{"id":"3fbf59aa-dc98-45f2-a46c-b6dab9067e63","arxiv_id":"2507.13078","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New dilation protocols on IBM quantum hardware provide the first experimental access to the dynamical metric of non-Hermitian quantum mechanics.","lead":"This paper shows how to simulate non-Hermitian quantum systems, including the metric formalism that preserves probabilities, by embedding them in a larger Hermitian quantum circuit with helper qubits. The authors demonstrate the protocol on an IBM quantum processor, reporting the first direct measurement of the evolving metric that defines the modified inner product.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (9) GBoNd dilation is not unitary unless ρ=I: with C=tr[ρ_G+ρ_G^{-1}]/D, M†M≠I for λ≠1, so the total evolution is not a closed Hermitian circuit and the metric-formalism state η|ψ> is not accessed as claimed.","rationale":"The reader identified the unitarity of Eq. (9) as a load-bearing but unproved assumption. My analysis shows the issue is stronger: Eq. (9) is not merely unproved, it is algebraically false for the stated C(t) whenever ρ_G ≠ I. The simplification to Eq. (9) requires ρ_G+ρ_G^{-1}=C I, which fails for the GBoNd construction because the identity block on the second ancilla gives a different eigenvalue (2) than the ρ-block (λ+1/λ). This makes the total evolution non-unitary and the claimed final state Eq. (10) incorrect and unnormalized. Since GBoNd is the only protocol that accesses the metric-formalism state η(t)|ψ(t)>, the paper's central claim of simultaneously realizing all three formalisms on a unitary circuit is not supported. The BoNd protocol is valid and provides direct access to ρ(t), so the metric measurement in Fig. 3 may stand, but the GBoNd-based results in Fig. 2 and the 'simultaneous access' claim need to be revised or removed. The verdict should therefore move from CONDITIONAL to REJECT, unless the authors can supply a corrected unitary dilation and rerun the experiment.","tokens_in":9268,"tokens_out":30372,"duration_ms":280956,"concrete_test":"Construct M in Eq. (9) for the qubit with ρ=diag(λ,1/λ), ρ_G=ρ⊗|0><0|+I⊗|1><1|, C=(λ+1/λ)/2+1, and evaluate M†M−I. For λ=3 (PT-broken regime; λ grows with time), the first diagonal block of M†M has eigenvalues (λ+1/λ)/C ≈ 1.25 and 2/C ≈ 0.75, so M†M≠I. Equivalently, check that the RHS of Eq. (10) for |ψ>=|0> has norm squared (1+λ)^2/C = 6 at λ=3, not 1. If either check fails, the GBoNd dilation is not unitary and the central claim collapses.","verdict_should_be":"REJECT","load_bearing_attack":"The GBoNd protocol's central object is ζ_G^{-1}(t) in Eq. (9), claimed to be a unitary dilation of η_G with C(t)=tr[ρ_G+ρ_G^{-1}]/D. For the 8×8 block matrix M=1/√C [[η_G^{-1},η_G],[η_G,-η_G^{-1}]] to be unitary, the columns of the first block column must be orthonormal, requiring η_G^{-2}+η_G^2=C I, i.e., ρ_G+ρ_G^{-1}=C I. With ρ_G=ρ⊗|0><0|+I⊗|1><1|, one has ρ_G+ρ_G^{-1}=(ρ+ρ^{-1})⊗|0><0|+2I⊗|1><1|. This is proportional to identity only if ρ+ρ^{-1}=2I, i.e., ρ=I. For the qubit H in Eq. (11), ρ has eigenvalues λ and 1/λ, so ρ+ρ^{-1}=(λ+1/λ)I, but the A=|1> block contributes 2I, and the condition fails whenever λ≠1. Hence M is not unitary for any time with ρ≠I, and U_tot=ζ^{-1}(U_h⊗I)ζ(t0) is not a closed Hermitian evolution. Direct application of M(t) after U_h yields (up to ancilla relabeling) 1/√(2C)(|ψ>|00>+η|ψ>|10>+ρ|ψ>|01>+η|ψ>|11>), not the claimed Eq. (10), and this state is not normalized because 2C≠1+2⟨ρ⟩+⟨ρ^2⟩ in general. The BoNd protocol (Eq. (13)) is correct because there ρ+ρ^{-1}=C I for a qubit, but the GBoNd scheme, the only one accessing η|ψ>, is invalid as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two operator-dilation protocols, the 'Biorthogonal Naimark dilation' (BoNd) and the 'Generalized Biorthogonal Naimark dilation' (GBoNd), to realize non-Hermitian quantum dynamics inside a larger unitary circuit. The BoNd protocol is designed to give access to the norm-evolved state |ψ(t)> and the left state ρ(t)|ψ(t)>; the GBoNd protocol is designed to additionally give access to the metric-formalism state η(t)|ψ(t)>. The authors implement both on IBM Quantum for the qubit Hamiltonian H=σx+i r σz, compare tomographic reconstructions with analytical solutions, and report the first experimental access to the dynamical metric ρ(t).","tokens_in":9724,"tokens_out":19336,"duration_ms":201184,"significance":"Access to the dynamical metric ρ(t) in a closed-system embedding would be a valuable methodological advance, because the metric is the central object of the self-consistent non-Hermitian formalism and is generally not directly measurable in a closed non-Hermitian system. The BoNd construction is sound for qubits (det ρ=1 makes ρ+ρ^{-1} proportional to the identity, so the dilation is unitary), and the experimental data in Fig. 3 show good agreement without fitted parameters. However, the GBoNd protocol, which is the only part claiming access to η(t)|ψ(t)> and to observables in the metric formalism, is not a unitary dilation as written. Since this undermines the paper's strongest claim, the manuscript cannot be accepted in its present form.","major_comments":[{"comment":"The block matrix in Eq. (9) is not a unitary dilation of η_G for the stated C(t). For M = 1/√C [[η_G^{-1}, η_G],[η_G, -η_G^{-1}]] to be unitary, the first column must satisfy η_G^{-2} + η_G^2 = C I_{H⊗A}, i.e. ρ_G^{-1} + ρ_G = C I. With ρ_G = ρ⊗|0><0|_a + I⊗|1><1|_a, this requires both ρ^{-1}+ρ = C and 2I = C on the two blocks, forcing ρ = I. For the implemented Hamiltonian (11) ρ(t) ≠ I for t>0, so ζ_G^{-1}(t) is non-unitary. Consequently U_tot = ζ_G^{-1}(t)(U_h⊗I)ζ_G(t0) is not a closed Hermitian evolution, and the embedding claim for GBoNd collapses.","section":"GBoNd protocol, Eq. (9)"},{"comment":"Even taken as a formal linear map, Eq. (9) does not yield the normalized state (10). Applying Eq. (9) to |ψ(t0)>|+>_a|+>_b gives, up to relabeling of the ancillas, (1/√(2C))[|ψ>|00> + ρ|ψ>|10> + η|ψ>(|01>+|11>)]. This is proportional to Eq. (10) but with prefactor 1/√(2C), not 1/√C; the norm equals 1 only if 2C = 1 + 2⟨ρ⟩ + ⟨ρ^2⟩, which is not the chosen C(t)=tr[ρ_G+ρ_G^{-1}]/D. Thus the postselection probabilities and the tomographic reconstruction based on Eq. (10) are not those of a unitary circuit, and the Fig. 2 data cannot be interpreted as the advertised metric-formalism observables.","section":"GBoNd protocol, Eq. (10)"},{"comment":"The assertion that C(t) ∝ tr[ρ(t)+ρ^{-1}(t)] 'ensures the validity of our dilation at all times' is not proved and is false in general. The off-diagonal block √(C I - ρ^{-1}) in Eq. (7) requires C(t) ≥ ||ρ^{-1}(t)||. For a three-dimensional example with ρ eigenvalues 100,1,1, the proposed C = tr[ρ+ρ^{-1}]/D ≈ 104.01/3 < 100, so the square root is not real. The simplified BoNd case works only because a qubit with det ρ = 1 satisfies ρ+ρ^{-1} = (λ+1/λ)I; this special structure should be stated as the reason, rather than presenting the trace choice as generally valid.","section":"Metric formalism, Eq. (7)"}],"minor_comments":[{"comment":"D is used inconsistently: in the BoNd section D=tr[Is], while in the GBoNd section D=tr[Is,a]; please define D explicitly in each context and ensure the formula for C(t) is dimensionally consistent.","section":"Notation"},{"comment":"The supplemental material is cited for the circuit decomposition and for the Hermiticity of the total Hamiltonian, but as submitted it is only a placeholder URL; these implementation details are not verifiable in the current manuscript.","section":"Supplemental material [39]"},{"comment":"The time axis in Fig. 2 is labelled 'in units of [s]=1'; this is unclear, since the circuit time is not physical time. Please specify the relation between the plotted parameter t and the actual gate parameters used in the compiled circuit.","section":"Fig. 2"},{"comment":"The notation ∮ for the time average over a period is not defined; please define it explicitly or replace it with a standard time-average symbol.","section":"Conclusion"},{"comment":"The normalization denominator ⟨χ(t)|χ(t)⟩ assumes the state is nonzero; at exceptional points or at special times some of the states may vanish, so the condition for the denominator to be nonzero should be stated.","section":"Eq. (12)"},{"comment":"The dilation gates in both protocols are constructed from the analytical ρ(t); the authors should state explicitly that the experiment is an emulation that uses the theoretical metric to build the circuit, rather than a measurement that is independent of the solution.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the GBoNd unitarity error. The BoNd results appear valid and could form the basis of a revised paper, but the authors must either provide a correct unitary construction that yields the metric-formalism state or substantially weaken the claims and remove the Fig. 2 interpretation. Please also ask them to supply the missing supplemental material before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about this paper. The one-ancilla BoNd scheme is a clean, legitimate way to access the right and left states of a qubit and to tomograph the dynamical metric directly. The two-ancilla GBoNd scheme, presented as the route to the metric-formalism state η|ψ>, has a unitarity bug: the matrix in Eq. (9) is not unitary unless ρ = I, so the claim that the total evolution is a closed Hermitian circuit collapses.\n\nThe paper does identify a genuine gap: the standard Naimark dilation with C = 1 is ill-defined in the PT-broken regime, and the general dilation formula in Eq. (7) is a reasonable starting point. The BoNd implementation on ibm_kyiv, assuming the displayed data are representative, is a useful proof-of-principle, and the observation that the time-averaged metric recovers the stationary pseudo-Hermiticity transformation is a nice point.\n\nThe GBoNd claim fails because the simplification to Eq. (9) requires ρ_G + ρ_G^{-1} = C I. With ρ_G = ρ⊗|0⟩⟨0| + I⊗|1⟩⟨1|, this forces ρ = I on the |0⟩ block and C = 2 on the |1⟩ block; for any time with ρ ≠ I, the matrix is not unitary. Applying the non-unitary matrix gives a state that is not the normalized Eq. (10). Since GBoNd is the only scheme that accesses η|ψ⟩, the headline experimental evidence for the metric formalism is unsupported. The paper also asserts the unitarity of Eq. (7) without proof and hand-waves the choice of C. The experimental section shows only visual agreement, with no fidelities or error bars beyond raw shot noise, and the Supplemental Material—which should contain the circuit details and normalization analysis—is absent from the arXiv posting. The abstract's \"new paradigm\" and \"perfectly capture\" are overclaims.\n\nThis deserves a serious referee because the topic is significant and the BoNd part may survive, but as written the central new result is mathematically inconsistent. I would not cite the GBoNd claim. The authors need to fix the dilation (different C or a different block structure) or restrict the claim to BoNd, and they need to release the data and supplement.","headline":"The BoNd protocol is sound, but the GBoNd dilation that is supposed to access the metric-formalism state η|ψ> is not unitary as written, so the paper's central experimental claim is unsupported.","tokens_in":10243,"tokens_out":4038,"would_cite":false,"duration_ms":42872,"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":"A dilation-based protocol embeds non-Hermitian quantum dynamics in a larger closed system and yields the first direct tomographic measurement of the dynamical metric.","keywords":["non-Hermitian quantum mechanics","metric formalism","operator dilation","Naimark dilation","parity-time symmetry","dynamical metric","quantum simulation","biorthogonal quantum mechanics"],"falsifier":"Compute whether $\\zeta(t)$ from Eq. (7) with $C(t)\\propto\\mathrm{tr}[\\rho(t)+\\rho^{-1}(t)]$ satisfies $\\zeta^{\\dagger}(t)\\zeta(t)=I$ for a Hamiltonian of dimension three or higher in the parity-time-broken regime; finding any time where $U_{\\mathrm{tot}}^{\\dagger}U_{\\mathrm{tot}}\\neq I$ would show the embedding is not a closed Hermitian evolution.","tokens_in":9046,"feed_emoji":"⚛️","tokens_out":7835,"duration_ms":75120,"temperature":0.7,"pith_summary":"The paper sets out to show that the two self-consistent formulations of non-Hermitian quantum mechanics — the norm method and the metric (biorthogonal) formalism — can be realized on a closed, unitary quantum computer rather than remaining theoretical constructions. The proposed operator dilation schemes embed the non-Hermitian system in a larger Hermitian one, and projective measurements on the ancilla qubits make the normally inaccessible states $|\\psi(t)\\rangle$, $\\rho(t)|\\psi(t)\\rangle$, and $\\eta(t)|\\psi(t)\\rangle$ separately available. On a digital quantum simulator, the tomographically reconstructed dynamical metric $\\rho(t)$ matches the analytical solution in both the parity-time-symmetric and broken regimes for a two-level system. If it holds beyond the qubit proof of principle, the scheme makes the metric formalism experimentally testable and opens the way to probing non-Hermitian phenomena in many-body settings.","feed_headline":"First direct measurement of non-Hermitian quantum metric","feed_subtitle":"Embedding non-Hermitian dynamics in a larger closed system makes the metric formalism accessible on real hardware.","key_machinery":"The central machinery is the operator dilation $\\zeta(t)$, a time-dependent unitary embedding of the non-Hermitian time evolution into a closed Hermitian system with ancillas. Its block form in Eq. (7) is chosen with $C(t)\\propto\\mathrm{tr}[\\rho(t)+\\rho^{-1}(t)]$ for two-level systems, which simplifies to the block matrix in Eq. (9) and is claimed to keep the dilation valid at all times, including the PT-broken regime. The metric itself, $\\rho(t)=\\eta^2(t)$, evolves by $i\\dot{\\rho}=H^{\\dagger}\\rho-\\rho H$, and $\\eta$ maps the non-Hermitian state to the equivalent Hermitian state via $|\\Psi\\rangle=\\eta|\\psi\\rangle$. The dilation is the object that makes the normally inaccessible metric accessible through ancilla postselection and tomography.","core_discovery":"The paper's central claim is that a single total evolution $U_{\\mathrm{tot}}(t,t_0)=\\zeta^{-1}(t)(U_h(t,t_0)\\otimes I)\\zeta(t_0)$, built from a time-dependent operator dilation $\\zeta(t)$ of the metric's square root $\\eta(t)$, realizes non-Hermitian dynamics inside a purely Hermitian closed system. Postselecting on the ancilla outcomes yields three time-evolved states: the norm-method state $|\\psi(t)\\rangle$, the biorthogonal left state $\\rho(t)|\\psi(t)\\rangle$, and the metric-formalism state $\\eta(t)|\\psi(t)\\rangle$. The paper reports the first experimental access to the metric operator $\\rho(t)$ via state tomography on the latter two states, finding good agreement with analytics for the Hamiltonian $H=\\sigma_x+i r\\sigma_z$ in both PT-symmetric and PT-broken regimes. It also observes that in the PT-symmetric regime the dynamical metric is time-periodic and its time average $\\rho_C$ satisfies $H^{\\dagger}\\rho_C=\\rho_C H$, recovering the stationary pseudo-Hermiticity transformation.","pith_inferences":["The paper leaves implicit that the same dilation circuit could provide a synthetic-laboratory test of information-theoretic bounds: in the metric picture, faster-than-Hermitian evolution and Lieb-Robinson violations of the norm method are tamed, and the protocol gives a way to compare both pictures on a single platform.","If the unitarity of the dilation survives in higher dimensions with a suitable $C(t)$, a natural extension is to many-body or lattice systems, where the dynamical metric could be used to observe the non-Hermitian skin effect or geometry-driven defect freezing directly.","The authors' analogy between the self-consistent metric and general relativity suggests a testable toy model: treat the ancilla qubits as a synthetic background whose 'geometry' ($\\rho$) is back-reacted by the system's evolution, and look for observable signatures in the postselected branch probabilities.","Because the qubit demonstration has $\\det\\rho=1$, the simple choice $C(t)\\propto\\mathrm{tr}[\\rho+\\rho^{-1}]$ is exact; testing BoNd on a system with $\\det\\rho\\neq 1$ would pin down whether the protocol's general validity claim needs additional assumptions."],"forward_implications":["One circuit realizes all three relevant non-Hermitian states simultaneously, so the norm method and the metric formalism can be compared on identical experimental conditions.","The metric operator $\\rho(t)$ becomes a directly measurable quantity rather than an inferred theoretical object, since BoNd reconstructs it from tomographic data.","Both GBoNd and BoNd remain well-defined in the PT-broken regime for arbitrary times, whereas the original Naimark dilation breaks down when $\\rho(t)-I$ ceases to be positive semidefinite.","The measured difference at the exceptional point — metric-formalism observables tend to $\\langle\\sigma_y\\rangle\\to 0$, $\\langle\\sigma_z\\rangle\\to -1$, while the norm method gives the reverse — demonstrates that the two formalisms are experimentally distinguishable.","The time-averaged metric in the PT-symmetric phase obeys the pseudo-Hermiticity relation, connecting the dynamical metric to the static similarity transformation usually used to define PT-symmetric quantum mechanics."],"supporting_citations":[{"why":"Supplies the operator dilation theorem that justifies embedding a non-unitary evolution inside a unitary one.","marker":"[33]"},{"why":"Defines the metric evolution equation $i\\dot{\\rho}=H^{\\dagger}\\rho-\\rho H$ that the protocols are designed to realize.","marker":"[27]"},{"why":"Provides the biorthogonal quantum mechanics framework in which $|\\psi(t)\\rangle$ and $\\rho(t)|\\psi(t)\\rangle$ are the right and left states.","marker":"[28]"},{"why":"Earlier experimental Naimark dilation in a single-spin system; the scheme the new protocols extend and improve upon in the PT-broken regime.","marker":"[12]"},{"why":"Prior superconducting implementation of Naimark dilation; its time-step redefinition is the baseline that the new protocols avoid.","marker":"[36]"},{"why":"Shows physical consequences of the metric formalism in the quantum brachistochrone problem, motivating experimental access to the metric.","marker":"[7]"},{"why":"Prior work relating the quantum metric to defect freezing in non-Hermitian systems, supporting the phenomenological significance of the metric.","marker":"[32]"},{"why":"Best-approximate quantum compiling method used to translate the total unitary into a circuit on the digital simulator.","marker":"[40]"}],"fun_headline_variants":["Non-Hermitian quantum metric measured on a qubit","Hermitian embedding unlocks non-Hermitian physics","First qubit experiment for dynamical metric formalism","Metric formalism comes to life on digital quantum simulator","Non-Hermitian dynamics via Hermitian dilation: first proof"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the block matrix used in the dilation is a norm-preserving (unitary) operation at every time, including after parity-time symmetry breaking; the paper asserts this for two-level systems with the chosen $C(t)\\propto\\mathrm{tr}[\\rho+\\rho^{-1}]$ but supplies no general proof.","fun_headline_variants_meta":{"raw":{"variants":["Non-Hermitian quantum metric measured on a qubit","Hermitian embedding unlocks non-Hermitian physics","First qubit experiment for dynamical metric formalism","Metric formalism comes to life on digital quantum simulator","Non-Hermitian dynamics via Hermitian dilation: first proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1260,"prompt_tokens":937,"completion_tokens":323,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":248}},"tokens_in":553,"tokens_out":323,"duration_ms":4202,"temperature":1.0,"reasoning_tokens":248,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:32:36.101061+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute whether $\\zeta(t)$ from Eq. (7) with $C(t)\\propto\\mathrm{tr}[\\rho(t)+\\rho^{-1}(t)]$ satisfies $\\zeta^{\\dagger}(t)\\zeta(t)=I$ for a Hamiltonian of dimension three or higher in the parity-time-broken regime; finding any time where $U_{\\mathrm{tot}}^{\\dagger}U_{\\mathrm{tot}}\\neq I$ would show the embedding is not a closed Hermitian evolution.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the operator dilation theorem that justifies embedding a non-unitary evolution inside a unitary one."},{"cited_title":"Mostafazadeh, Time-dependent pseudo-hermitian hamiltonians and a hidden geometric aspect of quantum mechanics, Entropy 22, 10.3390/e22040471 (2020)","cited_arxiv_id":null,"evidence_quote":"Defines the metric evolution equation $i\\dot{\\rho}=H^{\\dagger}\\rho-\\rho H$ that the protocols are designed to realize."},{"cited_title":"Dogra, A","cited_arxiv_id":null,"evidence_quote":"Prior superconducting implementation of Naimark dilation; its time-step redefinition is the baseline that the new protocols avoid."},{"cited_title":"Mostafazadeh, Quantum brachistochrone problem and the geometry of the state space¡? format?¿ in pseudo- hermitian quantum mechanics, Physical review letters 99, 130502 (2007)","cited_arxiv_id":null,"evidence_quote":"Shows physical consequences of the metric formalism in the quantum brachistochrone problem, motivating experimental access to the metric."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior work relating the quantum metric to defect freezing in non-Hermitian systems, supporting the phenomenological significance of the metric."},{"cited_title":"Madden and A","cited_arxiv_id":null,"evidence_quote":"Best-approximate quantum compiling method used to translate the total unitary into a circuit on the digital simulator."}],"review_version":1}