Entrapped Pulse Coincidence Detection in QKD

• EPCD is a coincidence-detection strategy that uses 'entrapped' pulses combined with timing validation to assess photon statistics in quantum key distribution.
• It enhances QKD protocols by allowing both single- and two-photon contributions into the secure-key calculation when measured coincidences match expected models.
• EPCD and related methods integrate pulse preservation, digital timing windows, and waveform retention to effectively manage accidental coincidences and pulse overlaps.

Entrapped Pulse Coincidence Detection (EPCD) denotes a coincidence-detection strategy in which specially designated “entrapped” pulses are introduced or preserved and then evaluated jointly with coincidence observations to test whether measured event statistics remain compatible with an expected source or channel model. In current arXiv usage, the term appears explicitly in quantum-key-distribution work, where EPCD is layered onto BB84 and used to monitor coincidences so that, when observed statistics match expectation, both single-photon and two-photon contributions can enter the secure-key calculation (Chandravanshi et al., 14 Aug 2025). A broader adjacent literature studies closely related mechanisms without adopting the EPCD label, including digitally retained gamma–electron coincidence spectroscopy (Gladen et al., 2019), FPGA-centered digital coincidence spectrometers (Hai et al., 2013), high-resolution time-tagging with coincidence-vector formation (Hidvegi, 2020), and superconducting on-chip pulse-overlap logic (Guay et al., 23 Apr 2026).

1. Terminology and conceptual scope

In its explicit QKD formulation, EPCD is not merely a narrow timing gate. It is a protocol-level combination of extra optical pulses, coincidence monitoring, and post-processing logic. Alice sends ordinary signal pulses together with additional “entrapped” pulses, and Bob monitors coincidences to determine whether observed photon statistics remain unchanged. If the coincidence statistics are consistent with the expected model, EPCD permits the secure-key calculation to include both one-photon and two-photon contributions rather than restricting the useful term to single photons alone (Chandravanshi et al., 14 Aug 2025).

The surrounding arXiv literature suggests that EPCD belongs to a wider family of coincidence-validation methods rather than a single settled hardware architecture. In some systems, the distinctive feature is delayed software validation of stored waveforms rather than hardware latching. In others, the crucial mechanism is a programmable digital coincidence window, a time-sorted stream of timestamps, or transient overlap in superconducting logic. This suggests that “entrapped” can refer either to specially injected protocol pulses, as in QKD, or more loosely to the preservation of event information long enough for coincidence-based acceptance or rejection, depending on context (Gladen et al., 2019).

A common misconception is to equate EPCD with ordinary coincidence counting. The QKD papers instead treat coincidence monitoring as a discriminator on photon-number statistics. Conversely, much adjacent hardware work performs coincidence detection without any explicit entrapment mechanism. The distinction between these two uses of the concept is central to the literature.

2. Protocol formulation in quantum key distribution

The clearest in-article statement of EPCD occurs in a free-space BB84 implementation. There, Alice sends a signal pulse of mean photon number $\mu$ together with additional entrapped pulses of mean photon number $\nu$, while Bob monitors coincidence events to test whether the received photon statistics have been altered. In that implementation, the general BB84 timing relation remains

$t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$

with the EPCD-specific addition that Alice logs the timestamps of the entrapped pulses and Bob records coincidence events (Chandravanshi et al., 14 Aug 2025).

The secure-key expression presented for EPCD extends the standard decoy-state form by adding a two-photon term: $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$ with

$$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$

This is the mathematical step that differentiates EPCD from a standard one-photon-only decoy analysis: the coincidence test is used to justify inclusion of $Q_2$ and $e_2$ in the rate expression (Chandravanshi et al., 14 Aug 2025).

A 200 m free-space realization integrates this protocol into a practical polarization-BB84 platform with four $808\ \mathrm{nm}$ laser diodes, a $5\ \mathrm{MHz}$ repetition rate, $1\ \mathrm{ns}$ optical pulses, FPGA clocks at $\nu$0, and four Excelitas SPCM-AQRH-14-FC SPADs. In the EPCD portion of that implementation, entrapped pulses with $\nu$1 are used while $\nu$2 is varied from $\nu$3 to $\nu$4, and the reported key-rate comparison places EPCD above BB84, ordinary coincidence detection, and decoy-state operation over the explored $\nu$5 range (Chandravanshi et al., 14 Aug 2025).

The associated review material for “Enhancing key rates of QKD protocol by Coincidence Detection” presents the same core idea in sharper security terms: Bob-side coincidence monitoring is intended to detect photon-number-splitting behavior and to permit a key-rate formula with an additional $\nu$6 contribution when observed coincidences match expectation. The same material also records a substantial controversy: reviewers argued that the security argument is restricted to specific PNS-style attack models and that a two-photon privacy term is not generally justified by standard GLLP reasoning, because bit and phase errors need not coincide for the two-photon component (Sharma et al., 2024).

3. Pulse handling and coincidence architectures

Outside QKD, EPCD-adjacent hardware spans several distinct pulse-handling models. One line of work keeps digitized detector waveforms available for later coincidence validation rather than performing irreversible online reduction. In positron-annihilation spectroscopy, a LeCroy HD 4096 oscilloscope acquires MCP and NaI(Tl) waveforms simultaneously, MATLAB performs FFT filtering, digital CFD timing for the MCP, ELET timing and shaping for the gamma channel, and a two-dimensional coincidence matrix $\nu$7 is formed offline. The system explicitly records “all of the coincident pulses” for later repeated analysis, but no explicit pretrigger buffer, pulse queue, or deferred hardware accept/reject mechanism is described; its relevance to EPCD is therefore methodological rather than terminological (Gladen et al., 2019).

A second line implements fully digital coincidence spectrometers in FPGA logic. A two-channel DSP-based coincidence spectrometer uses comparator-derived timing marks, a high-frequency counter, and a programmable time window $\nu$8 to store $\nu$9 event tuples, while its spectroscopy chain performs digital shaping, baseline restoration, and pile-up rejection. The coincidence window is reported as programmable from $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$0 to $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$1 with $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$2 steps, but the architecture does not describe any explicit entrapped-pulse recovery beyond rejection of overlapped events in the shaping interval (Hai et al., 2013).

More elaborate time-tagging hardware pushes the same principle to many channels. A 32-channel FPGA unit assigns timestamps with about $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$3 resolution, forms a time-sorted stream of tags, groups them into coincidence vectors in real time, and applies a programmable coincidence window in $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$4 steps up to $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$5. That system is directly relevant to EPCD at the architectural level because it supports both hardware coincidence detection and offline analysis from raw time tags, yet it still operates on timestamped events rather than an explicit pulse-entrapment model (Hidvegi, 2020).

Superconducting nanowire logic shows a different limit case: coincidence by transient overlap rather than pulse storage. A bias-programmable three-nTron gate implements AND, XOR, and OR directly on detector-like electrical pulses at $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$6. In coincidence mode, currents $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$7 and $t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$8 diverted from two side nTrons add at a central gate,

$t_{b_i}=t_{a_0}+t_d+n_iT\pm T_f,$9

and output occurs if $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$0. This is a self-resetting combinational overlap detector rather than a device that traps a first pulse until a second arrives, so it is adjacent to EPCD but not an implementation of strict pulse entrapment (Guay et al., 23 Apr 2026).

A related superconducting detector-array architecture encodes coincidence information into propagation delays and reflected pulse shapes on a nanowire transmission line. A 16-element detector resolves all $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$1 single-photon and two-photon event classes using only two terminals, while a 4-element variant uses waveform fingerprints to distinguish up to four-photon cases. Here again the information resides in timing logic and pulse-shape discrimination rather than in a trapped-pulse memory state (Zhu et al., 2017).

4. Temporal selection as an event-space operation

A major conceptual theme across EPCD-adjacent work is that coincidence selection does not merely suppress noise; it defines the effective event space. In Franson interferometry, the coincidence-time variable

$$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$2

selects the central slot $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$3, in which the two nonlocal alternatives $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$4 and $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$5 remain indistinguishable while $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$6 and $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$7 are excluded. Under that selection, the coincidence rate becomes

$$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$8

even though each local detector shows a uniform mean intensity. This is not called EPCD in the paper, but it supplies a precise model of how a coincidence slot can redefine the interfering basis (Ham, 2022).

Time-resolved coincidence detection in integrated microresonators realizes an analogous principle with cavity-stored pump energy. When the pump photon lifetime satisfies $$R \geq \frac{1}{2}\Big\{- Q_{\mu} H_{\mathrm{bin}}(E_{\mu}) f(E_{\mu}) + Q_1 \left( 1 - \Phi(2 e_1 - 1) \right) + Q_2 \left( 1 - \Phi((2 e_2 - 1)^2) \right)\Big\},$$9, where $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$0 is the $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$1 pump-pulse duration, pairs generated in the resonator become temporally distinguishable from pairs generated in the bus waveguide. In one configuration the paper reports $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$2, and temporal post-selection of the resonator region yields a measured purity of $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$3. This is closely aligned with EPCD in the sense that cavity storage creates a time domain in which only certain coincidences are retained (Borghi et al., 2024).

Coincidence gating in positron-annihilation spectroscopy provides a non-quantum-optical example of the same selectivity. There the central observable is a matrix $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$4, and projections gated on timing windows or gamma-energy windows selectively enhance or suppress physical channels. Gating on the delayed secondary-electron tail yields a gamma spectrum with a valley-to-peak ratio of $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$5, whereas gating on the prompt secondary peak reduces that ratio to $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$6. The method therefore uses coincidence sorting to change the spectral composition in a physically meaningful way (Gladen et al., 2019).

Ultrafast type-II SPDC supplies a further caution for EPCD-like interpretation: the conditional coincidence spectrum itself can contain hidden multi-branch structure. For a short-pulse pump, the coincidence distribution becomes a sum of two $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$7 terms associated with $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$8 and $$\Phi(x):= H_{\mathrm{bin}}\!\left(\frac{1}{2} + \frac{x}{2}\right).$$9, and the observed spectrum may split into two peaks because Bose symmetrization forces two distinct polarization-dispersion assignments into the same coincidence measurement (Fedorov et al., 2011). This shows that a single coincidence feature need not correspond to a single physical process.

5. Statistical inference, accidentals, and pulse-overlap limits

A recurring requirement in EPCD-like systems is explicit modeling of accidental coincidences. In ns-pulsed SPDC, true and accidental events can occur within the same pump pulse, so temporal filtering alone is insufficient. For whole-pulse counting the accidental rate is given as

$Q_2$0

and for sub-pulse analysis the paper uses

$Q_2$1

An empirical overlap factor $Q_2$2 is then introduced through

$Q_2$3

to account for real pulse shapes and timing response (Agüero et al., 2013). These formulas are directly transferable to any EPCD setting in which the pulse envelope itself confines both signal and accidental events.

When clocks are independent, coincidence identification must also solve for affine timing distortion. Remote correlated-photon timing is modeled by

$Q_2$4

and coarse alignment is obtained by cross-correlation

$Q_2$5

or, computationally,

$Q_2$6

The expected peak significance is

$Q_2$7

and the paper reports that $Q_2$8 keeps the wrong-peak probability below about $Q_2$9 for $e_2$0 (0901.3203). For EPCD in distributed systems, this establishes that coincidence extraction is fundamentally an inference problem on sparse time series, not just a gate-width choice.

A Bayesian version of the same inference problem appears in $e_2$1-fold photon-triplet characterization. There, a latent true event $e_2$2 and an observed coincidence declaration $e_2$3 are related through a binary noisy channel, and useful detection requires

$e_2$4

In the instrument-limited regime, the minimum detectable $e_2$5-fold source rate becomes

$e_2$6

which cleanly separates detector false-coincidence floor from total efficiency (Chen et al., 2024). This is particularly relevant to EPCD implementations that rely on coincidence logic but operate near the dark-count limit.

Waveform overlap imposes a separate identifiability limit. In frequency-domain multiplexed scintillator readout, two pulses in one digitized record are modeled as

$e_2$7

Sequential deconvolution can recover the first pulse well when the separation is at least the scintillator pulse width, about $e_2$8 for the EJ-309 signals used there, but when the delay shrinks to $e_2$9 the second pulse is contaminated by the unrecovered tail of the first. In the overlap case, the first-pulse timing uncertainty remains about $808\ \mathrm{nm}$0, but the second-pulse timing uncertainty degrades to $808\ \mathrm{nm}$1 with a positive bias of $808\ \mathrm{nm}$2, and its charge uncertainty degrades to $808\ \mathrm{nm}$3 (Mishra et al., 2020). For EPCD, this establishes that later pulses in an overlap sequence are intrinsically disadvantaged unless a stronger model than sequential subtraction is available.

6. Limitations, controversies, and current status

The present literature does not support a single canonical meaning of EPCD across fields. In QKD, EPCD is a protocol enhancement built around entrapped pulses and coincidence statistics. In most adjacent detector and electronics work, by contrast, the central mechanism is waveform retention, programmable digital windows, or transient current overlap, not literal pulse entrapment. A direct implication is that references to “EPCD” and references to “coincidence detection” cannot be treated as interchangeable without examining the actual pulse-handling semantics (Chandravanshi et al., 14 Aug 2025).

The most prominent controversy concerns security claims in the QKD application. The review-associated discussion of (Sharma et al., 2024) records explicit objections that the proposed two-photon key term rests on a restricted PNS-attack model and that the corresponding privacy term is not generally valid under standard unconditional-security reasoning. The free-space implementation paper therefore demonstrates practical integration and a reported rate advantage, but it does not settle the theoretical status of EPCD as a universally accepted secure-QKD primitive (Chandravanshi et al., 14 Aug 2025).

A second limitation is architectural. Several strongly related systems are useful precursors to EPCD yet explicitly stop short of implementing pulse entrapment in the strict sense. The positron coincidence spectrometer records coincident waveforms for later analysis but provides no hardware hold-and-wait logic and no explicit accidental-coincidence subtraction model in digital mode (Gladen et al., 2019). The superconducting nTron gate performs coincidence through transient overlap, but detector loading, reflections, and afterpulsing degraded detector-driven bit-error rates to below $808\ \mathrm{nm}$4, well above its electrically driven performance (Guay et al., 23 Apr 2026).

A third limitation is intrinsic recoverability. When pulses overlap strongly, later events become progressively less identifiable. The sequential deconvolution study shows that earliest-pulse timing can remain precise while later-pulse amplitude, charge, and shape become unreliable (Mishra et al., 2020). This suggests that any EPCD architecture intended for dense pulse traffic must specify whether it will reject, flag, or attempt to decompose overlapped events; coincidence logic alone does not resolve that ambiguity.

Taken together, the literature suggests that EPCD is best understood, at present, as a family of coincidence-validation strategies centered on selective event retention under timing, waveform, or protocol constraints. Its explicit, named formulation is currently strongest in QKD, where it aims to turn coincidence statistics into a security-relevant observable. Its broader methodological foundation lies in digitally retained waveforms, programmable coincidence windows, sparse-event synchronization, and overlap-aware inference. Whether future work will consolidate these strands into a single, hardware-explicit entrapment architecture remains an open question.