{"id":"bca95eca-f1c6-41f7-b7bf-21635852d322","arxiv_id":"2505.23581","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Proposes a 'quantum Hilbert transform' via state-dependent phase shifts in the Fourier basis, but the rule does not reproduce the classical Hilbert transform and is not a fixed quantum operation.","lead":"This paper proposes a quantum version of the Hilbert transform by shifting the phases of a quantum state's Fourier coefficients by 90 degrees, with the direction of the shift based on the sign of each coefficient's phase. The proposed rule does not match the classical Hilbert transform, which shifts based on the sign of frequency, and the paper's steganography application is under-specified.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proposed QHT in Eq. (16) is not a well-defined unitary quantum operation: the phase-sign map is nonlinear and not invariant under global phase, so the central analogy to the classical Hilbert transform fails.","rationale":"The manuscript's central contribution is the claim that Eq. (16) defines a quantum analogue of the classical Hilbert transform. The reader correctly identifies the weakest assumption: the sign of the Fourier coefficient phase theta_j is used in place of the sign of frequency. My stress-test goes further and shows that this choice is not merely a weak analogy but is internally fatal. The map defined in Eq. (16) is nonlinear, so it cannot be implemented as a fixed unitary circuit on the Fourier-basis amplitudes. It is also not invariant under global phase, meaning the operation is not defined on the physical state but depends on an arbitrary phase convention. These are not aesthetic objections; they break the central claim that QHT is a coherent, unitary phase-shift operation. The steganography application does not repair the issue because it presumes that Bob can meaningfully compare phase signs before and after QHT, without any protocol for establishing a common phase reference. The security analysis is also decoupled from the QHT-specific encoding and merely restates known properties of the three-stage protocol. Given that the central construction fails as a quantum transform, the paper does not establish its claimed result. The reader's verdict of REJECT is therefore appropriate, and the identified concern is the same one in substance: the phase-sign rule cannot carry the weight assigned to it.","tokens_in":7682,"tokens_out":3271,"duration_ms":35003,"concrete_test":"Perform a direct algebraic/numerical test of Eq. (16) as a map T on single Fourier coefficients. For N=1 (or for a single coefficient), compute T(1), T(i), and T(-i). Since T(1)=1, T(i)=1, T(-i)=1, linearity requires T(i + (-i)) = T(i) + T(-i) = 2, but T(i + (-i)) = T(0) = 0. This contradiction proves that no fixed linear unitary implements the map. Separately, for a two-qubit state |psi> = (|0> + |1>)/sqrt(2), compute the QFT coefficients beta, apply Eq. (16), and invert QFT. Then repeat for |psi_phi> = e^{i pi/4}|psi>. If QHT(e^{i phi}|psi>) is not equal to e^{i phi} QHT(|psi>) (up to numerical precision), the operation is not global-phase covariant and therefore is not a well-defined quantum operation on physical states.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim requires the QHT to be a coherent, unitary operation that mirrors the classical Hilbert transform. The classical transform is linear: in the Fourier domain it multiplies every coefficient by the fixed factor -i sgn(f), with the sign determined by frequency. Eq. (16), by contrast, multiplies each Fourier coefficient beta_j = r_j e^{i theta_j} by a factor that depends on the sign of the coefficient's argument theta_j: e^{-i pi/2} for theta_j > 0, e^{+i pi/2} for theta_j < 0, and 1 for theta_j = 0. This map T is not linear. For example, T(1) = 1, T(i) = 1, and T(-i) = 1, so T(i) + T(-i) = 2, while T(i + (-i)) = T(0) = 0. Hence T cannot be represented by any fixed linear unitary acting on the Fourier-basis amplitudes. Implementing Eq. (16) would require knowing the sign of each theta_j before choosing the phase shift, i.e., knowing the state being transformed, which is not a quantum transform in the usual sense. Moreover, the operation is not even well-defined on physical states: multiplying |psi> by a global phase e^{i phi} changes each theta_j to theta_j + phi, which can flip the signs of theta_j for different j. Since global phase is physically unobservable, QHT(e^{i phi}|psi>) cannot be a fixed phase multiple of QHT(|psi>). The classical Hilbert transform, depending only on frequency sign, does not have this defect. The steganography protocol inherits the same problem: it assumes Alice and Bob can reliably compare phase signs before and after QHT, but no mechanism is provided for defining or preserving these signs across the protocol. The Holevo-bound calculation in Eqs. (24)-(26) is generic to the three-stage protocol and does not validate the QHT encoding. Therefore the central claim that Eq. (16) is a quantum analogue of the Hilbert transform is unsupported by the argument in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quantum analogue of the classical Hilbert transform, which it calls the quantum Hilbert transform (QHT). The definition in Section 2 consists of applying the quantum Fourier transform (QFT), modifying each Fourier coefficient's phase by ±π/2 according to the sign of the coefficient's phase angle (Eq. 16), and then applying the inverse QFT. The paper presents examples of this operation on random quantum states and describes a quantum steganography protocol in which a reference state is used to recover phase modifications that encode a secret message. The central claim is that this phase-modification rule mirrors the classical Hilbert transform's frequency-dependent phase shifts.","tokens_in":8025,"tokens_out":5622,"duration_ms":49964,"significance":"A genuine quantum Hilbert transform would be a valuable addition to quantum signal processing, since the classical Hilbert transform is a foundational tool in communications and sensing. However, the proposed QHT is not what it claims to be: it does not implement the classical Hilbert transform, it is not a linear or unitary quantum operation, and it is not even well-defined on physical states because it depends on the signs of amplitude phases, which are global-phase dependent. The steganography protocol inherits these defects. The paper correctly reviews the classical Hilbert transform and QFT, but the central construction is unsound. The significance of the work, as a contribution to quantum information, is therefore very limited unless the core definition is fundamentally reworked.","major_comments":[{"comment":"The phase-modification rule in Eq. (16) does not implement the classical Hilbert transform. In the classical case, Eq. (4) multiplies the Fourier transform G(f) by -i sgn(f), where the sign depends on the frequency f. In contrast, Eq. (16) shifts the phase of each coefficient beta_j = r_j e^{i theta_j} by -pi/2 if theta_j > 0 and +pi/2 if theta_j < 0, replacing the sign of the frequency with the sign of the coefficient's phase. This is a load-bearing difference: the resulting map on the amplitudes is nonlinear, as T(1) = 1, T(i) = 1, and T(-i) = 1, so T(i) + T(-i) = 2 while T(i + (-i)) = T(0) = 0. A fixed linear unitary operator on the Fourier-basis amplitudes cannot produce this behavior, so the proposed QHT is not the quantum analogue of the Hilbert transform in any standard sense.","section":"Section 2, Eq. (16)"},{"comment":"The operation defined by Eq. (16) is not well-defined on physical states because it is not invariant under global phase. If |psi> is replaced by e^{i phi}|psi>, the QFT coefficients beta_j are all multiplied by e^{i phi}, so every theta_j is shifted by phi. For a generic phi, the signs of some theta_j flip, so the output of the QHT is not simply multiplied by a fixed global phase. Since global phase is physically unobservable, the proposed QHT is not a valid quantum operation. Moreover, step 3 assumes that the theta_j can be 'extracted' directly, but amplitude phases are not directly measurable without state tomography, which would require multiple copies and would disturb the state; the paper provides no mechanism for performing the phase-dependent shift coherently on a single copy.","section":"Section 2, Steps 3–4"},{"comment":"The steganography protocol inherits the defects of the QHT definition. It requires Alice to apply QHT to a state and Bob to compare the phase signs of the reference and message states, but since the QHT is state-dependent and not a fixed unitary, it cannot be applied as a quantum gate without first knowing the full coefficient phases. The security analysis in Eqs. (24)–(26) bounds the information accessible to an eavesdropper in the three-stage protocol, but it does not account for information that might leak through the QHT modification itself, so it does not support the claimed security of the embedding.","section":"Section 4"}],"minor_comments":[{"comment":"The filter is defined as H[k] in Eq. (6) but then used as H_an[k] in Eq. (7); please define the analytic-signal filter explicitly and clarify the treatment of even and odd N.","section":"Section 1.1.1, Eqs. (6)–(7)"},{"comment":"The examples in Figs. 2–4 only illustrate the effect of the ad hoc rule in Eq. (16); they do not compare the output against the classical Hilbert transform of a corresponding signal, so they do not provide evidence for the claimed analogy.","section":"Section 3.2"},{"comment":"The statement that 'the length of the msg sequence is l = 2s + 2, where s is the length of the ref sequence' is confusing because the example uses a 6-bit reference, a 3-bit secret, a 6-bit identifier, and a 2-bit encoding, totaling 14 bits, but the formula does not include the secret length; this should be clarified.","section":"Section 4"},{"comment":"There are several typographical and grammatical errors, including 'it's' for 'its' in the abstract, 'QHTand' missing a space, 'This gives usHilbert Spectrum' missing a space, and 'score phase information' which should likely be 'store phase information'.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central definition of the QHT is not a linear or unitary quantum operation and does not implement the classical Hilbert transform; this is a load-bearing error that cannot be fixed within the scope of the manuscript. The paper also needs substantial revision in presentation and in the steganographic security analysis, but the fundamental issue is the definition itself."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline is simple: the proposed \"quantum Hilbert transform\" is not a quantum analogue of the Hilbert transform. The classical transform multiplies Fourier coefficients by -i sgn(f), where f is the frequency. The paper instead multiplies each coefficient by a factor that depends on the sign of the coefficient's own phase θ_j. That map is not linear (e.g., T(i)=1, T(-i)=1, but T(i)+T(-i)≠T(0)), so it cannot be a fixed unitary gate. Worse, it is not even well-defined on physical states: a global phase rotation changes θ_j and can flip the signs that trigger the shift, so QHT(e^{iφ}|ψ⟩) is not a fixed multiple of QHT(|ψ⟩). That kills the central claim.\n\nWhat is genuinely new is narrow: someone has written down a phase-modification rule of the form \"shift by ±π/2 depending on the sign of θ_j.\" It is a trivial combination of QFT and a diagonal phase operator, with no new machinery, and no comparison to prior work on quantum analogues of analytic signals. The paper's review of the classical Hilbert transform and QFT is accurate, and the Q-sphere plots do show that something happens to the state. But that something is a state-dependent phase modulation, not a transform.\n\nThe steganography application inherits the same problem. Alice and Bob are supposed to compare phase signs before and after QHT, but there is no mechanism for defining or preserving those signs across the protocol. The identifier-sequence example is under-specified, and the Holevo-bound calculation (Eqs. 24–26) is generic to the three-stage protocol and does not validate the specific encoding. The amplitude observations in Section 3 are just plots, not validation.\n\nMinor issues: the paper says for θ_j>0 we see an addition of π/2 to α'_x, but the Fourier-basis shift is subtraction; Fig. 1(a) refers to an AM signal without giving parameters. These are minor compared to the load-bearing flaw.\n\nThis paper is not ready for refereeing. A serious editor would desk reject it because the central definition fails on first inspection. I would not cite it, and I would not bring it to reading group except perhaps as a cautionary example of why phase sign and frequency sign are different.","headline":"The proposed QHT is a state-dependent phase modulation, not a Hilbert transform: it is nonlinear, not unitary, and not invariant under global phase.","tokens_in":8592,"tokens_out":3584,"would_cite":false,"duration_ms":31975,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68"],"pacs":["03.67.-a","03.67.Hk"],"model":"deepseek-v4-flash","headline":"A quantum Hilbert transform is defined by shifting each QFT phase by ±π/2 according to its sign.","keywords":["quantum Hilbert transform","quantum Fourier transform","phase modulation","quantum steganography","quantum signal processing","Holevo bound","three-stage quantum protocol"],"falsifier":"Take any state whose QFT coefficients correspond to a single positive-frequency tone, apply QHT, and check whether the output is the inverse QFT of the same coefficients multiplied by −i; if the result differs, eq (16)'s sign-of-phase rule fails to reproduce the classical Hilbert transform.","tokens_in":7456,"feed_emoji":"⚛️","tokens_out":8644,"duration_ms":78304,"temperature":0.7,"pith_summary":"This paper introduces a quantum Hilbert transform (QHT) as a three-step operation: apply the quantum Fourier transform, change each Fourier coefficient's phase by −π/2, 0, or +π/2 according to the sign of the phase angle, and apply the inverse transform. The paper's claim is that this reproduces, on multiqubit states, the defining behavior of the classical Hilbert transform, which rotates frequency components by ±π/2 depending on the sign of frequency. The authors then embed the transform in a quantum steganography protocol where a receiver who already holds a reference state reads hidden message bits from the sign and size of phase shifts. If the analogy holds, QHT would fill a gap among quantum transforms and give a coherent, unitary tool for phase-based quantum signal processing and covert communication.","feed_headline":"Quantum Hilbert transform shifts Fourier phases by ±π/2","feed_subtitle":"A sign-dependent π/2 phase shift after QFT mirrors the classical Hilbert transform and hides bits in qubit phases.","key_machinery":"The operative object is the phase-modification rule in eq (16), attached to the QFT machinery of eqs (12)–(15). Each Fourier coefficient β_j = r_j $e^{{iθ_j}}$ is first decomposed into magnitude and phase; the sign of θ_j then decides whether the coefficient is multiplied by $e^{{−iπ/2}}$ or $e^{{+iπ/2}}$ (or left untouched at zero phase), and the inverse QFT reassembles the modified coefficients into a new computational-basis state. This single rule carries the whole claimed analogy to the classical Hilbert transform's frequency-sign-dependent π/2 phase shift, and the reference-state comparison in the steganography protocol is what turns the resulting phase deviations into readable bits.","core_discovery":"The central claim is eq (16): after expressing the QFT output state as |φ⟩ = Σ_j β_j |j⟩ with β_j = r_j $e^{{iθ_j}}$, the quantum Hilbert transform alters each phase to θ_j − π/2 when θ_j > 0, leaves θ_j unchanged when θ_j = 0, and sets θ_j + π/2 when θ_j < 0; the new state is then returned to the computational basis by the inverse QFT. The paper presents this rule as the quantum counterpart of the classical frequency-domain identity Ĝ(f) = −j sgn(f) G(f), with the sign of the Fourier-coefficient phase standing in for the sign of frequency. The authors illustrate the operation on random 3- and 5-qubit states, showing that amplitudes and phases both change after the inverse transform, and use the ±π/2 phase deviations as the information carrier in their steganography scheme.","pith_inferences":["Beyond the paper: the same sign-of-phase modulation could be tested as a general quantum signal filter, independent of steganography, by checking whether it yields quadrature pairs for arbitrary encoded signals.","Beyond the paper: the zero-leakage bound assumes one intercepted stage; an attacker who observes multiple stages or who knows the reference state would reduce the scheme to a classical phase-difference code.","Beyond the paper: an alternative QHT that keys the ±π/2 shift to the sign of the frequency index k in the QFT basis would map more directly onto eq (4); comparing that version with eq (16) on random states would reveal whether the phase-sign rule is necessary or merely conventional."],"forward_implications":["QHT is a unitary operation: QFT, a diagonal phase gate, and inverse QFT can in principle be implemented with standard quantum circuits, so the transform is executable rather than merely formal.","Any quantum signal encoded in amplitudes now has a quadrature-style partner state, opening a quantum analogue of envelope extraction and single-sideband processing.","In the steganography protocol, an eavesdropper intercepting one stage of the three-stage transmission learns nothing in the ideal case (zero Holevo information), and leakage is bounded by O(δ) when noise δ is small, as the paper's analysis states.","Because the inverse QFT mixes the modified phases back into amplitudes, the transform visibly changes both phases and magnitudes of computational-basis states, which the paper's numerical plots display."],"supporting_citations":[{"why":"Defines the continuous Hilbert transform and its Cauchy principal-value form, the classical object QHT claims to quantize.","marker":"[1]"},{"why":"Supplies the discrete Hilbert transform implemented through the DFT, the classical model whose structure QHT mimics with QFT.","marker":"[12]"},{"why":"Provides the QFT formalism in the computational basis used in eqs (10)–(12).","marker":"[18]"},{"why":"Earlier phase-shift-based information hiding in audio that motivates using Hilbert-type phase changes for covert data.","marker":"[9]"},{"why":"The three-stage quantum communication protocol over which the reference and message states are transmitted in the steganography application.","marker":"[22]"},{"why":"Provides the quantum information correlation bound used to quantify eavesdropper information leakage for the steganography protocol.","marker":"[23]"},{"why":"Original Holevo bound for the information capacity of a quantum channel, used by the paper to state the zero-leakage ideal case.","marker":"[24]"}],"fun_headline_variants":["Quantum Hilbert transform shifts phases by ±π/2 for steganography","First quantum Hilbert transform hides bits in qubit phases","Quantum Hilbert transform: sign-dependent ±π/2 phase shift","Quantum Hilbert transform enables phase-based steganography","Quantum Hilbert transform: π/2 phase shifts for hidden data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the sign of a Fourier coefficient's phase angle θ_j plays the same role as the sign of frequency in the classical Hilbert transform, an identification the paper asserts rather than derives.","fun_headline_variants_meta":{"raw":{"variants":["Quantum Hilbert transform shifts phases by ±π/2 for steganography","First quantum Hilbert transform hides bits in qubit phases","Quantum Hilbert transform: sign-dependent ±π/2 phase shift","Quantum Hilbert transform enables phase-based steganography","Quantum Hilbert transform: π/2 phase shifts for hidden data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000812,"raw_usage":{"total_tokens":3486,"prompt_tokens":798,"completion_tokens":2688,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":414,"completion_tokens_details":{"reasoning_tokens":2605}},"tokens_in":414,"tokens_out":2688,"duration_ms":20094,"temperature":1.0,"reasoning_tokens":2605,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:42:15.719241+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any state whose QFT coefficients correspond to a single positive-frequency tone, apply QHT, and check whether the output is the inverse QFT of the same coefficients multiplied by −i; if the result differs, eq (16)'s sign-of-phase rule fails to reproduce the classical Hilbert transform.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the QFT formalism in the computational basis used in eqs (10)–(12)."},{"cited_title":"The Basic Discrete Hilbert Transform with an Information Hiding Application","cited_arxiv_id":"0907.4176","evidence_quote":"Earlier phase-shift-based information hiding in audio that motivates using Hilbert-type phase changes for covert data."},{"cited_title":"A three-stage quantum cryptography protocol.F oundations Phys","cited_arxiv_id":null,"evidence_quote":"The three-stage quantum communication protocol over which the reference and message states are transmitted in the steganography application."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Original Holevo bound for the information capacity of a quantum channel, used by the paper to state the zero-leakage ideal case."}],"review_version":1}