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REVIEW 3 major objections 5 minor 22 references

On Optimal Quantum Data Hiding and Maximal Separable Ball

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Exact data-hiding ratios for PPT/SEP/LOCC are credible and new; the LO upper bound is conditional on an unpublished companion. read the letter →

arxiv 2608.06308 v1 pith:G74CHQ6S submitted 2026-08-06 quant-ph

Zhi Li This is my paper · ORCID
classification quant-ph
keywords quantumdatahidingLOCCseparablemeasurementspositivepartialtransposeSchattenp-ballsdistinguishabilitynormsone-waymaximalball
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $4/\pi$.

What would settle it

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.

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Extended reading notes

Core claim

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 $(\pi\sqrt3/4+o(1))d$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $(\pi\sqrt3/4)d$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 5, Eq. (5.1)] 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.
  2. [Section 2.2, after Eq. (2.25)] 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.
  3. [Lemma 4.3 and Eq. (4.12)] 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.
minor comments (5)
  1. [Section 2] 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.
  2. [Section 4.1, proof of Proposition 4.1] 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 ...'.
  3. [Section 5.2, Proposition 5.2] 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.
  4. [Introduction, after Theorem 1.2] 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.
  5. [Section 5.1, proof of Proposition 5.1] There is a typo: 'Hermtian' should be 'Hermitian'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central ratio theorems are derived from explicit endpoint constructions and standard duality; the only caveats are external proof gaps and a minor unpublished self-citation, neither of which reduces a prediction to its input by construction.

full rationale

The main derivation chain is non-circular. Theorem 1.1 is obtained from the upper bound in Corollary 3.1, which is a duality consequence of Theorem 1.2, and from the SWAP-operator lower bound that is an external witness. Theorem 1.2 itself is built from the two endpoint theorems 2.3 and 2.4, which are proved explicitly: Theorem 2.3 uses a Perron-Frobenius rescaling plus the explicit two-dimensional LOCC block protocol of Lemma 2.1, and Theorem 2.4 uses a direct row-sum estimate. The interpolation between the endpoints is a standard polar/bipolar argument. No fitted parameter is introduced and no quantity is renamed as a prediction. The skeptical concern about the trace-norm closedness of the two-round-LOCC set K, imported from [CLM+14, Cor. 3], is a real verification gap but is not a circular reduction: it is an external topological input, not a definition of the target radius in terms of itself. The only self-citation is [LS26], used for the block-matrix inequality (5.1) in the proof of Theorem 1.4. That inequality has independent content and does not state the target data-hiding ratio, so it is not a fitted-input or definitional circularity, but because [LS26] is an unpublished companion manuscript by the present author, it creates a verification burden for the secondary LO bound. The central claims of the paper remain self-contained against explicit constructions and standard arguments, so any circularity is at most minor and confined to a non-central dependency.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical free parameters are fitted; all constants in the proofs are determined by the arguments. The main imported inputs are standard probability and linear algebra tools, the compactness of fixed-round LOCC, and one unpublished companion estimate [LS26] used for the no-communication bound. No new physical particles, forces, dimensions, or ledger entities are introduced.

assumptions (4)
  • domain assumption Fixed-round LOCC instruments with a fixed number of outcomes form a compact set, even when intermediate measurements may have unbounded outcomes (CLM+14, Corollary 3).
    Invoked in the proof of Theorem 1.2 after eq (2.25) to make the set K closed so the bipolar theorem applies.
  • standard math Wick's formula for complex Gaussian moments and the Weingarten/Haar integration formulas for unitary groups.
    Used in Appendix A for the Gaussian rank-one POVM moment estimates and in Appendix B for the fourth-moment estimates in the LO proof.
  • standard math Perron-Frobenius theorem for nonnegative matrices.
    Used in Theorem 2.3 to construct the row-stochastic matrix A satisfying the dominance condition eq (2.8).
  • domain assumption The block-matrix estimate ||h||_1 <= sqrt(n) ||(h_ij)||_{L1[R+C]} of [LS26] (eq 5.1).
    Load-bearing for Theorem 1.4; the companion [LS26] is unpublished and self-authored, so the estimate is not independently verifiable from this paper.

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Pith. "Pith review of On Optimal Quantum Data Hiding and Maximal Separable Ball." pith.science (2026). https://pith.science/paper/G74CHQ6S

@misc{pith2026260806308,
  author       = {Pith},
  title        = {Pith review of: On Optimal Quantum Data Hiding and Maximal Separable Ball},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G74CHQ6S}},
  note         = {Machine review of arXiv:2608.06308}
}
abstract

Quantum data hiding asks how much distinguishing power can be lost when global measurements are restricted to local measurements and classical communication. In this work, we establish sharp results and improved bounds for several natural classes of restricted measurements. For bipartite systems on $\mathbb C^n\otimes\mathbb C^m$, we prove that the optimal data-hiding ratios against separable and LOCC measurements are both $\min\{n,m\}$. This result follows from a stronger result that, for every $2\le p\le\infty$, the largest centered Schatten $p$-ball whose associated binary measurements are implementable by finite-round LOCC has radius $\min\{n,m\}^{2/p-1}$. This strengthens the classic separable-ball theorems, while also providing an explicit finite-round LOCC implementation. For Alice-first one-way LOCC with Alice's local dimension equal to $n$, we prove that the optimal ratio is $(1+o(1))n$, with the upper bound obtained from a Gaussian rank-one POVM. For local operations without communication, we improve the universal upper bound to $(\pi\sqrt3/4+o(1))\min\{n,m\}$.

Figures

Figures reproduced from arXiv: 2608.06308 by the authors.

Figure 1
Figure 1. The tensor network for Oσ for the σ in eq. (A.13), with contraction proceeding from right to left. The dashed lines indicate the cuts K0, . . . , K4. We estimate the tensor network by moving a vertical cut from right to left, across one block at a time. This expresses Oσ as a composition of four linear maps. Note that the domain and codomain at each step depend on the Alice and Bob index lines crossing the cut, and … view at source ↗

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