{"id":"a1d317b9-5118-445e-9088-5db55f8970f9","arxiv_id":"2607.26474","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Phase retrievability for rank-≤k positive semidefinite operators is equivalent to separating pairs with orthogonal supports, and this yields Lipschitz trace-norm stability and Bures-Wasserstein Hölder/Lipschitz stability.","lead":"This paper extends phase-retrieval theory to quantum states of bounded rank and to general super-operators, not just quantum channels. It shows that distinguishing only orthogonal-support states is enough, and derives stability bounds in trace and Bures-Wasserstein distances.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central Theorem 6.2 depends on Lemma 6.1, whose proof cites a false inequality; the lemma is true but needs a corrected proof, so the paper's main criterion is as yet unproved.","rationale":"The reader identified Lemma 6.1 as the weakest assumption, and I agree that its proof is invalid. However, the reader's proposed repair via Q≤B and P≤A is itself false; the lemma can be repaired by a different rank-based argument. This does not overturn the paper's central claim, because the lemma is true, but it means the paper as written does not yet rigorously establish Theorem 6.2 and the subsequent stability theorems. Other proof flaws exist (e.g., factor-2 algebra in Theorem 4.1, a missing division in Theorem 8.4's proof chain, and an unproved rank condition in Theorem 5.5), but Lemma 6.1 is the most load-bearing because it underpins the main orthogonal-pair reduction and the trace-norm bi-Lipschitz stability. The verdict should remain conditional: the theorems are likely correct, but a corrected proof of Lemma 6.1 and the other fixable slips are required before acceptance.","tokens_in":31375,"tokens_out":29864,"duration_ms":287430,"concrete_test":"Randomly search (or use a small semidefinite program) over A,B∈Pos(C^4), rank(A),rank(B)≤2, and compute the ranks of the positive and negative parts of A−B; a violation of Lemma 6.1 would refute Theorem 6.2. If no violation appears (expected), the issue is purely a missing proof; then verify that the corrected proof above actually establishes the lemma, and patch §6 accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 6.1 claims that for A,B≥0 with rank≤k, the Hahn–Jordan parts P,Q of A−B have rank≤k. The proof invokes [ZCLW22, Prop. 2.2(ii)] with the direction n_-(A−B)≤n_-(A), but the cited result does not give this; for scalars A=1,B=2, n_-(A−B)=1 > n_-(A)=0. The reader's suggested repair 'Q≤B' is also false: take A=[[1,1],[1,1]], B=diag(1,0); then Q=(B−A)_+ has a negative (2,2) entry in B−Q, so Q≦B. The lemma is nevertheless true: if (B−A)_+ has rank r, then on its r-dimensional range S the operator B−A is positive definite, so A<B on S, forcing dim S≤rank(B) (otherwise B would vanish on a nonzero vector in S, contradicting A≥0). Similarly rank(P)≤rank(A). But this argument is absent. Since Theorem 6.2 (orthogonal-pair reduction), Lemma 6.5 (minimization characterization), and Theorem 6.9 (bi-Lipschitz bound) all hinge on Lemma 6.1, the central stability claims are not justified as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies phase retrieval for super operators and measurements on finite-dimensional spaces, generalizing the pure-state framework of Liu-Han to the set P_k of positive semidefinite operators of rank at most k. The main results are: a correspondence between injectivity on P_k and phase retrievability (Theorem 3.2 and §4); existence of quantum channels that are P_k-phase retrievable but not injective, and measurements that separate P_k from P_{k+1} (Section 5); a reduction of P_k-phase retrievability to distinguishing orthogonal-support pairs (Theorem 6.2); trace-norm bi-Lipschitz stability (Theorem 6.9); and Bures–Wasserstein stability via Fuchs–van de Graaf inequalities (Section 8). The paper also contains block-diagonal characterizations inspired by the complement property in frame theory.","tokens_in":31740,"tokens_out":14590,"duration_ms":142529,"significance":"If the results are correct, this is a genuinely useful contribution: it places quantum phase retrieval in a broader operator-theoretic setting, gives a surprisingly simple orthogonal-pair criterion for bounded-rank retrievability, and extends Lipschitz stability from frame theory to the quantum trace-norm and Bures–Wasserstein settings. The paper is refreshingly free of fitted parameters and circularity: the equivalences are biconditional statements proved from definitions and cited external theorems. However, the current exposition contains several load-bearing proof errors, most importantly in Lemma 6.1, Theorem 4.1, Theorem 5.5, and Theorem 8.4. These do not appear to invalidate the main claims—the errors are local and fixable—but they must be corrected before the paper can be accepted.","major_comments":[{"comment":"The proof of the bound rank(Q)≤k is invalid. The invocation of [ZCLW22, Prop. 2.2(ii)] to conclude n_-(A−B)≤n_-(A) is false, e.g. for scalars A=1, B=2 one has n_-(A−B)=1 but n_-(A)=0. The sentence 'Θ_1 ⊆ Θ_1' is also a typo. The lemma is true (one can prove n_+(A−B)≤rank(A) and n_-(A−B)≤rank(B) by interlacing or by a range argument), but the supplied proof is not. Since Theorem 6.2, Lemma 6.5, Theorem 6.9, and all measurement analogues depend on this lemma, the central stability criterion is unsupported as written.","section":"Section 6, Lemma 6.1"},{"comment":"The proof asserts that the constructed four-dimensional subspace B spanned by B1,...,B4 satisfies condition (d) of [HMW13, Thm. 1], namely rank(B)≥2k+1 for every nonzero B∈B. This is not verified. It is a nontrivial statement about arbitrary linear combinations of a shift matrix, its transpose, and two diagonal matrices, and it is exactly the condition needed to apply [HMW13]. Without a proof, the existence of a P_k-phase retrievable but not P_{k+1}-phase retrievable measurement is not established.","section":"Section 5, Theorem 5.5"},{"comment":"The proof contains factor-2 algebra errors. In (a)⇒(b), with U=1/2(X+Y) and V=1/2(X−Y), one has UU^*−VV^* = 1/2(XY^*+YX^*), not XY^*+YX^*. In (b)⇒(a), with X=1/2(U+V) and Y=1/2(U−V), one has XY^*+YX^* = 1/2(ρ−σ), not ρ−σ. The theorem's statement is still salvageable because the factor 1/2 cancels in the implications, but as written the proof is algebraically incorrect.","section":"Section 4, Theorem 4.1"},{"comment":"The first displayed inequality has the wrong scaling. From Corollary 8.3 one obtains d_BW(ρ,σ)^2 ≤ C d_BW(Φρ,Φσ)(∥Φρ∥_1^{1/2}+∥Φσ∥_1^{1/2}). Combining with the rank-one identity ∥ρ−σ∥_1 = d_BW(ρ,σ)√(Trρ+Trσ+2F) gives a factor √(Trρ+Trσ+2F) in the denominator on the right, not in the numerator. The final line also drops this factor entirely. A correct proof is possible by bounding TrΦρ+TrΦσ+2F(Φρ,Φσ) by 2∥Φ∥_{1→1}(Trρ+Trσ) and using S≥Trρ+Trσ, but the argument as printed does not establish the claimed Lipschitz bound.","section":"Section 8, Theorem 8.4"}],"minor_comments":[{"comment":"The compactness of D is asserted without detail. It is true because P_k is closed (rank≤k is closed under limits) and the normalization makes D bounded, but this should be stated.","section":"Section 6, Lemma 6.5"},{"comment":"There are many stray minus signs in function arrows, e.g. 'L(X)− →L(Y)' in Theorem 6.7, Theorem 7.5, and elsewhere. Also the definition of P_k in Theorem 6.7 has a misplaced brace: it reads '{ρ∈Pos(X)}: rank(ρ)≤k}'.","section":"Throughout"},{"comment":"The final displayed derivation of the Hölder constant is dimensionally inconsistent: the right-hand side first has a linear factor d_BW(Φρ,Φσ), then is declared equal to K√(d_BW(Φρ,Φσ)). The correct conclusion is d_BW(ρ,σ) ≤ K√(d_BW(Φρ,Φσ)) with K = √(2C√L√∥Φ∥_{1→1}).","section":"Section 8, Corollary 8.3"},{"comment":"In the proof, the notation alternates between Herm(R^r) and Herm(C^r) in several displayed lines. These should be consistently C^r in the complex theorem.","section":"Section 4, Theorem 4.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is not circular and contains no fitted parameters; the issues are local proof errors rather than conceptual ones. The central claims appear plausible and fixable, so rejection is not warranted. The rank-condition gap in Theorem 5.5 is the one I would ask the authors to check most carefully: if condition (d) fails for some linear combination, the existence construction may need to be replaced. The incorrect proof of Lemma 6.1 is especially important because it supports the paper's central orthogonal-pair reduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper before the next round: it extends phase retrievability from pure states to PSD operators of rank at most k, and from quantum channels to arbitrary superoperators. The new results are real. Theorem 6.2 (orthogonal-pair reduction) is a clean structural insight, and it feeds the bi-Lipschitz trace-norm stability of Theorem 6.9 and the Bures–Wasserstein bounds in Section 8. The block-diagonal complement characterization in Theorems 4.7/4.8 also meaningfully generalizes the frame-theoretic complement property. The paper is a continuation of [LH25], not a breakthrough, but it is a substantial step for the subfield.\n\nThe main problem is Lemma 6.1, the Hahn–Jordan rank bound. The proof cites [ZCLW22, Prop. 2.2(ii)] for an inequality direction that is simply false — for scalars A=1, B=2, n_-(A−B)=1 > n_-(A)=0. The reader's suggested repair via Q≤B is also false, as the stress-test note shows by explicit counterexample. The lemma itself is true, provable via a range argument: if (B−A)_+ has rank r, then on its range B−A is positive definite, forcing dim of that range ≤ rank(B), and similarly for the positive part. But that argument is absent. Since Theorem 6.2, Lemma 6.5, and Theorems 6.7 and 6.9 all hinge on Lemma 6.1, the central stability claims are not justified as written. This is a fixable gap, not a fatal one, but it needs a corrected proof before the paper can stand.\n\nTwo smaller issues. Theorem 4.1 has factor-of-two algebra mistakes: UU*−VV* is (XY*+YX*)/2, not XY*+YX*, and similarly in the converse. These are harmless because the nonzero factor cancels in the implications, but they should be corrected. Theorem 5.5 asserts that the constructed 4-dimensional space B satisfies the rank condition rank(B)≥2k+1 for every nonzero B, with no verification. Given the paper's reliance on [HMW13], that is a real gap, probably fillable but not trivial.\n\nCitation pattern looks fair — they build on [LH25] and [HMW13] and acknowledge precedents like [KW15]. No fitted parameters, no circularity. The writing is clear even if occasionally verbose.\n\nWho is this for? People working on phase retrieval, quantum tomography, and low-rank quantum state recovery. It deserves a serious referee: the results are plausible and valuable, and the flaws are specific and addressable. I'd accept it for review with the expectation of major revision, mainly to fix Lemma 6.1 and Theorem 5.5.","headline":"Useful generalization of phase retrievability to bounded-rank states and general superoperators, but the key stability results rest on a lemma whose proof is wrong; the lemma is true and repairable, so the paper is salvageable.","tokens_in":32193,"tokens_out":3257,"would_cite":true,"duration_ms":34957,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","81P16","15A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a super operator is phase-retrievable on positive semidefinite operators of rank at most k if and only if it separates every pair of such operators with orthogonal supports, and from this criterion derives trace-norm a","keywords":["quantum phase retrieval","super operators","rank-k positive semidefinite operators","orthogonal supports","trace norm stability","Bures–Wasserstein distance","quantum measurements","Hahn–Jordan decomposition"],"falsifier":"Construct two rank-2 positive semidefinite matrices A and B whose difference A − B has a positive part of rank 3, which would disprove the key rank lemma. Alternatively, exhibit a super operator that separates all orthogonal-support rank-k pairs but fails to separate a non-orthogonal pair, which would falsify the central criterion.","tokens_in":31314,"feed_emoji":"⚛️","tokens_out":6540,"duration_ms":67746,"temperature":0.7,"pith_summary":"This paper extends quantum phase retrieval from pure states to all positive semidefinite operators of rank at most k, and from quantum channels to arbitrary super operators. Its central result is that a super operator can distinguish every pair of rank-k positive operators if and only if it distinguishes all pairs with orthogonal supports. From this single criterion the authors derive several stability theorems: nearby super operators preserve the property, the map is bi-Lipschitz with respect to the trace norm on rank-k states, and—via generalized Fuchs–van de Graaf inequalities—stable in the Bures–Wasserstein distance. This recovers and generalizes the classical stability results for phase retrieval in frame theory, and connects phase retrieval to quantum tomography.","feed_headline":"Phase retrieval for rank-k states reduces to orthogonal pairs","feed_subtitle":"Separating disjoint-support rank-k states is necessary and sufficient, and yields trace-norm and Bures–Wasserstein stability.","key_machinery":"The load-bearing mechanism is the Hahn–Jordan decomposition of Hermitian operators, which splits any difference ρ − σ into mutually orthogonal positive and negative parts P and Q. The paper's Lemma 6.1 asserts that when ρ and σ have rank at most k, the parts P and Q also have rank at most k; this rank bound is the delicate step that turns injectivity on the entire set P_k into separation of orthogonal-support pairs. A second piece of machinery is the correspondence between measurements and diagonal-valued super operators, which lets the authors translate every result between the two settings. A third piece is the generalized Fuchs–van de Graaf inequalities, which bridge trace-norm stability","core_discovery":"The paper's central claim is Theorem 6.2: a super operator Φ is P_k-phase retrievable—meaning injective on positive semidefinite operators of rank at most k—if and only if for every pair ρ, σ in P_k, not both zero, with orthogonal supports (ρσ = 0), one has Φ(ρ) ≠ Φ(σ). The proof reduces a general pair ρ, σ to the orthogonal pair (P, Q) obtained from the Hahn–Jordan decomposition of ρ − σ; if the positive and negative parts remain of rank ≤ k, then separating those parts is enough. The authors use this criterion to prove a compactness-based minimization lemma, openness of the class of P_k-phase retrievable super operators, a trace-norm bi-Lipschitz equivalence, and then translate these to Bu","pith_inferences":["The orthogonal-pair criterion suggests a practical certification strategy: instead of checking all pairs of rank-k states, one could enumerate or sample only pairs with disjoint supports, potentially giving efficient verification for fixed k.","The Bures–Wasserstein connection indicates that classical stability of phase retrieval in frame theory is a special case of a quantum metric inequality; analogous stability might hold for other quantum distances, though the paper does not explore that.","The gap in Lemma 6.1's proof is likely repairable because the statement follows directly from the inclusions P ≤ A and Q ≤ B, so the main theorems are expected to survive once the proof is corrected.","The reduction to orthogonal pairs may inspire new algorithms for quantum state tomography with bounded-rank priors, since it decouples the rank constraint from the injectivity check."],"forward_implications":["Testing whether a super operator can retrieve rank-k states reduces to checking a smaller test set: pairs with orthogonal supports.","The set of P_k-phase retrievable super operators is open, so small perturbations of a retrievable map remain retrievable.","A P_k-phase retrievable super operator is bi-Lipschitz with respect to the trace norm on P_k, so output differences control input differences up to a constant.","Via the generalized Fuchs–van de Graaf inequalities, retrieval is 1/2-Hölder stable in the Bures–Wasserstein distance for general rank k, and Lipschitz stable for k = 1.","The orthogonal-pair reduction and stability results carry over to measurements, giving an open set of retrievable measurements with analogous bounds.","The symmetry-equivariant version extends the stability results to sets of states invariant under a group representation."],"fun_headline_variants":["Orthogonal pairs decide phase retrieval for rank-k states","Rank-bounded phase retrieval reduces to disjoint supports","Phase retrieval for PSDs of bounded rank: key is orthogonality","Trace-norm stable phase retrieval from orthogonal-separation test","Phase retrievability for rank-k: only orthogonal pairs need separating"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central reduction assumes that splitting the difference of two rank-k positive semidefinite operators into positive and negative parts keeps those parts within rank k; the paper's proof of this lemma cites an eigenvalue inequality in the wrong direction, so the lemma's proof is incomplete as written, although the lemma itself can be proven by a more direct argument.","fun_headline_variants_meta":{"raw":{"variants":["Orthogonal pairs decide phase retrieval for rank-k states","Rank-bounded phase retrieval reduces to disjoint supports","Phase retrieval for PSDs of bounded rank: key is orthogonality","Trace-norm stable phase retrieval from orthogonal-separation test","Phase retrievability for rank-k: only orthogonal pairs need separating"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000253,"raw_usage":{"total_tokens":1402,"prompt_tokens":748,"completion_tokens":654,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":570}},"tokens_in":492,"tokens_out":654,"duration_ms":7092,"temperature":1.0,"reasoning_tokens":570,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T15:03:12.814722+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct two rank-2 positive semidefinite matrices A and B whose difference A − B has a positive part of rank 3, which would disprove the key rank lemma. Alternatively, exhibit a super operator that separates all orthogonal-support rank-k pairs but fails to separate a non-orthogonal pair, which would falsify the central criterion.","supporting_citations":[],"review_version":1}