Weight of informativeness, state exclusion games and excludible information

Andr\'es F. Ducuara; Paul Skrzypczyk

Weight of informativeness, state exclusion games and excludible information

T0 review · 0 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The weight of informativeness of a measurement equals its optimal advantage in state exclusion games.

desk verdict This paper proves the weight-exclusion dual of the robustness-discrimination result and is sound enough to referee. read the letter →

arxiv 1908.10347 v1 pith:VBTNZW7X submitted 2019-08-27 quant-ph

Andrés F. Ducuara , Paul Skrzypczyk This is my paper · ORCID

classification quant-ph

keywords weightofinformativenessstateexclusiongamesexcludibleinformationmeasurementsimulabilityquantumresourcetheorysingle-shotPOVMconvexoptimisation

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 gives an operational meaning to a geometric resource quantifier called the weight of informativeness of a quantum measurement: the minimal weight with which a general measurement must be mixed with an uninformative one to reproduce it. It proves that this number exactly equals the best possible advantage the measurement can provide over purely classical guessing in a state exclusion game, where the player must name a state that was not sent. The paper also introduces a single-shot information quantity, excludible information, and shows that for the quantum-to-classical channel associated to a measurement it is determined by the same weight. This creates a three-way correspondence between a resource quantifier, a game, and an information-theoretic quantity, and it shows that exclusion games form a complete set of monotones for measurement simulation.

What carries the argument

The central object is the weight of informativeness, $\mathrm{WoI}(M)=\min\{w : M_a = wN_a + (1-w)q(a)\mathbb{1}\}$ with $N$ a POVM and $q$ a probability distribution, which has the closed form $\mathrm{WoI}(M)=1-\sum_a \lambda_{\min}(M_a)$. Its dual semidefinite programme yields the extremal ensemble that saturates the bound in Result 1. The second engine is the quantum-to-classical channel $\Lambda_M$ and the conditional exclusion entropy $H_{-\infty}(X|G)$, which is manipulated into $-\log P^{\mathrm{Q}}_{\mathrm{err}}(\mathcal{E}, M)$ by optimising over classical post-processings. The simulability order $N_x=\sum_a p(x|a)M_a$ connects the game's minimisation to the resource order, and minimax reasoning is used to prove the completeness result.

What would settle it

Take a POVM $M$, compute $\mathrm{WoI}(M)=1-\sum_a \lambda_{\min}(M_a)$, and obtain the optimal ensemble from the dual SDP in Appendix B; if the ratio $P^{\mathrm{Q}}_{\mathrm{err}}(\mathcal{E}^*,M)/P^{\mathrm{C}}_{\mathrm{err}}(\mathcal{E}^*)$ is not exactly $1-\mathrm{WoI}(M)$, Result 1 is false. Similarly, one can compute the excludible information of $\Lambda_M$ directly from the minimisation in Appendix C and compare it with $-\log[1-\mathrm{WoI}(M)]$.

Watch

Extended reading notes

Core claim

Result 1 is the exact identity $\min_{\mathcal{E}} P^{\mathrm{Q}}_{\mathrm{err}}(\mathcal{E}, M)/P^{\mathrm{C}}_{\mathrm{err}}(\mathcal{E}) = 1 - \mathrm{WoI}(M)$, where the left side is the smallest ratio, over all ensembles, between the best exclusion-error probability achievable using the measurement $M$ and the best classical error probability, and the right side is a number computed from $M$ alone. Result 2 adds that the single-shot excludible information of the quantum-to-classical channel $\Lambda_M(\rho)=\sum_a |a\rangle\langle a| \operatorname{Tr}(M_a\rho)$ equals $-\log[1-\mathrm{WoI}(M)]$. Result 3 states that $M$ can simulate $N$ if and only if $P^{\mathrm{Q}}_{\mathrm{err}}(\mathcal{E}, M) \le P^{\mathrm{Q}}_{\mathrm{err}}(\mathcal{E}, N)$ for every ensemble $\mathcal{E}$. The paper's claim is that these form one coherent correspondence: a weight-based resource measure, an exclusion task, and a single-shot information quantity are the same object from three viewpoints.

Load-bearing premise

The proof assumes that the best a player can do with a fixed measurement in an exclusion game is to classically process its outcomes, and that the relevant optimisation problems can be swapped in the standard convex way; if some strategy outside this description performed better, the exact equality with the weight of informativeness would fail.

Editorial extensions

If this is right

For any measurement and any state exclusion game, the measurement can reduce the classical error probability by at most a factor $1-\mathrm{WoI}(M)$, and there is a game where this bound is attained.
The single-shot excludible information of the channel induced by a measurement is exactly $-\log[1-\mathrm{WoI}(M)]$, so a purely geometric quantifier computes a communication-theoretic quantity for exclusion tasks.
A measurement $M$ can simulate a measurement $N$ exactly when $M$ is never worse than $N$ in any state exclusion game, giving a second complete set of monotones for measurement simulation.
The weight of informativeness can be evaluated efficiently: it is the solution of an SDP and has the explicit eigenvalue formula $1-\sum_a \lambda_{\min}(M_a)$.

Reading between the lines

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

The authors conjecture that the weight-exclusion correspondence is generic across quantum resource theories; if that holds, weight-based quantifiers in other resource theories would acquire operational interpretations as optimal advantages in exclusion tasks rather than remaining purely geometric measures.
Because $\mathrm{WoI}$ has a closed form, Result 1 turns an optimisation over all ensembles into an eigenvalue computation, which could make exclusion advantages easy to estimate for large or noisy POVMs.
The if-and-only-if in Result 3 implies that any failure of simulation is witnessed by some exclusion game, but the proof does not identify a finite set of games that would certify simulation; finding such a finite witnessing set would turn the criterion into a practical test.

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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

Dear colleague,

You should know that this is the dual result to Skrzypczyk-Linden: instead of robustness and discrimination, it is weight and exclusion. The main result, Eq. (5), says the optimal state-exclusion advantage of a measurement over classical strategies is exactly 1−WoI(M). That is a real theorem, proven with SDP duality, and it gives weight-based quantifiers the operational meaning they were missing. The paper also introduces excludible information and shows it equals −log(1−WoI) for the c-q channel, making a genuine three-way correspondence.

What is good: the proofs are self-contained and largely rigorous. I checked the lower bound via the weight decomposition and the upper bound via the dual SDP; both go through. The complete-set-of-monotones result (Result 3) is a nice bonus, and the minimax step is legitimate once you fix the outcome-count issue. No fitted parameters, no circularity. The parallel to [23] is appropriately acknowledged; this is not a re-derivation.

The soft spots are all minor. The monotonicity lemma has a typo (WoI(N) ≤ WoI(N)). Appendix C assumes, without stating it, that for c-q channels the optimal decoding is the computational basis followed by classical post-processing; the stress-test note is right that this is valid because only diagonal entries matter, but it should be written as a lemma. Appendix D does not explain how outcome counts are matched when simulating M′; fixable by padding or coarse-graining, as you noted. None of these affect the main claims.

The generic conjecture in Section VII is clearly labeled as future work; it does not detract from what is proven. This is not a branch-reshaping paper, but it is exactly the kind of clean, useful result that belongs in the literature.

I would send it to a serious referee. With a careful revision touching the appendices, it should be accepted.

Best, [You]

Referee Report

0 major / 3 minor

Summary. The paper introduces the weight of informativeness (WoI) as a resource quantifier for the convex quantum resource theory of measurement informativeness, proves its basic properties, and establishes two main operational results. First, for any measurement M, the optimal advantage over classical strategies in state exclusion games, quantified by the ratio of quantum to classical error probabilities minimized over all ensembles, equals 1 - WoI(M). Second, the single-shot excludible information of the quantum-to-classical channel associated with M is -log[1 - WoI(M)], extending the known robustness-based three-way correspondence to weight-based quantifiers and exclusion tasks. The paper also proves that error probabilities in state exclusion games form a complete set of monotones for measurement simulation, giving a necessary and sufficient condition for M to simulate N in terms of these probabilities. The central proofs are analytic, using an explicit dual SDP for the WoI, strong duality, and a minimax argument.

Significance. If the results hold, the paper makes a significant conceptual contribution to quantum resource theories: it provides the first operational interpretation of a weight-based quantifier, establishes a parallel three-way correspondence between resource quantifiers, operational tasks, and single-shot information-theoretic quantities, and identifies a second complete set of monotones for measurement simulation. The proofs are mathematically sound and self-contained: the dual SDP in Appendix B is explicit, the reduction in Appendix C is valid for quantum-to-classical channels, and the monotonicity proof in Appendix D uses a correct minimax step. There is no circularity: the state exclusion game and excludible information are defined independently of WoI, and the equalities are proven rather than assumed. The conjecture that the weight-exclusion correspondence holds for arbitrary resource theories is clearly stated and appropriately supported by a forthcoming companion result.

minor comments (3)

[Section II, Lemma (iii)] The monotonicity statement in the main text reads "N≼ M→ WoI(N)≤ WoI(N)", which is a typo: the second WoI should be evaluated on M. Appendix A states the correct inequality WoI(M') ≤ WoI(M). Please correct the main-text statement.
[Appendix C, proof of Result 2] The proof silently restricts the decoding POVM to the computational basis by setting D_g = |g><g|, then optimizes over classical post-processings p(x|g). This reduction is valid for a quantum-to-classical channel because any decoding POVM D_g enters only through its diagonal entries <a|D_g|a>, which form a conditional probability distribution, but this justification is not stated. Since Eq. (C6) and hence Eq. (8) rely on this reduction, please add a brief explanation. Also, the Kronecker delta immediately after "Choosing D_g = |g><g|" should read δ_{a,g}, not δ_a^x.
[Appendix D, sufficiency proof] In the sufficiency proof, the argument treats M' as a k-outcome measurement in Eq. (D3), but M' has l' outcomes. The assumption (9) holds for all ensembles, so one can take k = l' (or pad/coarse-grain accordingly) to make the comparison well-defined. The text should state this explicitly so that the subsequent minimax step is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: WoI, exclusion games, and excludible information are defined independently, and the equalities are proven via SDP duality and minimax rather than assumed.

full rationale

Result 1 is proven directly from Definitions 2 and 3: the lower bound uses the defining decomposition of WoI, and the upper bound uses the dual SDP of the explicit form WoI = 1 - sum_a lambda_min(M_a), constructing the adversarial ensemble from the dual variables. Neither direction assumes the equality being proved; the state exclusion game and the weight are defined independently. Result 2 follows from Result 1 after reducing the decoding optimization for the c-q channel Lambda_M to post-processing of M, which is valid because any decoding POVM on the classical output only enters through its diagonal entries, which form a conditional probability distribution; no fitted parameter is renamed as a prediction. Result 3 is a direct minimax argument over the simulability polytope, with no importation of a uniqueness theorem. The only author-overlap citation is [23] (Skrzypczyk-Linden), used for context, notation, and the parallel robustness/discrimination/accessible-information correspondence; none of the proofs in this work invoke [23] as a premise. The unstated decoding reduction in Appendix C and the outcome-count handling in Appendix D are presentation gaps, not circular steps. I therefore find no significant circularity.

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

The central results rest on standard convex resource theory, SDP duality, and the c-q channel decoding reduction. No free parameters are fitted; WoI is defined as an optimization and then related to game values by proof. No new physical entities are introduced.

assumptions (4)

domain assumption Uninformative measurements {q(a)1} form the free set, and classical strategies in the exclusion game are exactly uninformative measurements followed by classical post-processing.
Section II Definition 1 and Section III define the baseline; the value P_C_err(E)=min_x p(x) depends on this choice.
standard math Strong duality holds for the SDP (A14) and its dual (B6).
Appendix B uses the dual solution to construct the optimal exclusion ensemble; the SDP is convex with strictly feasible points, but this is not checked in the text.
domain assumption For quantum-to-classical channels, arbitrary decoding POVMs can be reduced to computational-basis measurement plus classical post-processing.
Appendix C uses this implicitly when setting D_g=|g><g| and then optimizing over p(x|g).
standard math The minimax theorem applies to the exchange in Eq. (D6).
Appendix D's sufficiency proof requires max_E min_N = min_N max_E; compactness and continuity conditions are not discussed.

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Cite this review

Pith. "Pith review of Weight of informativeness, state exclusion games and excludible information." pith.science (2026). https://pith.science/paper/VBTNZW7X

@misc{pith2026190810347,
  author       = {Pith},
  title        = {Pith review of: Weight of informativeness, state exclusion games and excludible information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBTNZW7X}},
  note         = {Machine review of arXiv:1908.10347}
}

read the original abstract

We consider the quantum resource theory of measurement informativeness and introduce a weight-based quantifier of informativeness. We show that this quantifier has operational significance from the perspective of quantum state exclusion, by showing that it precisely captures the advantage a measurement provides in minimising the error in this game. We furthermore introduce information theoretic quantities related to exclusion, in particular the notion of excludible information of a quantum channel, and show that for the case of quantum-to-classical channels it is determined precisely by the weight of informativeness. This establishes a three-way correspondence which sits in parallel to the recently discovered correspondence in quantum resource theories between robustness-based quantifiers, discrimination games, and accessible information. We conjecture that the new correspondence between a weight-based quantifier and an exclusion-based task found in this work is a generic correspondence that holds in the context of quantum resource theories.

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A. F. Ducuara and P. Skrzypczyk, “In preparation,” (2019). APPENDICES Appendix A: Lemma Lemma: (Properties of WoI) The weight of informa- tiveness (2) satisﬁes the following properties: (i) Faithfulness: WoI(M) = 0↔ M ={Ma =q(a)1}. (A1) (ii) Convexity: given two measurements M...

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We now consider the quantities for p∈ [0, 1]: pM1a + (1−p)M2a = =p [ (1−w∗ 1)q∗ 1(a)1 +w∗ 1N∗ 1a ] + + (1−p) [ (1−w∗ 2)q∗ 2(a)1 +w∗ 2N∗ 2a ]

satisfying: M1a = (1−w∗ 1)q∗ 1(a)1 +w∗ 1N∗ 1a, M2a = (1−w∗ 2)q∗ 2(a)1 +w∗ 2N∗ 2a. We now consider the quantities for p∈ [0, 1]: pM1a + (1−p)M2a = =p [ (1−w∗ 1)q∗ 1(a)1 +w∗ 1N∗ 1a ] + + (1−p) [ (1−w∗ 2)q∗ 2(a)1 +w∗ 2N∗ 2a ] . (A7) We now deﬁne the variables: ˜w =pw∗ 1 + (1−p)w∗...
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(B1) Let us start with the weight of informativeness of a mea- surement as given by (2)

First part In this ﬁrst part we prove that: [1− WoI(M)]P C err(E)≤P Q err(E, M), ∀E, M. (B1) Let us start with the weight of informativeness of a mea- surement as given by (2). Consider that the minimum is achieved with the triple (q∗, N∗,w∗) so that∀a: Ma− (1−w∗)q∗(a)1 =w∗Na≥...
[59]

(B3) 8 This will be done by considering the dual formulation of the primal SDP for the weight of informativeness [45]

Second part In this second part we prove that ∀M,∃E M such that: [1− WoI(M)]P C err ( E M) ≥P Q err(E M, M), ∀M. (B3) 8 This will be done by considering the dual formulation of the primal SDP for the weight of informativeness [45]. a. Deriving the dual SDP We start by addressi...
[60]

Necessary condition Let us address the necessary condition: M≽ M′ =⇒ P Q err(E, M)≤P Q err(E, M′) ∀E. (D1) Let us consider the probability of error in state exclusion: P Q err(E, M′) = min M′≽N′ k∑ x=1 Tr (N′ x ˜ρx), = min {p(x|b)} k∑ x=1 Tr l′ ∑ b=1 p(x|b)M′ b ˜ρx, = min {p(x...
[61]

(D2) 10 Let us start by assuming that the right-hand side is true

Suﬃcient condition We now address the suﬃcient condition: M≽ M′ ⇐= P Q err(E, M)≤P Q err(E, M′) ∀E. (D2) 10 Let us start by assuming that the right-hand side is true. We now want to prove that M≽ M′ which is equivalent to∑ aq(x|a)Ma = M′ x. Let us continue by considering the i...
[2016]

acknowledges support from a Royal Society URF (UHQT)

P.S. acknowledges support from a Royal Society URF (UHQT)

Pith tools

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

[1] [23]

Skrzypczyk and N

P. Skrzypczyk and N. Linden, Phys. Rev. Lett. 122, 140403 (2019)

2019

[2] [1]

Chitambar and G

E. Chitambar and G. Gour, Rev. Mod. Phys. 91, 025001 (2019)

2019

[3] [2]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996)

1996

[4] [3]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu- macher, Phys. Rev. A 53, 2046 (1996)

1996

[5] [4]

Oszmaniec, L

M. Oszmaniec, L. Guerini, P. Wittek, and A. Ac´ ın, Phys. Rev. Lett. 119, 190501 (2017)

work page 2017

[6] [5]

Oszmaniec and T

M. Oszmaniec and T. Biswas, Quantum 3, 133 (2019)

2019

[7] [6]

A resource theory of quantum measure- ments,

T. Guﬀ, N. A. McMahon, Y. R. Sanders, and A. Gilchrist, “A resource theory of quantum measure- ments,” (2019), arXiv:1902.08490

arXiv 2019

[8] [7]

J. I. de Vicente, Journal of Physics A: Mathematical and Theoretical 47, 424017 (2014)

2014