Randomness cost of masking quantum information and the information conservation law

Hyunseok Jeong; Seok Hyung Lie

REVIEW 4 major objections 5 minor 23 references

Randomness cost of masking quantum information and the information conservation law

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that masking quantum information costs randomness at least equal to the unevenness of how information splits between two parties, tightening the earlier lower bound and linking the cost to conservation of quantum…

desk verdict New lower bound on randomness cost of masking and a counterexample to the disk conjecture; the bound is sound if you grant the decomposition theorem imported from the authors' earlier work. read the letter →

arxiv 1908.07426 v1 pith:4EELDODJ submitted 2019-08-20 quant-ph

Seok Hyung Lie , Hyunseok Jeong This is my paper · ORCID

classification quant-ph

keywords quantummaskingrandomnesscostinformationconservationlawchannelmixingentanglement-assistedcapacitysecretsharingblackholeparadoxno-hidingtheorem

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

This paper tries to show that the randomness needed to mask quantum information is not governed by the total correlation between the two parties, but by how unevenly information flows to them. It proves that any universal quantum masking process must consume randomness at least as large as a regularized measure of this unevenness, a quantity between $\log d$ and $2\log d$ for $d$-dimensional systems. It also disproves a geometric conjecture about which states can be masked unitarily, replacing the geometric picture with an algebraic, information-conservation-based one. If correct, the result gives a quantitative consistency test for black-hole fast scrambling and a lower bound on entanglement needed for quantum secret sharing.

What carries the argument

The central object is the decomposition of a universal quantum masker as a probabilistic mixture of bipartite embeddings, $\Phi_M(\rho)=\sum_i p_i M_i\rho M_i^\dagger$, with a 'safe state' $\sigma_S$ whose entropy $R(\Phi_M)=S(\sigma_S)$ is the randomness cost. The argument's engine is the information conservation law, which says that the mutual information between a reference system and the two output systems is conserved, so masking can only push information from one party to the other. This is combined with a channel-mixing tradeoff: if a subchannel $N_i$ has entanglement-assisted classical capacity exceeding the mixed channel's capacity by more than $-\log p_i$, then the mixture could transmit more information than an erasure channel should allow, a contradiction. The measure $I_1$ and its regularization $I_\infty$ capture the largest 'unevenness' of information flow across subchannels, and Theorem 4 bounds the randomness cost from below by $I_\infty$.

What would settle it

Find a $d$-dimensional universal masking process whose safe-state entropy $R(\Phi_M)$ is strictly smaller than the regularized unevenness $I_\infty(\{M_i\})$ computed from its bipartite embeddings. Concretely, perform complete process tomography on a candidate masker, reconstruct the subchannels and probabilities, compute $I_1$ on many copies and take the regularized limit, and compare it with $S(\sigma_S)$; any violation would refute Theorem 4.

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Extended reading notes

Core claim

For any $d$-dimensional universal quantum masking process $\Phi_M$ with decomposition into random bipartite embeddings $\{M_i\}$ with orthogonal images, the randomness cost $R(\Phi_M)$ satisfies $I_\infty(\{M_i\}) \le R(\Phi_M)$, where $I_\infty$ is the regularized version of a measure $I_1$ quantifying how unevenly information is distributed between the two parties. This improves the previously known $\log d$ lower bound to a quantity that can reach $2\log d$. The paper further shows that the set of maskable states of an isometric quantum masker need not form a 'disk' in the sense conjectured earlier, by exhibiting a $d^2$-dimensional counterexample where the maskable set is not tied to any preferred product basis. All of these results are derived from a channel-mixing tradeoff combined with the information conservation law $2S(R)=I(R:A)+I(R:B)$, and they tolerate incomplete masking up to a small error.

Load-bearing premise

The proofs assume that every universal quantum masker can be represented exactly as a probabilistic mixture of isometries with a safe state whose von Neumann entropy is the true randomness cost; if some masker required a different accounting of randomness, the lower bounds would not apply to it.

Editorial extensions

If this is right

The randomness cost of masking is minimal when each subchannel distributes information as evenly as possible, saturating the bound as in quantum one-time pad and the four-qubit masker.
A masking process in which information flows entirely to one party in half of the subchannels and entirely to the other in the other half must spend close to $2\log d$ bits of randomness.
For approximate masking with error $e$, the bounds remain valid after replacing each $I_i$ by $I_i-e$, so the result applies to realistic imperfect masking devices.
Every quantum masker induces a $(2,3)$-threshold quantum secret sharing scheme, so the lower bound estimates the sizes of unauthorized sets in pure $(k,2k+1)$-threshold protocols.
For black-hole evaporation modeled as masking, the inequality $I_\infty(\{M_i\}) \le c(t_*)S(T)$ provides a consistency check between the scrambling time, the evaporation dynamics, and the black hole's entanglement entropy.

Reading between the lines

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

The channel-mixing suppression theorem may be a general principle beyond masking: in any probabilistic mixture of quantum channels, a highly capable subchannel forces its mixing probability to be exponentially small, which could constrain randomized encoding and decoupling protocols.
The counterexample to the geometric conjecture suggests that masking can hide arbitrary quantum correlations, not just phase information relative to a fixed classical basis, so purely algebraic characterizations may be needed for general maskers.
Theorem 4 is directly testable: perform process tomography on a candidate masker, reconstruct the subchannels and their probabilities, compute the regularized unevenness measure, and compare it with the entropy of the safe state; a violation would refute the bound.
One could extend the black-hole consistency test to concrete evaporation models by numerically estimating $I_\infty$ from proposed internal unitaries, yielding quantitative predictions for how many qubits can be reflected at a given entanglement entropy.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper proves a genuinely new lower bound on the randomness cost of quantum masking up to 2 log d instead of the previous log d, and gives a convincing counterexample to the disk conjecture of Modi et al. The bound is notable, and the counterexample is clean. The thing to know: the result is conditional on a decomposition theorem (Fact 1) taken from the authors' previous work, and the paper does not prove it. If that theorem holds, the core here is sound.

What is actually new: the counterexample works for d≥4 and is a different family from the Ding-Hu qutrit counterexample; the measure I_infty of information unevenness is new; and Theorem 4's inequality R ≥ I_infty strengthens the old log d bound. The e=0 case of Theorem 2 is a neat coding argument and is enough for the masking application. The information conservation law route is clean and explains the log d floor.

Soft spots: the e>0 part of Theorem 2 is heuristic—the authors assert a 2^{ne} probability enhancement without formal proof. That is fine for the main result, since masking is exactly the e=0 case. More substantively, Theorem 4 inherits Fact 1: every universal masker can be written as a probabilistic mixture of isometries with orthogonal images, and the entropy of the safe state equals the randomness cost. The paper states this as known from [4] but gives no proof. If some invertible masker has Kraus operators that are not proportional to orthogonal isometries, the main inequality would not cover it. I cannot find an actual counterexample, and the examples in the paper satisfy the decomposition, but the assumption is load-bearing. A referee should push for a proof or a precise statement of the theorem in [4].

The appendix proof of Theorem 4 is dense and has a few leaps, particularly in the step from the minimization over probability distributions to the Shannon entropy of a subset. It looks fixable, but it needs more detail. The black hole discussion is speculative, as the authors admit; it is a consistency test, not a prediction.

Bottom line: this paper is for the quantum information community, especially people working on masking, secret sharing, and the randomness cost of quantum operations. It deserves serious peer review. I would send it out and ask the authors to make the dependency on Fact 1 explicit and tighten the appendix.

Referee Report

4 major / 5 minor

Summary. The paper studies the minimal amount of classical randomness needed for universal quantum masking, the encoding of an unknown quantum state into a bipartite system such that each local reduced state is input-independent. The authors define the randomness cost R(Phi_M) as the entropy of the 'safe state' in a Stinespring-type representation Phi_M(rho)=M(rho otimes sigma_S)M^dagger, and under the assumption that any universal masker can be decomposed as a random mixture sum_i p_i M_i rho M_i^dagger of bipartite isometries with orthogonal images, they prove lower bounds on R(Phi_M) in terms of how unevenly information flows to the two parties. The main result (Theorem 4) is I_infty({M_i}) <= R(Phi_M), where I_infty is a regularized measure of information unevenness; this lower bound can be as large as 2 log d, improving on the earlier log d bound. The paper also disproves a geometric conjecture on unitarily maskable states, gives a channel-mixing theorem (Theorem 2) that bounds a subchannel's capacity by its mixing probability, and discusses applications to quantum secret sharing and black-hole fast-scrambling scenarios.

Significance. If the main theorem holds, it identifies the relevant resource measure for universal masking: not the average information flow but the evenness of information distribution across subchannels. The bound I_infty <= R is a genuine improvement over the previous log d bound and is saturated by known examples such as the quantum one-time pad. The paper is commendable for stating precise entropic inequalities and for providing a concrete counterexample to the geometric conjecture of Modi et al. The main results, however, are conditional on the imported decomposition theorem (Fact 1), and several proof steps need clarification. With those points addressed, this would be a useful contribution to quantum information theory.

major comments (4)

[Introduction, Fact 1] Fact 1 is the load-bearing assumption for Theorems 3 and 4, but it is only cited, not proved. The paper assumes that every invertible universal masker can be written as Phi_M(rho)=sum_i p_i M_i rho M_i^dagger with M_i isometries having orthogonal images and with R(Phi_M)=S(sigma_S), and that this decomposition is unique up to degeneracy. If this structural theorem fails for some invertible constant-marginal channel, then Eq. (9) and the subsequent inequalities are not well defined for that channel. The authors should either provide a proof of Fact 1 in the appendix or cite a specific theorem in [4] (or [7,8]) and verify that its hypotheses match exactly.
[Appendix, proof of Theorem 2] The e>0 part of the proof of Theorem 2 is not rigorous. The statement that 'one can only have up to 2^{ne}-fold probability enhancement' is asserted without a derivation, and the displayed inequality following the negation of the assumption appears to have a sign error: as printed it reads p_i^n(1-delta)>2^{ne}2^{n(1-epsilon)CEA(N_i)}, which cannot follow from CEA(N_i)-e>-log p_i; the intended comparison is with 2^{ne}/2^{n(1-epsilon)CEA(N_i)}. Since the exact masking application uses e=0, the main theorem survives, but the claimed robustness to incomplete masking is not established.
[Main text, (Counter) Example] The example intended to disprove speculation (13) is not valid as written. The embeddings M_{A,i} map the input into |i+d>_B for i=1,...,d, which is outside a d-dimensional H_B as used elsewhere in the paper. More seriously, the reduced state on A from the M_{A,i} subchannels is (1/(d+1))sum_i Z^i rho Z^{-i}, which equals (1/(d+1)) times the diagonal of rho in the Z basis, not a constant state; the contribution from the M_j subchannels is constant, so the total A marginal depends on the input. Thus the construction does not have constant margins and is not a universal masker. This example should be corrected or removed; it does not affect the proof of Theorem 4, but it weakens the discussion of the tightness of the bound.
[Appendix, proof of Theorem 4] The derivation of the one-shot bound (25) from Eq. (27) is too terse. In particular, the argument connecting the minimizing probability distribution in Eq. (27) with the optimizing subset S in the definition of I1 needs a more explicit proof that the greedy assignment attains the minimum and that its value is bounded below by H({2^{-I_i}}_{i in S0}). Please expand this step so that the proof of the main theorem is self-contained.

minor comments (5)

[Appendix, theorem numbering] The statement labeled 'Theorem 3' in the appendix is actually the main text's Theorem 4; renumber to avoid confusion.
[Throughout] Several typos should be corrected, including 'with with' in the statement of Theorem 3 and 'hiden' and 'naively' in the introduction.
[Eq. (11)] The use of the Shannon entropy H for a subnormalized set {t_i} is unusual; the authors should emphasize that H({t_i}) is defined by the formula -sum t_i log t_i for nonnegative numbers that do not necessarily sum to one.
[Main text, (Counter) Example] The dimensions of H_A and H_B should be clarified; the notation |i+d>_B suggests that H_B has dimension 2d, which contradicts the d x d convention used elsewhere in the paper.
[References] The references [4] and [22] should be given with full publication details, as [22] currently appears only as a conference abstract.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the randomness-cost bound is derived from independent information-theoretic inequalities and a standard channel representation, not from the target quantity.

full rationale

The paper's central inequalities (Theorems 3 and 4) lower-bound R(Φ_M) by quantities built from mutual informations of the subchannels M_i. The derivation chain is: Eq. (2) represents an invertible channel via an isometry M and a safe state σ_S, cited to the external reference [6]; Theorem 2 is proved from the entanglement-assisted classical capacity achievability theorem and an erasure-channel guessing argument; Eq. (9) follows from the standard capacity formula (8); Theorem 3 follows by algebra using the pure-state identity I_i = log d + |S(A)_i − S(B)_i|; Theorem 4 follows by regularizing a one-shot entropy bound. I∞ is a function of the M_i's mutual informations and is not defined in terms of R. No parameter is fitted to the predicted quantity. The self-citation to [4] supplies context and saturation examples, and Fact 1's decomposition is a direct algebraic consequence of Eq. (2) plus spectral decomposition; the nontrivial size bound in Fact 1 is not used in the main proof. The reliance on Fact 1's full statement (including uniqueness) is an unproved structural premise from prior work, so the theorems are conditional on it, but this is an assumption, not a circular reduction. The black-hole application is speculative but not part of the derivation.

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

No free parameters or invented entities. The paper's only new definition is the evenness measure I1/I_infty, which is a function of the channel's sub-entropies and not fitted to data.

assumptions (4)

standard math Information conservation law (Lemma 1): for any isometry U: H_I -> H_A tensor H_B and maximally entangled |Gamma>_RI, 2S(R) = I(R:A) + I(R:B).
Follows directly from purity of the tripartite state and S(A)=S(B); stated without proof in the paper.
domain assumption Generalized quantum masking theorem (Fact 1): a universal quantum masker can be written as Phi_M(rho) = sum_i p_i M_i rho M_i^dagger with a safe state sigma_S and orthogonal embeddings M_i, with min{S(sigma_A), S(sigma_B), S(sigma_S)} >= log d.
Taken from prior work [4,7,8]; used as the starting point for all entropy calculations.
standard math Entanglement-assisted classical capacity formula: C_EA(N) = max_phi I(A:B)_tau for a channel N, and the corresponding achievability theorem (used in Theorem 2's proof).
Standard results [9,10] invoked in the proof of Theorem 2.
standard math No-hiding and no-masking theorems, and standard properties of von Neumann entropy (concavity, subadditivity).
Background results referenced in the introduction and used in the entropy proofs.

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Cite this review

Pith. "Pith review of Randomness cost of masking quantum information and the information conservation law." pith.science (2026). https://pith.science/paper/4EELDODJ

@misc{pith2026190807426,
  author       = {Pith},
  title        = {Pith review of: Randomness cost of masking quantum information and the information conservation law},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EELDODJ}},
  note         = {Machine review of arXiv:1908.07426}
}

read the original abstract

Masking quantum information, which is impossible without randomness as a resource, is a task that encodes quantum information into bipartite quantum state while forbidding local parties from accessing to that information. In this work, we disprove the geometric conjecture about unitarily maskable states [K. Modi et al., Phys. Rev. Lett. 120, 230501 (2018)], and make an algebraic analysis of quantum masking. First, we show a general result on quantum channel mixing that a subchannel's mixing probability should be suppressed if its classical capacity is larger than the mixed channel's capacity. This constraint combined with the well-known information conservation law, a law that does not exist in classical information theory, gives a lower bound of randomness cost of masking quantum information as a monotone decreasing function of evenness of information distribution. This result provides a consistency test for various scenarios of fast scrambling conjecture on the black hole evaporation process. The results given here are robust to incompleteness of quantum masking.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

S. H. Lie, H. Kwon, M. Kim, and H. Jeong, arXiv preprint arXiv:1903.12304 (2019)

work page Pith review arXiv 2019

[2] [1]

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work page 1949

[3] [2]

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work page 2007

[4] [3]

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work page 2018

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P. Hayden and J. Preskill, Journal of High Energy Physics 2007, 120 (2007). 7

work page 2007

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A. Nayak and P. Sen, Quantum Information & Compu- tation 7, 103 (2007)

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H. Imai, J. M¨ uller-Quade, A. C. Nascimento, P. Tuyls, and A. Winter, Quantum Information & Computation 5, 69 (2005)

work page 2005