{"id":"783382cc-82c8-498c-9fa9-1c08861d47bd","arxiv_id":"2608.10674","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"When at least one of two quantum states is pure, the Uhlmann fidelity can be estimated with Θ(1/ε) queries and Θ(1/ε²) samples without knowing which state is pure, matching the optimal lower bounds.","lead":"This paper gives a quantum algorithm that estimates the fidelity between two quantum states when at least one of them is pure, and it does not need to know which one is pure. It achieves the optimal query and sample complexity, improving on methods that required knowing the pure state in advance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the cited unitary-dilation identity Eq. (5), the only load-bearing external step, checks out under direct computation, and the complexity claims follow.","rationale":"I read the paper as a compact theoretical contribution: it specializes the algorithmic Uhlmann transform to the one-pure-state setting and proves F = max{a0,a1}, yielding Θ(1/ε) query and Θ(1/ε²) sample optimality without knowing which state is pure. The reader's weakest-assumption identification is accurate in locating the load-bearing point at Eq. (5), the external unitary-dilation lemma. I independently re-derived Eq. (5) by direct computation with explicit Schmidt decompositions of the two purifications, and the zero block indeed reproduces tr_A(|ψ0><ψ1|). The rest of the proof is sound: Lemma 3.1 correctly computes the rank-one structure and norm of the Uhlmann cross operator; Proposition 3.2 correctly shows that at least one of a0,a1 attains F and neither exceeds it; Theorem 3.3's union-bound argument and query count are correct; Corollary 3.4 follows from the cited sample-to-query lifting; and the lower bounds transfer because the known-pure and pure-pure instances are subproblems of the unknown-pure promise. The only minor caveat is that the central identity is not proved in the manuscript and is taken from recent preprints with overlapping authorship; this is a reproducibility concern rather than a mathematical flaw. Since the specific lemma is elementary and verifiable, and since no internal inconsistency or missing step was found, I do not see a load-bearing objection. The reader's ACCEPT verdict is appropriate; I would keep it unchanged.","tokens_in":10922,"tokens_out":35800,"duration_ms":335147,"concrete_test":"Independently verify Eq. (5) for an explicit n=1 example, e.g., ρ0 = p|0><0| + (1-p)|1><1| with purification |ψ0> = sqrt(p)|0>_A|0>_R + sqrt(1-p)|1>_A|1>_R and ρ1 = |0><0| with |ψ1> = |0>_A|0>_R, by constructing the 6-qubit unitary W = Q1†(I⊗SWAP)Q0 and numerically checking that the zero block (⟨0|_{AR}⊗I_S)W(|0>_{AR}⊗I_S) equals tr_A(|ψ0><ψ1|) to machine precision. If the matrices match, the cited dilation lemma is correct and the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The only potentially load-bearing step is Eq. (5), the unitary-dilation identity for the Uhlmann cross operator, which is cited to [UNWT25] and [LLW26a] rather than proved in this manuscript. If Eq. (5) failed, Proposition 3.2 and Theorem 3.3 would collapse, because the amplitudes a0 and a1 would not equal the claimed norms of X|ψ1> and X†|ψ0>, and the max identity would not yield F. Direct calculation, however, confirms the identity: writing |ψ0> = Σ_{i,k} α_{i,k}|i>_A|k>_R and |ψ1> = Σ_{i,s} β_{i,s}|i>_A|s>_R, the zero block of W = Q1†(I_A⊗SWAP_{R,S})Q0 maps |s>_S to Σ_{i,k} α_{i,k} β*_{i,s}|k>_S = (tr_A(|ψ0><ψ1|))|s>. The subsequent derivation in Lemma 3.1 and Proposition 3.2 is internally consistent, the union-bound argument in Theorem 3.3 is sound, and the lower bounds transfer from the known-pure and pure-pure subproblems. Therefore the reliance on the cited lemma is a provenance and reproducibility point, not a correctness gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers the problem of estimating the Uhlmann fidelity F(ρ0, ρ1) under the promise that at least one of the two states is pure, with no prior knowledge of which state is pure. The main contribution is Theorem 3.3, a quantum estimator with purified query access that achieves additive-error ε using O(1/ε) queries, matching an Ω(1/ε) lower bound, and Corollary 3.4, which gives sample complexity Θ(1/ε²) via quantum sample-to-query lifting. The technical core is Proposition 3.2, which expresses the fidelity as F = max{a0, a1}, where a0 and a1 are norms of states obtained by applying a unitary dilation of the Uhlmann cross operator X = tr_A(|ψ0⟩⟨ψ1|) to the respective purifications. The proof uses Uhlmann's theorem, a rank-one simplification of the Uhlmann transform when one state is pure, and a unitary-dilation identity (Eq. (5)) cited from prior work.","tokens_in":11136,"tokens_out":11132,"duration_ms":104375,"significance":"If the result holds, it removes the prior-knowledge requirement in one-pure-state fidelity estimation and achieves the same optimal query and sample complexities as the known-pure-side setting. This is a clean and natural closing of a gap in the fidelity-estimation literature. The estimator is explicit, the lower bounds transfer from prior work, and the proof of the central identity is short and verifiable. The paper is likely to be useful to researchers in quantum algorithms and quantum state discrimination, and it demonstrates a nice application of the algorithmic Uhlmann transform. The main caveat is that the key unitary-dilation identity is imported rather than proved, but the cited sources are appropriate and the identity is directly checkable.","major_comments":[],"minor_comments":[{"comment":"The identity (⟨0|_{AR} ⊗ I_S) W (|0⟩_{AR} ⊗ I_S) = tr_A(|ψ0⟩⟨ψ1|) is load-bearing for Proposition 3.2 and Theorem 3.3, but it is only cited to [UNWT25, Section 5.1] and [LLW26a, Lemma 4.8], not proved. A short proof sketch in an appendix or a footnote would make the paper substantially more self-contained; the calculation is straightforward and confirms the cited result.","section":"Section 3.1, Eq. (5)"},{"comment":"The register conventions are confusing: Eq. (4) uses R as the reference register for both Q0 and Q1, while Eq. (8) applies Q1 to A,S and W to A′,R′,S′. Please clarify that S plays the role of the reference register in the circuits U1 and U0 and state explicitly that dim(S) = dim(R) so that the SWAP operation is well-defined.","section":"Section 3.1, Eq. (4) and Section 3.2, Eq. (8)"},{"comment":"The notation in Eq. (9) is terse; writing the unnormalized states explicitly, e.g., |bι1⟩ = (I_A ⊗ X)|ψ1⟩_{AS}, would help the reader connect Eq. (9) to the discussion in Section 1.2 and to Lemma 3.1.","section":"Section 3.2, Proposition 3.2, Eq. (9)"},{"comment":"The sentence 'the matching lower bound follows immediately from [Wan24, Theorem V.4]' should be expanded: the authors should state that any estimator for the unknown-pure-side problem also solves the known-pure-side subproblem, and therefore the Ω(1/ε) query lower bound from [Wan24] applies.","section":"Theorem 3.3, proof of lower bound"},{"comment":"The term 'purified query access' is used throughout but never formally defined. A precise definition (including the allowed controlled and inverse accesses to the state-preparation circuits) should be added to the preliminaries.","section":"Section 2 (Preliminaries)"}],"recommendation":"minor_revision","confidential_remarks":"The paper relies on [UNWT25] and [LLW26a], both of which involve a coauthor of this manuscript. This is not a circularity problem in my view: the cited unitary-dilation identity is stated precisely, is directly verifiable, and is used only as a lemma from prior work. The result is a focused, clean contribution that closes a natural gap, and the presentation is mostly clear. The requested clarifications about registers and the proof sketch for Eq. (5) are local and should be straightforward to address."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, compact result. It removes the 'known which state is pure' condition from the optimal fidelity estimator of Fang–Wang, achieving Θ(1/ε) queries and Θ(1/ε²) samples under the sole promise that one of two unknown states is pure. The new piece is the identity F(ρ0,ρ1)=max{a0,a1} (Prop 3.2), which follows cleanly from Uhlmann's theorem via a rank-one simplification of the cross operator. The proof is easy to follow, the union-bound argument in Theorem 3.3 is correct, and the lower bounds transfer from the pure-pure subproblem, so optimality is safe.\n\nThe one load-bearing element not proved in the paper is the unitary dilation identity Eq. (5), cited to UNWT25 and LLW26a. Because one of those preprints shares an author, it's worth a moment's suspicion, but it is not circular: the identity is a parameter-free construction, and I did the direct calculation on the expansion of the purifications; it holds. So this is a provenance and reproducibility point, not a correctness gap. Ideally the authors would include the short calculation as an appendix or explicitly restate the lemma, but its absence does not threaten the result.\n\nOther soft spots are minor. The novelty is incremental by design: the paper reuses square-root amplitude estimation and sample-to-query lifting, and its contribution is the specific max identity and the observation that QSVT is unnecessary in the one-pure-state case. That is a fair and honestly framed contribution, not a flaw. The success probability is a constant 0.9, which is enough for the complexity statement. The sample complexity corollary follows by a black-box lifting result that is still a preprint (TWZ26), though it is widely cited; that is normal in this subfield but worth noting.\n\nIn short: for anyone working on quantum state discrimination or the complexity of estimating standard functionals, this paper fills a genuine missing cell in the table. I would cite it, and I'd encourage a serious referee to engage rather than desk-reject. The referee should verify Eq. (5) and check that the same-register purification assumption is stated, but both are fine.","headline":"A short, correct paper that closes a small but real gap in fidelity estimation complexity, with the main identity cleanly derived; the only load-bearing external lemma checks out.","tokens_in":11737,"tokens_out":1776,"would_cite":true,"duration_ms":17085,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","68Q12"],"pacs":["03.67.-a"],"model":"deepseek-v4-flash","headline":"When at least one of two quantum states is pure, their Uhlmann fidelity can be estimated to additive error ε with Θ(1/ε) queries and Θ(1/ε²) samples, without knowing which state is pure.","keywords":["Uhlmann fidelity","quantum state estimation","purified query access","Uhlmann cross operator","square-root amplitude estimation","query complexity","sample complexity","pure quantum state"],"falsifier":"On a two-qubit example, e.g. ρ0=|0⟩⟨0| and ρ1=(1−λ)|0⟩⟨0|+λ|1⟩⟨1|, implement U0 and U1 from Eq. (8) and count queries until the estimate is within ε of √(1−λ); query growth beyond O(1/ε) would falsify Theorem 3.3, and a direct projection of W|0⟩ onto |0⟩ comparing with tr_A(|ψ0⟩⟨ψ1|) would test the dilation identity.","tokens_in":10667,"feed_emoji":"⚛️","tokens_out":10129,"duration_ms":87654,"temperature":0.7,"pith_summary":"This paper proves that estimating the Uhlmann fidelity F(ρ0,ρ1) between two unknown n-qubit quantum states requires Θ(1/ε) queries and Θ(1/ε²) samples when at least one of the states is pure, and that this optimal performance is achievable without knowing in advance which state is pure. The previous optimal estimator of Fang and Wang required the user to know which side was pure; removing that requirement is the paper's contribution. The key reduction is an identity: F(ρ0,ρ1) equals the larger of two square-root amplitudes, obtained by applying the Uhlmann cross operator X=tr_A(|ψ0⟩⟨ψ1|) (or its adjoint) to one of the two purifications. Because these amplitudes can be estimated with O(1/ε) queries using quantum amplitude estimation, the estimator matches the lower bound and quadratically improves the O(1/ε²)-query SWAP-test-based baseline.","feed_headline":"One-pure-state fidelity hits optimal Θ(1/ε) queries, side unknown","feed_subtitle":"The max of two amplitude estimates gives query- and sample-optimal fidelity without knowing the pure side.","key_machinery":"The load-bearing object is the Uhlmann cross operator $X=\\operatorname{tr}_A(|\\psi_0\\rangle\\langle\\psi_1|)$, which becomes rank-one when one of the two states is pure, with operator norm equal to the fidelity. It is implemented through the unitary dilation $W=Q_1^\\dagger(\\mathbb{I}_A\\otimes\\mathrm{SWAP}_{R,S})Q_0$, whose zero block is exactly $X$, so each of the two quantities $a_0,a_1$ is a square-root amplitude of a simple circuit. Square-root amplitude estimation then estimates each amplitude to $\\varepsilon$ at cost $O(1/\\varepsilon)$, and the identity $F=\\max\\{a_0,a_1\\}$ turns the two estimates into a fidelity estimate.","core_discovery":"The central claim is that, under the sole promise that at least one of the two states is pure, the Uhlmann fidelity admits the alternative expression $F(\\rho_0,\\rho_1)=\\max\\{a_0,a_1\\}$, where $a_1=\\lVert(\\mathbb{I}_A\\otimes X_S)|\\psi_1\\rangle\\rVert$ and $a_0=\\lVert(\\mathbb{I}_A\\otimes X_S^\\dagger)|\\psi_0\\rangle\\rVert$, with $X=\\operatorname{tr}_A(|\\psi_0\\rangle\\langle\\psi_1|)$ the Uhlmann cross operator and $|\\psi_0\\rangle,|\\psi_1\\rangle$ the purifications prepared by the state-preparation circuits. Because a purification of a pure state factorizes, $X$ has rank at most one and $\\lVert X\\rVert=F(\\rho_0,\\rho_1)$, which makes the max of the two amplitudes equal to the fidelity. The estimator constructs the explicit unitary dilation $W=Q_1^\\dagger(\\mathbb{I}_A\\otimes \\mathrm{SWAP}_{R,S})Q_0$ whose zero block is $X$, estimates $a_0$ and $a_1$ to additive error $\\varepsilon$ with $O(1/\\varepsilon)$ queries using square-root amplitude estimation, and returns their maximum. This yields query complexity $\\Theta(1/\\varepsilon)$, and applying quantum sample-to-query lifting gives sample complexity $\\Theta(1/\\varepsilon^2)$; both match lower bounds and neither requires knowing which state is pure.","pith_inferences":["The max-of-two-amplitudes identity is a rank-one phenomenon: if the promise were relaxed to 'one state has rank at most k', the cross operator would have rank at most k and its norm might be estimated by a low-rank variant of amplitude estimation; the paper does not explore this.","The construction suggests a general recipe for other Uhlmann-type quantities: express the target as the operator norm of a low-rank cross operator between two purifications, implement a dilation of that operator, estimate the two induced amplitudes, and take the max.","Because the paper relies on a prior dilation lemma rather than proving it, a direct numerical check of Eq. (5) on a small system would be a quick way to gain confidence in the estimator's correctness.","The estimator's accuracy degrades only through the sub-dominant singular values of X when neither state is exactly pure; quantifying that robustness could extend the result to 'near-pure' promises, which is not analyzed here."],"forward_implications":["The optimal query complexity for one-pure-state fidelity estimation is Θ(1/ε) even when the pure side is unknown, matching the known-which-side-is-pure case.","The optimal sample complexity is Θ(1/ε²), obtained by sample-to-query lifting, so the no-prior-knowledge estimator matches the known-side sample complexity.","Compared with the SWAP-test-based estimator that works without prior knowledge, query complexity improves from O(1/ε²) to Θ(1/ε) and sample complexity from O(1/ε⁴) to Θ(1/ε²).","Because the estimator uses purified query access, it also works in settings where copies are replaced by state-preparation circuits with controlled inverses; the same circuit construction implements both U0 and U1.","The lower bound is inherited from pure-state fidelity estimation, so the result is tight in the constant-error regime ε<1/4."],"supporting_citations":[{"why":"defines the Uhlmann fidelity and states Uhlmann's theorem that F equals the maximum overlap between purifications.","marker":"[Uhl76]"},{"why":"gives the elementary proof of Uhlmann's theorem and the explicit form of the Uhlmann transform used in Eq. (2).","marker":"[Joz94]"},{"why":"provides quantum amplitude estimation, the subroutine underlying the square-root amplitude estimator.","marker":"[BHMT02]"},{"why":"supplies the square-root amplitude estimation routine (Lemma 2.1) and the matching query and sample lower bounds used in Theorem 3.3 and Corollary 3.4.","marker":"[Wan24]"},{"why":"states the unitary dilation of the Uhlmann cross operator in Section 5.1, the key identity in Eq. (5) that makes the estimator implementable.","marker":"[UNWT25]"},{"why":"gives a self-contained statement of the same dilation lemma (Lemma 4.8), supporting the zero-block identity.","marker":"[LLW26a]"},{"why":"supplies the quantum sample-to-query lifting (Theorem 1.5) that converts the query-optimal estimator into the sample-optimal one.","marker":"[TWZ26]"},{"why":"gives the prior optimal estimator for the case where the pure side is known, the baseline that this paper's no-prior-knowledge result improves upon.","marker":"[FW25]"}],"fun_headline_variants":["Blind pure-state fidelity: optimal Θ(1/ε) queries without knowing which side","Max of two amplitudes: optimal fidelity for one pure state, no side info","Uhlmann transform yields optimal fidelity when one state is pure, unknown","One-pure-state fidelity hits optimal Θ(1/ε) queries via max of amplitudes","Blind to purity: optimal fidelity estimation from two amplitude estimates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction depends on a previously established fact about quantum circuits: the swap-based circuit built from the two purification circuits has, in its zero block, exactly the partial trace that defines the Uhlmann cross operator; the paper cites this fact without reproving it.","fun_headline_variants_meta":{"raw":{"variants":["Blind pure-state fidelity: optimal Θ(1/ε) queries without knowing which side","Max of two amplitudes: optimal fidelity for one pure state, no side info","Uhlmann transform yields optimal fidelity when one state is pure, unknown","One-pure-state fidelity hits optimal Θ(1/ε) queries via max of amplitudes","Blind to purity: optimal fidelity estimation from two amplitude estimates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000822,"raw_usage":{"total_tokens":3737,"prompt_tokens":1226,"completion_tokens":2511,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":842,"completion_tokens_details":{"reasoning_tokens":2418}},"tokens_in":842,"tokens_out":2511,"duration_ms":16201,"temperature":1.0,"reasoning_tokens":2418,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:40:25.245931+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a two-qubit example, e.g. ρ0=|0⟩⟨0| and ρ1=(1−λ)|0⟩⟨0|+λ|1⟩⟨1|, implement U0 and U1 from Eq. (8) and count queries until the estimate is within ε of √(1−λ); query growth beyond O(1/ε) would falsify Theorem 3.3, and a direct projection of W|0⟩ onto |0⟩ comparing with tr_A(|ψ0⟩⟨ψ1|) would test the dilation identity.","supporting_citations":[],"review_version":1}