REVIEW 2 major objections 5 minor 88 references
Limits of heralded photonic Bell-state generation in the presence of loss
T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Loss regime, not loss size, sets the best Bell-state scheme.
desk verdict Solid comparative study with a clean lumped-loss core; the frequency-domain winner map is useful but rests on an imported device model that needs qualifying before it becomes design guidance. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is a hierarchy of loss models built on amplitude-damping channels. Because uniform loss commutes with linear optics and vacuum is its fixed point, source, circuit, and detector losses in the lumped model combine into two efficiencies, and every scheme's heralding probability and fidelity reduce to one polynomial $f(\gamma_h)=1+\mu_1\gamma_h+\mu_2\gamma_h^2$ with circuit-specific coefficients. A second load-bearing object is the zero-transmission law for multiport beamsplitters, which forbids certain high-photon-output events in the 5-mode and 3-mode DFT circuits and explains why $\mu_1=0$ for the 5P5M and 6P6M schemes. For frequency-bin implementations the model imports the coupled-microring beamsplitter efficiency $\eta(\Omega,\theta)=1-2/(1+\sqrt{1+\Omega^2\csc^2\theta})$, which turns nonuniform, angle-dependent loss into the deciding factor in the scheme ranking. The yield identity $Y=p_{\mathrm{her}}F=p_{\mathrm{ideal}}\eta_h^{N_h}\eta_b^2$, independent of the noise polynomial, quantifies the unavoidable tradeoff between generation rate and output fidelity.
What would settle it
Measure the fidelity-versus-$(\gamma_s,\gamma_{\mathrm{id}})$ maps of Fig. 7 in an actual frequency-bin chip with characterized per-beamsplitter efficiencies $\eta(\Omega,\theta)$. If the hybrid 4P8M scheme does not outperform the 6P6M$-$ scheme when beamsplitters are lossy ($\Omega\approx20$), or if the crossovers in $\Omega$ occur at materially different loss values, the transfer-matrix model and the predicted winner map are wrong.
Extended reading notes
Core claim
Taking dual-rail Bell states as the benchmark, the paper establishes a ranking of the five schemes named by photon number and mode number (e.g., 4P5M and 6P6M) that changes with the error structure. In the lumped-loss model the heralding probability and fidelity of every scheme reduce to a single polynomial $f(\gamma_h)=1+\mu_1\gamma_h+\mu_2\gamma_h^2$; the 6-photon 6-mode circuit, when heralded by a (13)/(31) click pattern (6P6M$-$), has the maximum fidelity for every $0<\gamma_h<1$ [Eq. (29)], and among four-photon circuits 5P5M wins for $\gamma_h<1/3$ while 4P5M wins for larger $\gamma_h$ [Eq. (30)]. With heralded single-photon sources, 4P6M and 4P8M never achieve the highest fidelity; as signal-mode loss grows relative to multiphoton errors the winner moves from 6P6M$-$ to 6P6M$+$, then 5P5M, then 4P5M. In integrated frequency-bin implementations, where driven coupled-microring beamsplitters have angle-dependent efficiency $\eta(\Omega,\theta)$, the hybrid spatial-frequency 4P8M scheme wins when beamsplitter loss dominates, 6P6M$-$ wins under near-uniform loss, and 6P6M$+$ wins only at very efficient beamsplitters with significant multiphoton errors. The robustness of the higher-order-interference schemes is traced to the zero-transmission law for multiport beamsplitters.
Load-bearing premise
The frequency-domain ranking depends on the coupled-microring transfer-matrix loss model imported from prior work, in particular on the assumption that active beamsplitter loss is the dominant nonuniform circuit loss; a different frequency-beamsplitter device with different loss scaling could change which scheme wins.
Editorial extensions
If this is right
- Under the lumped-loss model, the 6P6M$-$ scheme achieves the highest Bell-state fidelity for all nonzero detector-side loss, with fidelity degrading only quadratically in the small-loss limit because its false-positive term starts at $\gamma_h^2$.
- With heralded single-photon sources, the 4P6M and 4P8M circuits are never the best; the optimal scheme is 6P6M$-$ for loss-dominated regimes and shifts through 6P6M$+$, 5P5M, and 4P5M as the multiphoton-error rate grows relative to signal loss.
- In frequency-bin chips, the hybrid spatial-frequency 4P8M scheme wins when active beamsplitter loss dominates, a result driven by minimizing the number of lossy frequency beamsplitters; 6P6M$-$ takes over when loss becomes more uniform.
- The yield identity $Y=p_{\mathrm{her}}F=p_{\mathrm{ideal}}\eta_h^{N_h}\eta_b^2$ is independent of the noise polynomial, so every scheme pays the same unavoidable rate-quality price for a given photon budget.
- The 5P5M and 6P6M circuits accept a single extra input photon on any mode as a valid Bell-heralding input, which can roughly double the success probability when the source herald detectors can resolve two-photon events.
Reading between the lines
- If the zero-transmission-law mechanism transfers to larger seed-state circuits, the same robustness ranking likely appears in direct generation of 3-qubit GHZ states and other DFT-based graph-state factories, which the paper mentions but does not simulate.
- In spatial-mode platforms where multiplexed near-deterministic sources remove the per-photon cost penalty, the boundaries of Fig. 4 should shift in favor of higher-$N$ schemes; this is a testable prediction the paper leaves implicit.
- A simple device-level design rule follows: first measure whether circuit loss is uniform or concentrated in angle-dependent beamsplitters, and that single measurement selects between the 6P6M family and the hybrid 4P8M strategy.
- Extending the analysis to threshold or array detectors with dark counts could alter the winner map in the low-fidelity corners, since dark counts act as additional false-positive sources; the paper provides the POVM machinery but does not carry out that comparison.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper analyzes five heralded schemes for preparing dual-rail photonic Bell states (4P5M, 5P5M, 4P6M, 6P6M, 4P8M and variants) under three progressively more realistic loss models: a lumped-loss model, a heralded single-photon-source model with multiphoton errors, and a frequency-bin implementation with architecture-dependent active-beamsplitter loss. For the lumped model it derives closed-form heralding probabilities and fidelities (Eqs. (25)–(27), Table II), identifies 6P6M− as the highest-fidelity scheme (Eq. (29)), and explains robustness via the zero-transmission law. With HSPS inputs it uses Perceval simulations with an explicit photon-number truncation to produce regime maps of the winning scheme (Fig. 4). In the frequency-domain model it imports a coupled-microring beamsplitter efficiency from the authors' earlier work (Eq. (33)) and finds that the hybrid 4P8M scheme wins when beamsplitter loss dominates, while 6P6M± win for more efficient beamsplitters (Fig. 7).
Significance. If the results hold, the paper provides useful, analytically transparent guidance for choosing HBSG schemes under realistic loss, and it sharpens the intuition that higher-order multiphoton interference, via the zero-transmission law, protects fidelity against loss and multiphoton errors. The lumped-loss section is a genuine strength: the formulas are parameter-free in the loss rates, the derivation is self-contained, and the Perceval numerics match. The explicit truncation convergence study in App. F1 is commendable. The frequency-domain section is potentially the most impactful part for integrated photonics, because it identifies a concrete hybrid spatial-frequency implementation of 4P8M that minimizes active beamsplitter use. However, the frequency-domain ranking is conditional on a device model imported from an unpublished preprint, and the paper would need a sensitivity analysis or experimental validation before the specific winner map can be regarded as robust.
major comments (2)
- [Sec. VI, Eq. (33) and Fig. 7] The entire frequency-domain comparison rests on the efficiency η(Ω,θ) of Eq. (33), which is imported from the authors' own Ref. [20] via the transfer matrix of Eq. (32). App. G shows the algebraic reduction of that transfer matrix to Eq. (33), but it does not independently validate the device model or its loss-versus-angle dependence. The only experimental anchor is footnote [57], with Ω≈16, while Fig. 7 sweeps Ω up to 500. Because the schemes differ in the number and angles of frequency beamsplitters they require, a different loss scaling with θ—for example the EOM-based design of footnote [51], whose 50:50 efficiency is capped near 60%—could reorder the winners. The paper acknowledges this limitation in Sec. VI.C, but Sec. VII states design guidance such as the preference for the hybrid 4P8M scheme without that qualification. I recommend adding a sensitivity analysis over alternative efficiency models, or at minimum restricting the regime-map claims to the Ω range supported by the device model, and carrying the caveat into the conclusions.
- [App. F1, Fig. 4(b)] The convergence analysis in App. F1 states that nextra=4 is sufficient for γid≤0.10 to obtain fidelities to within about one percent, yet Fig. 4 uses nextra=4 over a grid that extends to γid=0.12. More importantly, the winner margins in Fig. 4(b) are reported to 0.1%, which is at or below the truncation error suggested by the convergence data. The boundaries between 6P6M+, 5P5M, and 4P5M can therefore shift if the truncation is not fully converged in that region. The paper should either restrict the regime map to the validated truncation region, perform convergence checks at the boundary points, or provide uncertainty estimates on the margins so that the reader can judge which winner regions are numerically robust.
minor comments (5)
- [Abstract and Sec. I] The paper would benefit from an early statement that the frequency-domain results in Sec. VI depend on the coupled-microring device model of Ref. [20], so that readers are not surprised when the hierarchy of schemes changes relative to the lumped-loss model.
- [Sec. IV, Eq. (27)] The yield Y is defined as a fidelity-probability product but does not include the source-generation probability p_N; the text later notes this, but a brief reminder near Eq. (27) would prevent misinterpretation when comparing N=4, 5, and 6 schemes.
- [Sec. VI.B, Fig. 7 caption] The caption lists the variants (i) and (ii) for 5P5M and 6P6M± but does not define them; the definitions appear only in App. H3. A one-sentence pointer in the caption would improve readability.
- [App. F1, Fig. 10(f)] The non-monotonic behavior of |Δp_rel| for the 6P6M+ scheme is reported but not explained. Since the paper cites this as an open issue, it would be helpful to state explicitly whether any of the reported heralding-probability differences (e.g., in App. H1 or Fig. 17) could be affected by this oscillation at the precision level used.
- [Eq. (15)] In Eq. (15d) the notation c_{n≥3}=n ξ_i^{n-1}+O(γ^n) is slightly confusing because the error term is not uniform in n; clarifying the intended asymptotics (e.g., for fixed n as γ→0) would help.
Circularity Check
No significant circularity: the derivation chain is self-contained, with the frequency-domain model imported as an external component model rather than as a fitted or definitional input.
full rationale
The lumped-loss analysis in Sec. IV is parameter-free in the loss rates: the coefficients in Table II are obtained by direct Fock-state expansion of the ideal circuit outputs (e.g., Eq. (23) for 4P5M), and the ranking in Eqs. (29)-(30) follows from those polynomials without fitting any parameter to the quantity being predicted. The HSPS analysis in Sec. V uses the analytically derived source coefficients of Eqs. (13)-(15) and strong linear-optical simulation with a documented truncation-convergence check (App. F1), so the Fig. 4 regime map is a computed consequence of the model, not an input. The only reliance on the authors' prior work is the frequency-beamsplitter efficiency of Eq. (33), imported from Ref. [20]; this is a component-level device model with stated parameters (Omega, theta) and does not by itself contain the HBSG scheme comparison. The Sec. VI winner map is a nontrivial computation over different circuit topologies and beamsplitter counts, so it is not equivalent to Eq. (33) by construction. The paper explicitly identifies the device model as coming from Refs. [20, 53] and does not claim to derive it here; any concern about its validity is a correctness or validation issue, not circularity. The zero-transmission-law explanation is an external citation from Ref. [24], not a self-citation. Overall, the central results are derived rather than assumed, and no fitted parameter is renamed as a prediction.
Assumptions & free parameters
free parameters (5)
- gamma_b =
swept 0 to 0.4
- gamma_id (or gamma_h) =
swept 0 to 0.12
- Omega =
20, 40, 60, 100, 200, 500
- C_s(Omega, theta) =
engineered via Eq. (G4)
- |lambda|^2 =
chosen to maximize F1 p1 via Eq. (E18)
assumptions (5)
- domain assumption Single-mode photon loss is modeled by bosonic amplitude damping channels with multiplicative efficiency and commutation with uniform linear optics.
- ad hoc to paper The coupled-microring frequency beamsplitter transfer matrix of Ref. [20] is correct and can be written in the lossy beamsplitter form of Eq. (32) with efficiency of Eq. (33).
- domain assumption The photons are fully indistinguishable apart from their frequency bins, and this indistinguishability is not degraded by the frequency beamsplitters.
- domain assumption The detectors are PNR detectors with negligible dark counts for the main results.
- standard math The zero-transmission law of Ref. [24] applies to the DFT subcircuits and explains the robustness of 5P5M and 6P6M.
Cite this review
Pith. "Pith review of Limits of heralded photonic Bell-state generation in the presence of loss." pith.science (2026). https://pith.science/paper/QAHIO5N4
@misc{pith2026260801549,
author = {Pith},
title = {Pith review of: Limits of heralded photonic Bell-state generation in the presence of loss},
year = {2026},
howpublished = {\url{https://pith.science/paper/QAHIO5N4}},
note = {Machine review of arXiv:2608.01549}
}
read the original abstract
High-quality entangled states of photons underlie quantum information science (QIS) applications across communication, sensing, and computing. In many discrete-variable photonic QIS architectures, large application-ready states (e.g., cluster states, repeater graph states) are constructed via fusion measurements on small entangled seed states, of which Bell states are the fundamental example. The quality of seed-state generation therefore sets a baseline for application performance, making it crucial to understand this process under realistic error mechanisms, particularly in integrated photonics experiments. In this work, we analyze five heralded schemes for generating event-ready photonic Bell states, contrasting their heralding probabilities, fidelities, and error robustness. We develop a hierarchy of error models, progressing from an analytically tractable lumped-loss model to realistic heralded single-photon sources with multiphoton emission errors and finally to integrated frequency-bin implementations with architecture-dependent loss. Across these models, we find that schemes based on higher-order multiphoton interference provide superior fidelity robustness in low-loss implementations, while lower-photon-number schemes can be preferable when probabilistic sources or lossy beamsplitters dominate the resource cost. Our results provide design guidance for integrated discrete-variable quantum photonics, with particular relevance for frequency-bin architectures where active beamsplitter loss can determine the optimal resource-state-generation strategy.
Figures
Figures from the paper (16 more)
Reference graph
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In the experimental demonstration of Ref. 53, the strongest electro-optic modulation they use isϵ≈ 25dBm which we estimate corresponds toϵ/2π≈ 2.72GHz [20] and the internal loss rate of the device with the largest splitting was reported to beκ int/2π≈0.17GHz. Accordingly, the max- imum dimensionless modulation strength used is Ω =ϵ/κ int ∼16.Thiscanbeincr...
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[51]
This assumes a sinusoidal modulation inducing Bessel-weighted sidebands about a central fre- quency. Then the best efficiency isη= 2J 2 0 (z∗)≈ 0.60049, whereJ n(z)is the Bessel function of the first kind of ordernandz ∗ ≈1.4347is the smallest positive solution to|J0(z)|=|J ±1(z)|, the condition needed for 50:50 splitting. 19
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19 and is discussed in the surround- ing App
5P5M The Clements decomposition of this circuit is given in Fig. 19 and is discussed in the surround- ing App. H3 text. The corresponding DFT-based approach of Ref. 30 generalizes to the heralded gen- eration circuits forN-photon,d-dimensional GHZ states
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31, however, with phase conventions altered so that the final 50:50 beamsplitter can be omitted
4P6M The4P6MschemeofFig.1(c)istakenfromRef.9, which is based on the circuit presented in Ref. 31, however, with phase conventions altered so that the final 50:50 beamsplitter can be omitted. Note that the prescribedπphase on the top beamsplitter is not necessary, it simply acts to ‘flip’ the output Bell-like state to−|χ −⟩, whereas with the phase omitted,...
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4P8M This scheme, first proposed in Ref. 33, contains a significant amount of symmetry and structure [see Eq. (H4) and the surrounding discussion]. Said structure allows one to effectively “shift” the whole H4 subunitary around relative to four transversal beamsplitters, denoted here asT. For instance, one will also herald Bell-like states of the same for...
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Detector PNR requirements The number of detectors needed for each circuit and their necessary PNR capabilities varies. By ex- plicitly calculating the output states of the ideal 4P6M and 4P8M schemes one can easily compute the respective probabilities of a false positive, as in- duced by using threshold (as opposed to0,1,≥2) detectors for the post-circuit...
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Implementation flexibility There is additional freedom in exactly how each of these circuits is implemented including via permu- tations of the modes (e.g., to herald different Bell states) as well as different decompositions in some cases such as for the 5P5M and 6P6M circuits (see App. H3). Moreover, different arrangements of the input photons herald su...
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William R Clements, Peter C Humphreys, Ben- jamin J Metcalf, W Steven Kolthammer, and Ian A Walmsley. Optimal design for universal multiport interferometers.Optica, 3(12):1460–1465, 2016. 20 APPENDIX CONTENTS A Beamsplitter conventions 20 B Additional HBSG circuit details 20 C...
2016
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[69]
One can easily verify that this is equivalenttothebeamsplitterlossmodelofEq.(5b)
Amplitude damping channels On a single bosonic mode with creation (annihila- tion) operatora † (a), the AD channel is: ρ→ Eη(ρ) = ∞X l=0 ElρE† l (C1) with Kraus operators El = γ η l/2 al √ l! ηa†a/2 (C2) andγ≡1−η. One can easily verify that this is equivalenttothebeamsplitterl...
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[70]
After a few lines of algebra one finds Eη2 (Eη1 (|n1⟩ ⟩ ⟨ ⟨n2|)) =E η1η2 (|n1⟩ ⟩ ⟨ ⟨n2|),(C4) 22 which by completeness of the Fock basis verifies property 1
Single-mode loss is multiplicative The action of an AD channel on the Fock basis is thus Eη(|n1⟩ ⟩ ⟨ ⟨n2|) = mX k=0 ηn1+n2 n1 k n2 k 1/2 (C3) × γ η k |n1 −k⟩ ⟩ ⟨ ⟨n2 −k| withm= min(n 1, n2). After a few lines of algebra one finds Eη2 (Eη1 (|n1⟩ ⟩ ⟨ ⟨n2|)) =E η1η2 (|n1⟩ ⟩ ⟨ ⟨n2...
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Uniform loss commutes with linear optics Anm×mlinear optical unitary,U, transforms an- nihilation operators asa i U → Pm j=1 Uijaj (and simi- larly for creation operators). Meanwhile, under the beamsplitter loss model, loss induces the transfor- mationa i L → √ηai + √γbi, wher...
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[72]
One can immediately see that these form a complete POVM and, upon a brief calculation, can show that they have the ex- pected Fock state representationˆΠn =|n⟩ ⟩ ⟨ ⟨n|
PNR detectors The standard photocounting formula of an ideal PNR detector gives the probability of detecting ex- actlynphotons in a single-mode radiation field of stateρto be pn = Tr(ρˆΠn)(D3) with ˆΠn =:e −a†a a†a n n! :,(D4) wherea † andaare the respective single-mode cre- a...
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Threshold detectors Meanwhile, a threshold detector cannot resolve photon number, it either clicks if any light is de- tected or does not with respective POVM elements ˆΠclick(η) = ∞X n=1 ˆΠn(η, d) =1− ˆΠ0(η, d) = ∞X k=0 1−e −dγk |k⟩ ⟩ ⟨ ⟨k|,(D9a) ˆΠno-click(η) = ˆΠ0(η, d) =e−...
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[74]
distributed
Threshold array detectors A class of effective detectors with intermediate photon number-resolving capabilities consists of re- placing a single threshold detector by an array of Nd threshold detectors. Namely, we assume the sig- nal to be measured is split up uniformly across...
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[75]
These produced photons are constrained by energy conservation (ωs +ω i = ωp for SPDC andω s +ω i = 2ω p for SFWM)
Biphoton source Hamiltonian SPDC (SFWM) involves the interaction of three (four) waves within aχ(2) (χ(3)) nonlinear medium, 24 in which, with some probability, one (two) pump photon(s) of frequencyωp is (are) converted into two new photons, a signal and an idler of frequencie...
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[76]
all-click
Heralded single photons from biphoton sources After the multi-mode squeezed vacuum stateρ0 = |ψ0⟩ ⟨ψ0|of Eq. (E3) is generated, both the signal and idler modes propagate through lossy components, such as unloading microrings and output waveg- uides. Then, the idler modes are m...
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D we now specify the weightsw (j) k of Eq
HSPS quality For each type of detector considered in App. D we now specify the weightsw (j) k of Eq. (E5), then evaluatep (j) c andσ (j) n of Eqs. (E10b) and (E12b), respectively. For brevity, we omit the idler mode indexj, takingp (j) c →p 1, and use the shorthands γid = 1−η ...
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[78]
We do, however, note that the dominant effect of dark counts depends on where they occur
Implications of dark counts In-depth analysis of dark counts and the potential for array detectors is beyond the scope of this work, though we have set up the basic machinery. We do, however, note that the dominant effect of dark counts depends on where they occur. Dark counts...
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(31) is in- creased as is needed to capture the impact of higher- order multiphoton contributions
Convergence in terms ofn extra Here we investigate how the circuits behave as the truncation parametern extra of Eq. (31) is in- creased as is needed to capture the impact of higher- order multiphoton contributions. The simulations get more expensive in terms of memory and run...
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11 we report the fidelities for each HBSG scheme using HSPSs under the lumped circuit loss model
Fidelity, average photon number, and relative yield In Fig. 11 we report the fidelities for each HBSG scheme using HSPSs under the lumped circuit loss model. Thecorrespondingdatawasusedtogenerate Fig. 4. We also report the average photon number of the output Bell state, n= Tr ...
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[81]
Higher-Ninput flexibility and multiphoton-error robustness As alluded to in App. B, if we do not restrict to the prescribed number of input photons,N, for each circuit, there are additional inputs| ⃗N⟩ ⟩ that can be passed through these circuits to herald the generation of Bel...
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0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0
0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0. 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0. 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 γb 0. 0.2 0.4 0.6 0.8 1.0 (b) Bell-state average photon number,n 1.6 1.8 2 2.2 1.6 1.8 2 2.2 1.6 1.8 2 1.6 1.8 2 2.2 2.4 1.6 1....
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0.04 0.08 0.12 0.16 0.2 0
0.04 0.08 0.12 0.16 0.2 0. 0.04 0.08 0.12 0.16 0.2 0. 0.04 0.08 0.12 0.16 0.2 γb 1.25 1.50 1.75 2.00 2.25 2.50 2.75 FIG. 11. For each HBSG scheme, we construct a density plot of the (a) fidelity and (b) average photon number of the output Bell state, as a function of multiphot...
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[84]
A schematic illustration of the device is given in Fig
Physical setup Using a pair of strongly coupled microring res- onators each driven by an EOM, one can engineer frequency beamsplitters that act on a pair of fre- quency bins on the bus waveguide that one of the rings is coupled to [20, 53]. A schematic illustration of the devi...
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[85]
20: Ξ = 1− Γκ κ2+ϵ2 ieiϕ Γϵ κ2+ϵ2 ie−iϕ Γϵ κ2+ϵ2 1− Γκ κ2+ϵ2 ≡ Ξ11 Ξ12 −Ξ∗ 12 Ξ11 can be put into the form of Eq
Transfer matrix We will now show that the transfer matrix derived in Ref. 20: Ξ = 1− Γκ κ2+ϵ2 ieiϕ Γϵ κ2+ϵ2 ie−iϕ Γϵ κ2+ϵ2 1− Γκ κ2+ϵ2 ≡ Ξ11 Ξ12 −Ξ∗ 12 Ξ11 can be put into the form of Eq. (32) with the ef- ficiency as given by Eq. (33). We note that−1≤ Ξ11 ≤1and|Ξ 12| ≤1. To d...
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[86]
0.05 0.04 0.03 0.02 0.01 0
Fidelities and heralding probabilities Now we present all of the fidelities and heralding probabilities for the frequency-domain implementa- tion across each circuit and for each value ofΩ,γs, 33 98 95 92 89 87 84 93 90 87 85 82 80 88 85 83 80 78 76 83 81 78 76 74 72 78 76 74 ...
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[87]
grid of grids
0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 γs FIG. 16. For each circuit and value ofΩ∈ {20,40,60,100,200,500}, we construct a density plot of the (averaged...
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[88]
transverse
Hybrid spatial-frequency encodings As noted in Sec. VIB1, several heralded schemes for frequency-encoded resource-state generation can be spatially hybridized. This was illustrated for the 4P8M HBSG scheme in Fig. 6 and is illustrated for certain GHZ generation schemes in Fig....
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0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.2 0.2 0.1 0.1 0.1 0.1 0.2 0.2 0.1 0.1 0.1 0.2 0.2 0.2 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.4 0.2 0.3 0.3 0.3 0...
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[90]
0. 0. 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.4 0.5 0.5 0.5 0.5 0.6
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[91]
0. 0.1 0.1 0.1 0.2 0.1 0.1 0.2 0.2 0.2 0.3 0.2 0.2 0.2 0.3 0.3 0.4 0.3 0.3 0.3 0.4 0.4 0.5 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.1 0.1 0.1 0.2 0.2 0.3 0.2 0.2 0.2 0.3 0.3 0.4 0.3 0.3 0.3 0.4 0.4 0.5 0.3 0.4 0.4 0.5 0.5 0.6 0.4 0.5 0.5 0.5 0.6 0.6 0.5 0.6 0.6 0.6 0....
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0. 0. 0. 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.4 0.5 0.5 0.5 0.5 0.6
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[93]
0. 0. 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.6
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[94]
1.1 0.7 0.8 0.9 1
0.1 0.1 0.1 0.2 0.2 0.1 0.1 0.2 0.2 0.3 0.3 0.2 0.2 0.3 0.3 0.4 0.4 0.3 0.3 0.4 0.4 0.4 0.5 0.4 0.4 0.5 0.5 0.5 0.6 0.5 0.5 0.5 0.6 0.6 0.7 0.1 0.1 0.2 0.2 0.3 0.3 0.2 0.2 0.3 0.3 0.4 0.4 0.3 0.3 0.3 0.4 0.4 0.5 0.4 0.4 0.4 0.5 0.5 0.6 0.4 0.5 0.5 0.6 0.6 0.7 0.5 0.6 0.6 0.7 0...
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[95]
1.1 1.2 1.3 1.4 1.5 1.1 1.2 1.3 1.3 1.4 1.5 0.3 0.5 0.8 1. 1.2 1.5 0.3 0.6 0.8 1.1 1.3 1.5 0.4 0.6 0.8 1.1 1.3 1.5 0.4 0.6 0.9 1.1 1.3 1.5 0.4 0.7 0.9 1.1 1.3 1.5 0.5 0.7 0.9 1.1 1.3 1.5 0.6 0.9 1.1 1.3 1.5 1.7 0.7 0.9 1.1 1.4 1.6 1.8 0.7 0.9 1.2 1.4 1.6 1.8 0.7 1. 1.2 1.4 1.6...
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1.3 1.5 1.7 1.8 2
1.2 1.4 1.6 1.8 2. 1.3 1.5 1.7 1.8 2. 2.1 1.6 1.7 1.9 2. 2.2 2.3 1.8 1.9 2.1 2.2 2.3 2.5
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[97]
2.1 2.2 2.4 2.5 2.6 2.1 2.3 2.4 2.5 2.6 2.7 1.6 1.7 1.9 2.1 2.2 2.3 1.8 1.9 2.1 2.2 2.4 2.5
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[98]
2.1 2.3 2.4 2.5 2.6 2.7 2.3 2.4 2.5 2.6 2.7 2.8 2.4 2.6 2.7 2.8 2.9 3
2.1 2.3 2.4 2.5 2.6 2.2 2.3 2.4 2.5 2.6 2.7 2.3 2.4 2.5 2.7 2.8 2.9 2.5 2.6 2.7 2.8 2.9 3. 2.1 2.3 2.4 2.5 2.6 2.7 2.3 2.4 2.5 2.6 2.7 2.8 2.4 2.6 2.7 2.8 2.9 3. 2.6 2.7 2.8 2.9 3. 3. 2.7 2.8 2.9 3. 3.1 3.1 2.8 2.9 3. 3.1 3.1 3.2 3.2 3.3 3.4 3.4 3.5 3.5 3.3 3.4 3.4 3.5 3.5 3.6...
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[99]
1.2 1.4 1.6 1.8 2
1.2 1.4 1.6 1.8 2. 1.2 1.4 1.6 1.8 2. 2.2 1.4 1.6 1.7 1.9 2.1 2.3 1.5 1.7 1.9 2.1 2.2 2.4 1.1 1.3 1.5 1.7 2. 2.1 1.3 1.5 1.7 1.9 2.1 2.3 1.4 1.6 1.8 2. 2.2 2.4 1.6 1.8 2. 2.2 2.3 2.5 1.7 1.9 2.1 2.3 2.4 2.6 1.9 2.1 2.2 2.4 2.5 2.7 1.7 1.9 2.1 2.3 2.4 2.6 1.8 2. 2.2 2.4 2.5 2.7
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2.3 2.5 2.6 2.8 2.9 3
2.1 2.3 2.5 2.6 2.8 2.1 2.3 2.4 2.6 2.7 2.9 2.2 2.4 2.5 2.7 2.8 3. 2.3 2.5 2.6 2.8 2.9 3. 2.3 2.5 2.7 2.8 3. 3.1 2.4 2.6 2.7 2.9 3. 3.2 2.5 2.7 2.8 3. 3.1 3.2 2.6 2.8 2.9 3.1 3.2 3.3 2.7 2.9 3. 3.1 3.3 3.4 2.8 3. 3.1 3.2 3.3 3.4 3.7 3.8 3.9 4. 4.1 4.2 3.8 3.9 4. 4.1 4.1 4.2 3....
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[101]
4. 4.1 4.2 4.3 4.3 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 4P5M 4P6M 4P8M 4P8M hybrid 5P5M 6P6M+ 6P6M- 0.05 0.04 0.03 0.02 0.01 0. 0.05 0.04 0.03 0.02 0.01 0. 0.05 0.04 0.03 0.02 0.01 0. 0.05 0.04 0.03 0.0...
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[102]
0.01 0.02 0.03 0.04 0.05 0
0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 0. 0.01 0.02 0.03 0.04 0.05 γs (b) O u t [ ] = 0 5 10 15 20 pideal (%) 0 0.02 0.04 0.06 pN pideal (%) FIG. 17. (a) Comparison of the HBSG s...
2000
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[103]
▲fash- ion
Circuit decomposition variants For the frequency-domain implementation of Sec. VIB, the place where the beamsplitter loss oc- curs matters, we must pick specific instantiations of the circuits. For the 4P5M, 4P6M, and 4P8M (as well as its hybrid variant) we simply use the cir-...
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[104]
When beamsplitter loss lumps Generally, the introduction of lossy beamsplit- ters, especially in the frequency domain, goes be- yond the scope of the lumped-loss model as seen in Fig. 5. However, the 4P6M and 4P8M circuits have the property that beamsplitter loss can faithfull...
Reviewed August 7, 2026 · model on record in the stance chip above.
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