{"id":"6b4a955a-0c3f-492f-9b9a-22789bd719c3","arxiv_id":"2608.06308","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For bipartite systems of local dimensions n and m, separable, PPT, and LOCC measurements all have the same optimal data-hiding ratio min(n,m), with explicit finite-round LOCC protocols.","lead":"Quantum data hiding hides information so local measurements cannot see it, even though a global measurement can. This paper finds the exact strongest hiding for several standard measurement classes and proves the answer is simply the smaller local dimension.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's interpolation from the p=2 and p=∞ endpoints to all p rests on the unverified trace-norm closedness of the two-round-LOCC set K; if [CLM+14, Cor. 3] does not cover unbounded intermediate outcomes, the Schatten-ball radius for 2<p<∞ is unsupported.","rationale":"The reader's weakest assumption correctly identifies the compactness of K as the pivotal step in the proof of Theorem 1.2. I found no independent fatal flaw in the main derivations: the Gaussian construction in Lemma 4.3 is consistent when the complex-Gaussian Wick pairings are interpreted correctly, and the estimates in Theorems 2.3 and 2.4 check out. The Theorem 1.1 ratio result does not itself require the interpolation, since its upper bound only needs the p=∞ endpoint; however the stronger geometric claim in Theorem 1.2 for all 2≤p≤∞, which is central to the paper's title and abstract, genuinely depends on the closedness of K. The proof imports this closedness from [CLM+14, Corollary 3] without reproducing it, so the concern is real but conditional: if the cited corollary is exactly as stated, the argument is sound. The additional dependence on the unpublished companion [LS26] for Theorem 1.4 is a separate completeness issue but does not affect the central separable-ball theorem. Therefore the appropriate verdict remains the reader's CONDITIONAL, and I recommend no change.","tokens_in":19768,"tokens_out":33247,"duration_ms":357926,"concrete_test":"Check the exact statement and proof of [CLM+14, Corollary 3]: does it apply to two-outcome LOCC instruments whose intermediate measurements may use an unbounded number of outcomes? If the corollary covers this case, reproduce the argument that the set K in Section 2.2 is trace-norm closed; if it requires bounded intermediate outcomes, the bipolar step after eq. (2.25) fails to establish H∈K for 2<p<∞.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 2.2, after eq. (2.25), the proof of Theorem 1.2 applies the bipolar theorem to the set K of Hermitian H whose binary POVM (I±H)/2 is two-round-LOCC-admissible. The bipolar conclusion H∈K requires K to be closed in the trace-norm topology. The paper's only justification is the footnote citing [CLM+14, Corollary 3], which is not reproduced. This is exactly the step that promotes the two direct endpoint results (Theorems 2.3 and 2.4) into the full family of Schatten p-balls for 2<p<∞. If the cited corollary does not apply to two-outcome instruments with unbounded intermediate outcomes, or if the set of finite-round LOCC with unbounded intermediate outcomes is not closed, then the polar argument only shows H lies in the closure of K, not in K itself. That would invalidate the claimed LOCC implementability for the intermediate-p balls, although the p=∞ endpoint alone would still support the Theorem 1.1 upper bound via Corollary 3.1 with q=1. Thus the load-bearing gap is the unproved compactness/closedness of K, on which the maximal separable ball result for 2<p<∞ depends.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies optimal quantum data hiding for restricted measurement classes on bipartite systems C^n⊗C^m. Theorem 1.1 claims R_PPT(n,m)=R_SEP(n,m)=R_LOCC(n,m)=min{n,m}. This is derived from Theorem 1.2, which states that for every 2≤p≤∞ the largest centered Schatten p-ball of Hermitian perturbations whose binary POVM is implementable by two-round LOCC has radius min{n,m}^{2/p-1}, and that this radius is optimal even for PPT measurements. Theorem 1.3 claims that the Alice-first one-way LOCC ratio, maximized over Bob's dimension, is (1+o(1))n, with the upper bound obtained from a Gaussian rank-one POVM. Theorem 1.4 improves the LO upper bound to R_LO(n,m)≤(π√3/4)min{n,m}+O(1). The proofs combine convexity/polarity arguments, Perron–Frobenius theory, explicit two-dimensional LOCC building blocks, Gaussian moment calculations, and noncommutative Khintchine-type inequalities.","tokens_in":20013,"tokens_out":34539,"duration_ms":306135,"significance":"If the main results are correct, Theorems 1.1 and 1.2 resolve the sharp data-hiding constants against PPT, separable, and LOCC measurements and strengthen the classical Gurvits–Barnum/Ando separable-ball theorems by upgrading separability to finite-round LOCC implementability. Theorem 1.3 gives the optimal asymptotic behaviour for Alice-first one-way LOCC, and Theorem 1.4 improves the known constant for LO from √2 to π√3/4. A notable strength is that the lower-bound witnesses (the SWAP operator and the Gaussian rank-one POVM) are external, parameter-free constructions rather than fitted to the target ratios. However, the proof of Theorem 1.4 currently depends on a block-matrix estimate imported from the unpublished companion [LS26], which is neither stated nor proved here; this makes that theorem not self-contained as written.","major_comments":[{"comment":"The proof of Theorem 1.4 begins with the block-matrix estimate ||h||_1 ≤ √n ||(h_ij)||_{L1[R+C]}, attributed to the unpublished manuscript [LS26]. This estimate is load-bearing: it is the only bridge between the Khintchine-type inequality (5.2) and the LO measurement bound (5.3) that produces the constant π√3/4. Since [LS26] is not available to the reader and the estimate is not stated precisely or proved in this paper, Theorem 1.4 is not self-contained as written. Please either prove Eq. (5.1) in an appendix or state the exact result from [LS26] with a full proof.","section":"Section 5, Eq. (5.1)"},{"comment":"The bipolar step that concludes H∈K from the two endpoint inclusions uses in an essential way that the set K of two-round-LOCC-admissible Hermitian operators is closed in the trace-norm topology. The only support is the footnote citing [CLM+14, Corollary 3], which is not reproduced. This closedness is exactly what promotes the endpoint results Theorems 2.3 and 2.4 to the full family 2≤p≤∞. Please state the precise corollary and explain why it covers two-outcome instruments whose intermediate measurements may use unboundedly many outcomes, or provide a direct compactness argument for K.","section":"Section 2.2, after Eq. (2.25)"},{"comment":"The proof of the representation nE[gg^*⊗S_V(g)]=V/n is not checkable as typeset. The Gaussian moment identity (4.12) is written as E[g_a g_j g_b g_i] with no conjugation bars, which is inconsistent with the complex Gaussian normalization E[g_a \\overline{g_b}]=δ_ab/n used in Eq. (4.7). Since Lemma 4.3 is the exact-representation basis for Theorem 1.3, the moment identity and the block calculation leading to Eq. (4.13) should be rewritten with explicit conjugates (e.g. E[g_a \\overline{g_j} \\overline{g_b} g_i]) so that the computation is verifiable.","section":"Lemma 4.3 and Eq. (4.12)"}],"minor_comments":[{"comment":"The text refers to 'theorem 2.1' and 'theorem 2.2' when the intended statements are Lemma 2.1 and Proposition 2.2; please fix the cross-references.","section":"Section 2"},{"comment":"The sentence 'The value of this protocol on his' is incomplete; it should be completed, e.g. 'The value of this protocol on his system is ...'.","section":"Section 4.1, proof of Proposition 4.1"},{"comment":"In the final displayed inequality, the first term should involve \\overline{a(w)}_{ij} or a(w)^*_{ij} rather than a(w)_{ij}, because the dual pairing is E tr(F(w)^* \\sum \\overline{a(w)}_{ij} h_{ij}). The estimate still goes through using the distributional symmetry a(w)∼a(w)^*, but the displayed equation as written is inconsistent.","section":"Section 5.2, Proposition 5.2"},{"comment":"The claim that Theorem 1.2 provides 'an explicit two-round LOCC protocol' for every p overstates what the proof gives for 2<p<∞, where membership in K is obtained via the bipolar theorem and is nonconstructive. Please qualify this statement.","section":"Introduction, after Theorem 1.2"},{"comment":"There is a typo: 'Hermtian' should be 'Hermitian'.","section":"Section 5.1, proof of Proposition 5.1"}],"recommendation":"major_revision","confidential_remarks":"The main gap is the reliance on [LS26] for Eq. (5.1) in Theorem 1.4. Since [LS26] is an unpublished companion by the same group, the editor should require that the estimate be proved in the paper or that the companion be made available before acceptance. The remaining theorems appear technically sound, but the proof of Lemma 4.3 needs a corrected, fully legible calculation. The paper fits the journal's scope and, if the gaps are repaired, would be a substantial contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Strong paper. It closes the PPT/SEP/LOCC data-hiding ratio at min{n,m}, and the sharper Schatten p-ball statement with explicit two-round LOCC is a genuinely new tool. I went through Theorems 1.1–1.3 and their proofs carefully; the arguments hang together, and the optimality witness (embedded SWAP) is correct. The Perron–Frobenius step in Theorem 2.3 and the block-matrix decomposition in Proposition 2.2 are clean. The one-way result (1+o(1))n with a fixed Gaussian rank-one Alice POVM is also a substantial step.\n\nThe soft spots are concentrated in two places. Theorem 1.4's upper bound relies on the block-matrix estimate (5.1), which is imported from the author's unpublished companion [LS26] and is neither stated nor proved here. That is a real dependency: the LO constant pi*sqrt(3)/4 is unverified in this manuscript as it stands. The author should either prove the estimate or make [LS26] available. This does not affect the main ratios.\n\nThe stress-test worry about Theorem 1.2's interpolation is, in my reading, overstated. The footnote at the closedness claim explicitly cites [CLM+14, Corollary 3] as covering fixed-round LOCC with unbounded intermediate outcomes. That is exactly the point the polar argument needs. The corollary is not reproduced, so a picky referee may ask for a restatement, but I don't see a missing step here.\n\nMinor but real: the Gaussian conventions in Section 4 are sloppy. The text writes E[g_a g_b] = δ_ab/n, then uses a Wick formula in Appendix A that looks like a real or non-circular covariance. Lemma 4.3 has a similar no-bar/bar ambiguity. This is repairable but needs fixing before publication.\n\nOverall: the main theorems are important, likely correct, and carefully argued. The paper deserves a serious referee. My recommendation: send it to review, and condition acceptance on the [LS26] estimate being incorporated or made public, plus a cleanup of the Gaussian notation.","headline":"Exact data-hiding ratios for PPT/SEP/LOCC are credible and new; the LO upper bound is conditional on an unpublished companion.","tokens_in":20624,"tokens_out":5218,"would_cite":true,"duration_ms":50625,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For bipartite quantum systems, the optimal data-hiding ratio against separable, PPT, and LOCC measurements is exactly the smaller local dimension, derived from a sharp maximal-ball theorem.","keywords":["quantum data hiding","LOCC","separable measurements","positive partial transpose","Schatten p-balls","distinguishability norms","one-way LOCC","maximal separable ball"],"falsifier":"Check the SWAP operator $F_d$ in small dimensions: Theorem 1.1 predicts $\\|F_d\\|_{\\rm LOCC}=d$, so the maximum bias any LOCC measurement can extract is $d$ while a global measurement gets $d^2$. A semidefinite-programming search over two-round protocols for $d=2$ or $d=3$ that produced bias greater than $d$, or any Hermitian $h$ with $\\|h\\|_1/\\|h\\|_{\\rm LOCC}>\\min\\{n,m\\}$, would falsify the theorem; likewise, finding $H$ with $\\|H\\|_p<d^{2/p-1}$ whose binary POVM is not LOCC-implementable would falsify Theorem 1.2.","tokens_in":19504,"feed_emoji":"🎭","tokens_out":19880,"duration_ms":167794,"temperature":0.7,"pith_summary":"Quantum data hiding asks how much more difficult it is to distinguish two quantum states when only local operations and classical communication are allowed instead of a global measurement. This paper determines the exact answer for the three standard restricted measurement classes: on $\\mathbb{C}^n\\otimes\\mathbb{C}^m$, the optimal data-hiding ratio against positive-partial-transpose (PPT), separable, and LOCC measurements is $\\min\\{n,m\\}$. In particular, the Werner-state protocol of [LPW18] reaches the maximum possible hiding factor, so no more hiding is possible under these restrictions. The decisive step is a stronger geometric statement: for every $2\\le p\\le\\infty$, the largest centered Schatten $p$-ball of Hermitian perturbations whose associated binary measurement can be implemented by two rounds of LOCC has radius $\\min\\{n,m\\}^{2/p-1}$, and this radius is optimal even for PPT measurements. That upgrades the classical separable-ball theorems and yields explicit finite-round LOCC implementations.","feed_headline":"Quantum data hiding tops out at the smaller local dimension","feed_subtitle":"For separable, PPT, and LOCC measurements, the hiding limit is exactly the smaller local dimension.","key_machinery":"The central object is the set $\\mathcal K$ of Hermitian $H\\in\\mathcal M_n\\otimes\\mathcal M_m$ whose binary POVM $((I+H)/2,(I-H)/2)$ is realizable by two-round LOCC. $\\mathcal K$ is convex, symmetric, and closed (the closure is imported from [CLM+14, Corollary 3]), so the bipolar theorem can be applied to it. The proof places the $p=2$ and $p=\\infty$ unit balls inside $\\mathcal K$ by explicit protocols: a two-dimensional block lemma writes each contraction $X$ as the average of two unitary polar factors, and a block-criterion proposition supplies an entrywise nonnegative row-stochastic matrix $A$ with $a_{ij}a_{ji}\\ge\\|H_{ij}\\|_\\infty^2$ that organizes the protocol branches; the endpoint conditions are then met using positivity of the leading eigenvector of a nonnegative matrix and a row-sum estimate. Interpolation uses the polar $\\mathcal K^\\circ$: for $Y\\in\\mathcal K^\\circ$ the endpoint inclusions force $\\|Y\\|_2\\le1$ and $\\|Y\\|_1\\le d$, and log-convexity of Schatten norms gives $\\|Y\\|_q\\le d^{1-2/p}$, so $|\\operatorname{tr}(YH)|\\le1$ whenever $\\|H\\|_p\\le d^{2/p-1}$. The one-way upper bound uses a different machine: a fixed Gaussian rank-one POVM $n\\,gg^*$ on Alice's space with the exact identity $n\\,\\mathbb E[gg^*\\otimes S_V(g)]=V/n$, completed by truncation and iteration; the no-communication bound uses Haar-random matrices $a(w)=wDw^*$ with $D=J_3\\otimes I_r\\oplus0$, a fourth-moment Khintchine estimate, and an explicit product measurement whose sign function isolates the constant $4/\\pi$.","core_discovery":"The paper's central claim is that for all $n,m\\ge1$, $R_{\\rm PPT}(n,m)=R_{\\rm SEP}(n,m)=R_{\\rm LOCC}(n,m)=\\min\\{n,m\\}$, where $R_{\\mathcal M}$ is the largest factor by which restricting measurements to a class $\\mathcal M$ can reduce the optimal bias between two states. The lower bound is exhibited by the SWAP operator on embedded $d$-dimensional subspaces: it has global trace norm $d^2$, while every PPT measurement extracts bias at most $d$. The upper bound is forced by Theorem 1.2, which states that any Hermitian $H$ with $\\|H\\|_p\\le d^{2/p-1}$ gives a binary POVM $((I+H)/2,(I-H)/2)$ that is implementable by two rounds of LOCC; taking $p=\\infty$ yields $\\|h\\|_1\\le d\\,\\|h\\|_{\\rm LOCC}$ for every Hermitian $h$. The proof establishes the $p=2$ and $p=\\infty$ endpoint balls as LOCC-implementable, interpolates between them using Schatten-norm log-convexity and the bipolar theorem, and uses the SWAP operator to show the radius cannot be enlarged even under PPT. The paper also determines the Alice-first one-way ratio as $(1+o(1))n$ uniformly in Bob's dimension and improves the no-communication bound to $(\\pi\\sqrt3/4+o(1))d$.","pith_inferences":["The same interpolation scheme should transfer to any convex, closed, symmetric measurement class that contains the $p=2$ and $p=\\infty$ endpoint balls, so the $d^{2/p-1}$ radius may hold for other natural families between LOCC and PPT.","The equality of the PPT, SEP, and LOCC ratios suggests a dimension-only answer to data hiding in the bipartite finite-dimensional setting; an analogous statement may hold multipartite once the smallest local dimension is fixed.","The fixed Gaussian rank-one POVM is a concrete physical prescription: Alice runs a Gaussian continuous measurement, and the truncation-and-iteration construction indicates that finite approximations should achieve the asymptotic ratio, which could be tested numerically for small $n$.","The $4/\\pi$ constant in the no-communication bound reflects the choice of the nilpotent block $D=J_3\\otimes I_r$ and the periodic POVM; optimizing that choice is a natural route to close the gap between $d$ and $(\\pi\\sqrt3/4)d$."],"forward_implications":["The Werner-state data-hiding protocol of [LPW18] is optimal: it realizes the maximal hiding factor $d$ against every PPT, separable, or LOCC measurement.","For every $2\\le p\\le\\infty$, the binary measurement of any Hermitian $H$ with $\\|H\\|_p\\le d^{2/p-1}$ is implementable by two rounds of LOCC, upgrading the classical separable-ball theorems from separability to explicit finite-round communication.","The sharp comparison $\\|h\\|_1\\le d\\,\\|h\\|_{\\rm LOCC}$ holds for every Hermitian $h$, and more generally $\\|h\\|_{\\rm LOCC}\\ge d^{1-2/q}\\|h\\|_q$ for $1\\le q\\le2$.","Alice-first one-way LOCC cannot hide more than $(1+o(1))n$, and the Gaussian rank-one POVM gives a constructive protocol that works uniformly for every dimension of Bob's system.","Without communication, the data-hiding ratio lies between $d$ and $(\\pi\\sqrt3/4+o(1))d$, so the exact constant is pinned to the interval $[1,1.360\\ldots]$ asymptotically."],"supporting_citations":[{"why":"Supplies the Werner-state protocol whose optimality is established, together with the earlier upper bound $R_{\\rm LOCC}(n,m)\\le 2d-1$ that the paper improves.","marker":"[LPW18]"},{"why":"Proves the Hilbert-Schmidt separable-ball theorem and the sharpness of the Schatten radius $d^{2/p-1}$, supplying the $p=2$ endpoint and the optimality argument.","marker":"[GB02]"},{"why":"Gives the operator-norm separable-ball result that forms the $p=\\infty$ endpoint of Theorem 1.2.","marker":"[And04]"},{"why":"Introduces the distinguishability-norm formulation of data hiding and proves the balanced-case result $R_{\\rm SEP}(n,n)=n$ that the paper extends to all dimensions and to LOCC.","marker":"[MWW09]"},{"why":"Provides the compactness of fixed-round LOCC instruments used to close the admissible set and apply the bipolar theorem in the interpolation step.","marker":"[CLM+14]"},{"why":"Supplies the truncation-and-iteration lemma that turns approximate bounded representations into exact bounded representations in the one-way and no-communication proofs.","marker":"[HM07]"},{"why":"Provides the block-matrix estimate and the sharp product-observable comparison on which the improved no-communication upper bound relies.","marker":"[LS26]"}],"fun_headline_variants":["Data hiding ratio equals the smaller local dimension","Quantum hiding caps at the smaller dimension","Hiding limit: exactly the smaller local dimension","Optimal hiding equals min(n,m)","Max hiding ratio is the smaller dimension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The interpolation step in Section 2.2 assumes that the set of Hermitian operators whose binary measurement can be realized by two-round LOCC is closed, meaning it contains its limits; the paper cites [CLM+14, Corollary 3] for this rather than proving it.","fun_headline_variants_meta":{"raw":{"variants":["Data hiding ratio equals the smaller local dimension","Quantum hiding caps at the smaller dimension","Hiding limit: exactly the smaller local dimension","Optimal hiding equals min(n,m)","Max hiding ratio is the smaller dimension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000656,"raw_usage":{"total_tokens":3062,"prompt_tokens":1064,"completion_tokens":1998,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":680,"completion_tokens_details":{"reasoning_tokens":1934}},"tokens_in":680,"tokens_out":1998,"duration_ms":14143,"temperature":1.0,"reasoning_tokens":1934,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:49:29.657417+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the SWAP operator $F_d$ in small dimensions: Theorem 1.1 predicts $\\|F_d\\|_{\\rm LOCC}=d$, so the maximum bias any LOCC measurement can extract is $d$ while a global measurement gets $d^2$. A semidefinite-programming search over two-round protocols for $d=2$ or $d=3$ that produced bias greater than $d$, or any Hermitian $h$ with $\\|h\\|_1/\\|h\\|_{\\rm LOCC}>\\min\\{n,m\\}$, would falsify the theorem; likewise, finding $H$ with $\\|H\\|_p<d^{2/p-1}$ whose binary POVM is not LOCC-implementable would falsify Theorem 1.2.","supporting_citations":[],"review_version":1}