defined as
\[
\mathcal V=\frac{N_{\max}-N_{\min}}{N_{\max}+N_{\min}}.
\]
The setup uses a $1064\,\mathrm{nm}$ pump, a PPLN crystal, signal photons at $1550\,\mathrm{nm}$, and idler photons at $3.4\,\mu\mathrm m$ [2311.13421].

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 [2210.16653]. Instead of spatial or temporal multiplexing, a distributed detector array coherently absorbs a single standing-wave mode. For $M$ sublayers in the counterpropagating geometry, the required per-layer response for perfect total absorption is
\[
t=\frac{M}{M+1},\qquad r=\frac{1}{M+1},\qquad A=\frac{2M}{(M+1)^2}.
\]
The paper gives a striking example: a 10-layer distributed detector can resolve two-photon states with about $90\%$ probability and three-photon states with about $74\%$ probability, even though individual sublayers may absorb only around $17\%$ each in a traveling-wave picture [2210.16653]. 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
\[
\rho=(1-P_E)|\alpha\rangle\langle\alpha|+P_E|0\rangle\langle 0|,
\]
so the mode is either transmitted intact or completely replaced by vacuum [1006.3941]. 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,
\[
x_o\rightarrow x_o+\sqrt{G}x_m,\qquad p_o\rightarrow p_o+\sqrt{G}p_m,
\]
or probabilistic postselection with the threshold condition
\[
|x_m|>0.8\quad \text{and}\quad |p_m|>0.8
\]
for discarding an event [1006.3941]. The paper reports a maximum deterministic fidelity of approximately $0.57\pm 0.2$, surpassing the classical benchmark of $0.50$, and in the probabilistic protocol reports direct transmission at $P_E=0.25$ with about $F=75\%$ versus corrected output $F=82\%\pm 0.02$ for the entangled four-mode code and $F=77\%\pm 0.02$ for an entanglement-free version [1006.3941].

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” [2406.14621]. There, the dual-rail qubit is encoded in the single-photon manifold
\[
\ket{0_L}\equiv \ket{1,0},\qquad \ket{1_L}\equiv \ket{0,1},
\]
and the main erasure state is the joint vacuum $\ket{0,0}$. 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 $\Delta=\chi/2$, the transition frequencies are
\[
\omega_{\delta m}=\frac{N\chi}{2}+\delta m\,\Omega,\qquad \Omega=\sqrt{g_{\mathrm{bs}}^2+\left(\frac{\chi}{2}\right)^2},
\]
so an ancilla pulse can be selective for $N=0$ [2406.14621]. The reported performance includes a missed erasure fraction of $(9.0\pm 0.5)\times 10^{-4}$ in the abstract, and the mechanism-focused results report a false negative rate $p_{\mathrm{FN}}=3.7\pm 0.2\%$ per erasure check, a total erasure rate $2.92\pm 0.01\%$, and a Pauli error rate $0.31\pm 0.01\%$, dominated by cavity errors [2406.14621].

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 [1510.04042]. The relevant operator is not the annihilation operator but
\[
\hat s=|0\rangle\langle 0|+\sum_{n>0}|n-1\rangle\langle n|,
\]
which removes one photon regardless of $n>0$ [1510.04042]. Using single-photon Raman interaction with a single $^{87}\mathrm{Rb}$ 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 $\alpha$ to $\beta$, after which subsequent photons are transmitted. The measured extraction efficiency for low input photon number was about $40\%$, compared to about $52\%$ 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 [1510.04042].

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 [1408.1769], source identity [1501.00817], spectral-temporal multimode structure [1205.4346], OAM or wavefront labels [1702.06888] [2406.04358], local branch information in a meter [1910.10936], color labels at detection [1905.01823], or the occupation of an entire transmission mode in an erasure channel [1006.3941].

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 [1905.01823]. 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 [1702.06888] [2406.04358], but others concern nonlocal measurements [1910.10936], distributed-mode annihilation [1408.1769], or error correction against full-mode loss [1006.3941], 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 [1408.1769]. 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 [2407.07185]. 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 [2311.13421] [2210.16653] [1510.04042] [2406.14621].

Source: https://www.emergentmind.com/topics/photon-mode-erasure