{"id":"d26a11a6-1c08-475b-bc59-a4eab223711a","arxiv_id":"2608.07253","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For lossy optical systems with scattering matrices whose singular values are only 0 or 1, the input state decomposes into deterministic photon-loss subspaces, each evolving coherently, with the output a statistical mixture.","lead":"The paper splits the quantum state of light entering a lossy optical device into parts that either get fully absorbed or pass through loss-free, then shows the output is a mix of these parts. It uses this picture to explain known interference effects and to design a lossy three-port device that outputs W-states for several inputs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"DPLS theory is well-defined only for singular values exactly 0 or 1; the claimed general-σ extension (Appendix C) reintroduces ancilla modes and coincides with earlier SVD embedding methods, so the advertised ancilla-free generality does not hold.","rationale":"The restricted claim is the strongest one: for M=UΣV† with Σ diagonal entries in {0,1}, Eq. (20) is correct. The proof follows from the standard SVD dilation in Appendix A; the lossy modes' photons are transferred to ancilla modes and traced out, yielding a mixture over the photon-number distribution n in the lossy modes. I checked the normalization: M_n maps a normalized DPLS state to a normalized output state, so Σ P_n = 1 and the trace is preserved. The example in Section II D reproduces the expected total Hilbert-space dimension (10) and yields a valid block-diagonal output density matrix in Table I. Thus there is no fatal mathematical error in the restricted setting. The load-bearing weakness is the paper's claim to be a general theory for lossy systems without extra degrees of freedom. For σ∈(0,1), deterministic photon-loss subspaces do not exist: the number of photons lost is random, so no label n can be assigned with certainty. Appendix C repairs this only by adding an ancilla per partial-loss mode, enlarging the scattering matrix so its singular values become 0 or 1, and then tracing the ancilla. That is exactly the embedding of Refs. [37-39], as the reader notes. The paper's own Appendix A also uses ancilla modes for the σ∈{0,1} case, which undercuts the 'extra degrees of freedom are unnecessary' statement even in the restricted case, though the final description in terms of DPLS labels is a legitimate compactification. The W-state generation result is a valid application but is probabilistic under post-selection for generic inputs; the paper states the success probabilities, so this is not hidden, but it should be featured in the abstract. Overall, the central theorem is correct but the scope is more limited than advertised. The reader's CONDITIONAL verdict, with revised scoping claims and clearer positioning relative to SVD embedding, is appropriate. My stress-test does not change that verdict.","tokens_in":26615,"tokens_out":22107,"duration_ms":213043,"concrete_test":"For the 2×2 matrix M = [[3/4,1/4],[1/4,3/4]] in Eq. (C6), compute the output state for input |10⟩ two independent ways: (i) follow the extended-DPLS construction of Appendix C, Eqs. (C7)-(C14), including the ancilla mode; (ii) use the SVD-embedding unitary of Ref. [38] directly on M with one vacuum ancilla and trace it out. If the two reduced 2-mode density matrices coincide, then the Appendix C procedure is identical to the prior embedding method, confirming that the general-σ extension does not provide an ancilla-free DPLS decomposition in the physical Hilbert space. Additionally, attempt to apply the main-text equations (3)-(20) directly without ancilla to this M; the absence of any singular value equal to 0 or 1 means no completely lossy or lossless input modes can be defined, so the main-text construction cannot even be started.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central output formula Eq. (20) is derived for scattering matrices whose singular values are exactly 0 or 1. For such matrices, the DPLS decomposition is mathematically sound: the SVD dilation in Appendix A provides a legitimate unitary embedding, and tracing the ancilla yields the mixture in Eq. (20). However, the paper advertises a 'general theory' and claims that 'extra degrees of freedom are unnecessary' (Section II, p. 6). This claim is not sustained for the generic case σ∈(0,1): no input mode is perfectly transmitted or absorbed, so the number of lost photons is random and no deterministic-loss subspace exists in the physical Hilbert space. Appendix C 'extends' the theory only by introducing an ancilla mode for each partial-loss mode, enlarging the Hilbert space so that the extended singular values are in {0,1}, and then tracing the ancilla out. This is precisely the SVD-embedding procedure of Refs. [37-39] that the paper claims to improve upon. Moreover, even the σ∈{0,1} derivation in Appendix A uses ancilla modes, so the 'no extra degrees of freedom' rhetoric in the main text is at least overstated. The consequence is that the central claim's scope is much narrower than the title and abstract suggest: it applies cleanly only to CPA/CPT-type systems with perfect absorption/transmission modes, not to generic lossy devices. The W-state application, while correct, is probabilistic under post-selection for generic inputs (50% and 33.3% success probabilities are quoted in Section III B). These scoping and novelty issues do not invalidate the restricted theorem, but they do limit the paper's contribution as a 'general theory'.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a 'deterministic photon-loss subspace' (DPLS) framework for linear lossy optical systems whose scattering matrix M has SVD singular values restricted to {0,1}. It defines complete-loss and lossless input modes via singular vectors, decomposes the input Hilbert space into orthogonal DPLSs labeled by the photon-number distribution in the lossy modes, and derives the output state as a statistical mixture of the coherently evolved projections, Eq. (20). The framework is applied to revisit anti-HOM interference and DV/CV state distillation in a two-port CPA-type system, and to design a three-port lossy system whose one-dimensional DPLSs allow post-selected W-state generation from several single-photon inputs. Appendix C sketches an extension to general singular values in [0,1] by enlarging the scattering matrix with ancilla modes, and Appendix A provides the ancilla-dilation derivation of Eq. (20).","tokens_in":26845,"tokens_out":10557,"duration_ms":96322,"significance":"For the sigma in {0,1} case the central derivation is coherent and self-contained: the projectors in Eq. (9) are orthonormal, the dimension formula Eq. (8) is correct, and the mixture formula Eq. (20) follows from tracing the ancilla dilation in Appendix A. The sigma in {0,1} regime covers coherent perfect absorption and transmission devices, and the article gives concrete, falsifiable predictions for the anti-HOM peak ratio and for post-selected W-state success probabilities; the W-state device is a genuine constructive design rather than a post hoc fit. The main weakness is that the advertised generality is not achieved: the extension to intermediate singular values in Appendix C reintroduces ancilla modes and is essentially the SVD-embedding procedure of Refs. [37-39], and even the sigma in {0,1} proof in Appendix A uses an ancilla dilation. The physical DPLS picture is therefore strongest if presented as a theory of perfect-absorption and perfect-transmission mode decompositions, not as a general ancilla-free theory of arbitrary lossy systems.","major_comments":[{"comment":"The final paragraph of Section II C states that the theory can be applied to scattering matrices with sigma in [0,1] by introducing an ancilla mode for each mode with sigma not in {0,1}, and Appendix C carries out this embedding. This contradicts the Section II claim that 'extra degrees of freedom are unnecessary' and places the general-sigma extension on the same footing as the SVD embedding methods of Refs. [37-39]. Because this is part of the advertised scope of the paper, please either restrict the generality claims to sigma in {0,1} throughout the title, abstract, and introduction, or supply a genuinely ancilla-free derivation for partial-loss modes.","section":"Section II C, last paragraph, and Appendix C"},{"comment":"In Eq. (14), the normalized state after the Sigma operation is written with a single amplitude D_mn and a ket with no summation over m; as printed, this is not the normalized projection |psi_n,mid2> that is used in Eqs. (15) and (20). Please correct the summation and normalization, or state explicitly that the equation defines an unnormalized component and that normalization is restored before Eq. (15).","section":"Eq. (14)"},{"comment":"The derivation of Eq. (20) in Appendix A explicitly introduces N ancilla modes and traces them out in Eq. (A5). The statement in Section II that 'extra degrees of freedom are unnecessary' is therefore also overstated for the sigma in {0,1} case: the DPLS output formula is obtained by an ancilla dilation, even though the final formula is on the physical Hilbert space. Please rephrase the claim to distinguish between the physical DPLS construction and the proof technique used to derive the loss mechanism.","section":"Appendix A"}],"minor_comments":[{"comment":"There are several typographical errors that should be corrected, including 'surpress' in Section I, 'rwo' in the discussion after Eq. (3), and 'resonable' in the data-availability statement.","section":"Throughout"},{"comment":"The sentence 'The input photons in modes b_nls^dagger_j (k+1 <= j <= N) ... will be completely dissipated' should refer to the lossy modes b_ls^dagger_j, not the lossless modes; as written it contradicts the definitions in Eqs. (3) and (4).","section":"Section II A, paragraph after Eq. (4)"},{"comment":"The amplitudes D_mn / sqrt(P_n) are typeset in a way that can be misread as products D_mn sqrt(P_n); please use explicit division markers or parentheses throughout.","section":"Eqs. (19) and (A5)"},{"comment":"The success probabilities of 100%, 50%, and 33.3% for W-state generation are post-selected single-photon output probabilities; although the main text does state that post-selection is used, the abstract and conclusion should make this explicit to avoid overclaiming deterministic generation.","section":"Section III B"},{"comment":"The 2x2 matrix S_i is presented with a row-vector-like layout in the displayed equation; please format it unambiguously as a two-row matrix so that the embedding into Sigma_ext is clear.","section":"Appendix C, Eq. (C1)"}],"recommendation":"major_revision","confidential_remarks":"The core sigma in {0,1} results are sound and the applications are concrete, so I would not reject the manuscript. The main issue is scope: the paper advertises a general ancilla-free theory, but Appendix C and Appendix A rely on ancilla dilations, and the general-sigma extension reproduces the embedding approach of Refs. [37-39]. If the authors are willing to reframe the contribution as a DPLS theory for perfect-absorption/transmission mode decompositions, with the ancilla-based extension clearly identified as a standard embedding, the paper could be suitable for publication after revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe DPLS paper is a mixed bag. The core theorem for scattering matrices with singular values exactly 0 or 1 is correct: the input Hilbert space decomposes into orthogonal subspaces labeled by photon number in lossy modes, each projection evolves unitarily after deterministic loss, and the output is the mixture in Eq. (20). That part is clean, and the derivation in Appendix A works. The paper also has two genuinely new applications: distillation of squeezed vacuum from squeezed coherent states with different coherent amplitudes (Eq. 37) and the three-port W-state preparation that works for a family of input states, albeit with post-selection.\n\nThe soft spots are real and mostly about scope. The main text says 'extra degrees of freedom are unnecessary,' but the proof of the central result already uses ancilla modes in Appendix A to model the loss channels. That is standard dilation, not a flaw by itself, but the rhetoric is wrong. More importantly, the advertised general theory for singular values in (0,1) collapses in Appendix C: you introduce one ancilla per partial-loss mode, embed the scattering matrix so its singular values become 0 or 1, and then trace out. That is exactly the SVD-embedding method of Refs. [37-39], which the paper claims to improve upon. The title and abstract promise a general theory, but the clean ancilla-free subspace picture only exists for CPA/CPT-type devices with perfect absorption/transmission modes.\n\nThe W-state generation is honest: it states success probabilities of 100%, 50%, and 33.3% for different inputs. It is still probabilistic under post-selection for generic inputs, which is fine, but the 'robust' language should be paired with that caveat. Similarly, the anti-HOM and distillation sections revisit known effects, and the SVD machinery is already in the literature. The new content is the subspace interpretation and the two applications.\n\nNo fatal mathematical error. The paper deserves a serious referee, but it needs major revisions: narrow the claims to the 0/1 case, position Appendix C as an embedding extension rather than a separate achievement, and cite the earlier SVD dilations properly.\n\nRecommendation: send to peer review with a request for major revision. I'd bring it to a reading group focused on lossy quantum optics, but not otherwise.\n\nBest,","headline":"A correct but narrowly scoped reformulation of known SVD loss machinery, with two worthwhile new applications and an overclaimed 'general theory' that falls back on ancilla embedding.","tokens_in":27461,"tokens_out":2400,"would_cite":true,"duration_ms":22356,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a theory of deterministic photon-loss subspaces for linear lossy optical systems: if the scattering matrix has singular values only in $\\{0,1\\}$, the input Hilbert space splits into orthogonal subspaces in which photon…","keywords":["deterministic photon-loss subspace","singular value decomposition","non-unitary scattering matrix","coherent perfect absorption","anti-HOM interference","quantum state distillation","W-state generation","quantum decoherence"],"falsifier":"Send the two-photon state $|11\\rangle$ into the two-port coherent perfect absorber described by Eq. (26) and count output photons per port: the theory predicts a 50/50 mixture of the vacuum and a two-photon entangled state, so the probability of detecting exactly one photon in total must be zero. Any detected single-photon output would contradict the deterministic-loss claim.","tokens_in":26353,"feed_emoji":"⚛️","tokens_out":11874,"duration_ms":109177,"temperature":0.7,"pith_summary":"The paper proposes a theory of deterministic photon-loss subspaces (DPLSs) for linear lossy optical systems: when the scattering matrix's singular values are only 0 or 1, the input Hilbert space splits into orthogonal subspaces labelled by how many photons sit in the fully absorbing modes. Inside each subspace, photon loss becomes deterministic (every photon in a lossy mode is absorbed, every photon in a lossless mode passes through unitarily) and quantum coherence is preserved; between subspaces, decoherence turns the output into a statistical mixture. This gives an exact, analytical account of how quantum coherence, decoherence, and photon-number reduction jointly shape the evolution of quantum light in lossy devices, without the Langevin noise operators or ancilla embeddings of earlier approaches. On that basis the paper re-derives anti-Hong-Ou-Mandel interference and squeezed-state distillation, and shows how a three-port lossy system with one lossless mode robustly produces W-states from several different inputs. If the theory is right, loss becomes a structural resource for preparing quantum states rather than only an obstacle.","feed_headline":"Loss turns deterministic inside engineered optical subspaces","feed_subtitle":"With perfectly absorbing modes, coherence survives each loss channel, turning loss into a design tool.","key_machinery":"The carrying object is the DPLS itself: the orthogonal decomposition of the input Hilbert space into subspaces $\\mathcal{H}^{\\mathrm{in}}_{\\vec n}$ labelled by the photon-number distribution in the completely lossy modes defined by the singular value decomposition of the scattering matrix. The machinery is the SVD $M=U\\Sigma V^\\dagger$ with $\\Sigma=\\mathrm{diag}(1,\\ldots,1,0,\\ldots,0)$, read as three stages --- unitary rotation $V^\\dagger$, diagonal non-unitary loss $\\Sigma$, unitary rotation $U$ --- so that loss acts only on the modes with singular value 0. Projectors $\\hat P_{\\vec n}=\\sum_{\\vec m}|\\varphi_{\\vec m\\vec n}\\rangle\\langle\\varphi_{\\vec m\\vec n}|$ decompose any input, and the operator $\\hat M_{\\vec n}=\\hat S_U\\big(\\prod_j (\\hat a_j)^{n_j}/\\sqrt{n_j!}\\big)\\hat S_{V^\\dagger}$ implements each DPLS's evolution as deterministic annihilation followed by unitary transformation. The key identity is Eq. (20), where the output is the incoherent sum over DPLSs of coherently evolved projections, with weights $P_{\\vec n}$ equal to the projection probabilities. This structure is what turns a non-unitary scattering problem into a solvable sum of fully coherent sub-evolutions.","core_discovery":"The central claim is a decomposition theorem for lossy linear optics. For an $N$-mode device with scattering matrix $M=U\\Sigma V^\\dagger$ whose singular values lie only in $\\{0,1\\}$, the lossless input modes $\\hat b^{\\mathrm{nls}\\dagger}_j$ are those mapped to singular value 1 and the completely lossy input modes $\\hat b^{\\mathrm{ls}\\dagger}_j$ are those mapped to singular value 0. Indexing the input Hilbert space by the photon-number distribution $\\vec n$ in the lossy modes produces orthogonal subspaces $\\mathcal{H}^{\\mathrm{in}}_{\\vec n}$ (the DPLSs) that form a direct sum of the whole space; each has dimension $\\binom{N_t+k-1}{N_t}$ fixed by the photon number $N_t$ in the $k$ lossless modes. The evolution of a projection onto $\\mathcal{H}^{\\mathrm{in}}_{\\vec n}$ is the operator $\\hat M_{\\vec n}=\\hat S_U\\big(\\prod_j (\\hat a_j)^{n_j}/\\sqrt{n_j!}\\big)\\hat S_{V^\\dagger}$ --- deterministic photon annihilation in the lossy modes followed by a unitary --- and because components in different DPLSs lose their relative coherence, the output is the mixture $\\hat\\rho_{\\mathrm{out}}=\\sum_{\\vec n} P_{\\vec n}\\hat M_{\\vec n}|\\psi_{\\vec n}\\rangle\\langle\\psi_{\\vec n}|\\hat M^\\dagger_{\\vec n}$ (Eq. (20)). The paper argues this mixture structure is exactly the joint effect of quantum interference, photon-number reduction, and decoherence, and that the geometry of the subspaces can be engineered to control all three.","pith_inferences":["One extension not made in the paper: the same one-dimensional-DPLS logic used for W-states generalizes to other targets, since a device whose DPLSs are all one-dimensional acts as a deterministic filter mapping any input's projection onto a prescribed output; higher-$N$ W-states or graph-type states could be prepared by choosing $V^\\dagger$ and $U$ accordingly.","Because the mixture weights $P_{\\vec n}$ are set by the input state and the DPLS geometry alone, the theory yields quantitative predictions for output photon-number statistics and second-order correlations that could be tested on tunable-loss devices such as metasurfaces or coupled waveguides with adjustable absorption.","The block-mixture structure of Eq. (20) implies that coherence between different photon-number sectors is never regenerated by unitary post-processing once loss has occurred, suggesting a practical bound on distillation in lossy channels: only within-sector coherence is recoverable."],"forward_implications":["For the two-port coherent perfect absorber, the input $|11\\rangle$ projects equally onto the lossless and fully-lossy DPLSs, which immediately explains anti-HOM interference and the zero-or-two-photon absorption rule: exactly one photon can never be absorbed.","Squeezed coherent inputs are distilled into a pure squeezed vacuum because every DPLS projection evolves to the same squeezed vacuum; the scheme extends to inputs with unequal coherent amplitudes provided $\\alpha_1/\\alpha_2=\\tan\\theta$, which amounts to engineering the DPLS geometry.","In a three-port lossy system with a single lossless mode, all one-dimensional DPLSs with one photon in the lossless mode evolve to the same W-state; postselecting the one-photon output yields $|W\\rangle$ for various inputs with success probabilities 100%, 50%, and 33.3% in the paper's examples.","Because no ancilla modes are needed when singular values are 0 or 1, the theory produces explicit analytical output density matrices (such as the block-form Table I) without tracing out extra degrees of freedom.","All DPLS dimensions and the direct-sum decomposition generalize to higher mode numbers and arbitrary input photon numbers, pointing toward high-dimensional entanglement preparation and multi-qubit gates in non-Hermitian systems."],"supporting_citations":[{"why":"Supplies the SVD-embedding method (embedding the non-unitary scattering matrix into a $2N$-dimensional unitary via ancilla modes) that the DPLS theory positions itself against and extends.","marker":"[37]"},{"why":"Provides the generalized SVD treatment of lossy $N$-port devices whose extended-system input-output relation is used in the paper's Appendix A when tracing out ancilla modes to derive the mixture output.","marker":"[38]"},{"why":"Gives the alternative ancilla construction (lossless two-mode beam splitters coupling each lossy mode) that DPLS claims to supersede, and is referenced for the zero-or-two-photon absorption effect.","marker":"[39]"},{"why":"Reported anti-coalescence of bosons on a lossy beam splitter; the peak ratio $g^{(2)}_{q,\\mathrm{out}}/g^{(2)}_{c,\\mathrm{out}}=2$ that the DPLS calculation reproduces.","marker":"[17]"},{"why":"Reported anti-Hong-Ou-Mandel interference from coherent perfect absorption of entangled photons; the phenomenon the paper re-derives from DPLS projections.","marker":"[12]"},{"why":"Demonstrated quantum coherent absorption of squeezed light; the distillation outcome the paper re-derives with DPLS and generalizes to unequal coherent amplitudes.","marker":"[30]"},{"why":"Exhibited anti-PT-symmetric selective filtering whose steady entangled states the DPLS theory reproduces while fully incorporating loss.","marker":"[22]"},{"why":"Supplies the beam-splitter and phase-shifter decomposition used to realize the unitary matrices $V^\\dagger$ and $U$ in the engineered W-state-generation system.","marker":"[44]"}],"fun_headline_variants":["Deterministic loss: turning dissipation into a design tool","Loss becomes deterministic in engineered subspaces","Quantum interferences tamed by perfectly lossy modes","Subspace theory: extracting coherence from loss","Engineered loss: deterministic subspaces for quantum control"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The picture rests on every mode of the device being either perfectly transmitting or perfectly absorbing, with nothing in between; for devices with partial absorption the paper's own appendix has to add extra auxiliary modes, reintroducing exactly the extra structure the main text claims to avoid.","fun_headline_variants_meta":{"raw":{"variants":["Deterministic loss: turning dissipation into a design tool","Loss becomes deterministic in engineered subspaces","Quantum interferences tamed by perfectly lossy modes","Subspace theory: extracting coherence from loss","Engineered loss: deterministic subspaces for quantum control"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000786,"raw_usage":{"total_tokens":3616,"prompt_tokens":1239,"completion_tokens":2377,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":855,"completion_tokens_details":{"reasoning_tokens":2319}},"tokens_in":855,"tokens_out":2377,"duration_ms":15945,"temperature":1.0,"reasoning_tokens":2319,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T11:41:34.484833+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Send the two-photon state $|11\\rangle$ into the two-port coherent perfect absorber described by Eq. (26) and count output photons per port: the theory predicts a 50/50 mixture of the vacuum and a two-photon entangled state, so the probability of detecting exactly one photon in total must be zero. Any detected single-photon output would contradict the deterministic-loss claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the generalized SVD treatment of lossy $N$-port devices whose extended-system input-output relation is used in the paper's Appendix A when tracing out ancilla modes to derive the mixture output."},{"cited_title":"Kn¨ oll, S","cited_arxiv_id":null,"evidence_quote":"Gives the alternative ancilla construction (lossless two-mode beam splitters coupling each lossy mode) that DPLS claims to supersede, and is referenced for the zero-or-two-photon absorption effect."},{"cited_title":"Ehrhardt, M","cited_arxiv_id":null,"evidence_quote":"Reported anti-coalescence of bosons on a lossy beam splitter; the peak ratio $g^{(2)}_{q,\\mathrm{out}}/g^{(2)}_{c,\\mathrm{out}}=2$ that the DPLS calculation reproduces."},{"cited_title":"Zhou, Characterization of PT-symmetric quantum interference based on the coupled mode theory, Opt","cited_arxiv_id":null,"evidence_quote":"Reported anti-Hong-Ou-Mandel interference from coherent perfect absorption of entangled photons; the phenomenon the paper re-derives from DPLS projections."},{"cited_title":"Jeffers, Nonlocal coherent perfect absorption, Phys","cited_arxiv_id":null,"evidence_quote":"Demonstrated quantum coherent absorption of squeezed light; the distillation outcome the paper re-derives with DPLS and generalizes to unequal coherent amplitudes."},{"cited_title":"Longhi, Quantum interference and exceptional points, Opt","cited_arxiv_id":null,"evidence_quote":"Exhibited anti-PT-symmetric selective filtering whose steady entangled states the DPLS theory reproduces while fully incorporating loss."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the beam-splitter and phase-shifter decomposition used to realize the unitary matrices $V^\\dagger$ and $U$ in the engineered W-state-generation system."}],"review_version":1}