Pith. sign in

REVIEW 3 major objections 3 minor 33 references

Optimal value-readout probability in quantum oracles equals a normalized Rényi-1/2 Fourier support

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 05:54 UTC pith:F2P3UOBR

load-bearing objection A correct and clean but largely non-novel reformulation of known SRM results in oracle language, with a fixable gap in the covariance reduction and a harmless-but-sloppy orthogonality typo. the 3 major comments →

arxiv 2607.13198 v1 pith:F2P3UOBR submitted 2026-07-14 quant-ph math-phmath.MP

An Information-Theoretic Characterization of Optimal Value-Readout in Response-Register Quantum Oracles

Milad Ghadimi , Hesam Soltanpanahi , Vahid Salari This is my paper
classification quant-ph math-phmath.MP MSC 81P4581P50 PACS 03.65.Ta03.67.Lx
keywords response-register oraclevalue readoutphase kickbackRényi entropyeffective Fourier supportcovariant POVMquantum state discriminationphase-value complementarity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves that, for response-register quantum oracles with a finite Abelian response group, the optimal single-query probability of reading out the oracle value is exactly the normalized Rényi-1/2 effective Fourier support of the response state. In other words, the best possible value-readout performance is completely determined by the Fourier-weight distribution of the response state, and not by the phases of its character expansion. This closes a quantitative gap between the two complementary interpretations of such oracles: value readout and phase kickback. The result yields a tight phase–value complementarity tradeoff, with an explicit family of response states that saturate it, and gives the Rényi-1/2 entropy an operational meaning in the oracle setting.

Core claim

The central claim is Theorem 1: for a finite Abelian response group of size d, given a response state |η⟩ with Fourier-weight distribution p over characters, the optimal probability Pval of correctly identifying the oracle value from a single query is Pval = 2^{H_{1/2}(p)}/d = (1/d)(Σ_χ √p_χ)². This holds regardless of the phases θ_χ in the character expansion. The proof works by recognizing value readout as a minimum-error discrimination problem over the geometrically uniform orbit ensemble {T_a|η⟩}, showing the Gram matrix is diagonalized by characters with eigenvalues d p_χ, then bounding and saturating the success probability via a covariant POVM. The quantity 2^{H_{1/2}(p)} acts as an e

What carries the argument

The central object is the Fourier-weight distribution p of the response state, together with the Rényi-1/2 effective Fourier support S_eff = 2^{H_{1/2}(p)} = (Σ_χ √p_χ)². The argument relies on the Gram matrix of the translated orbit being diagonalized by characters (with eigenvalues d p_χ), reducing optimal covariant discrimination to a Cauchy–Schwarz bound on the seed of a covariant POVM. The explicit saturating measurement is the square-root (pretty-good) measurement with seed (1/d)|μ⟩⟨μ|, where |μ⟩ has phases matching the response state. The tight phase–value complementarity follows from maximizing S_eff under a fixed target-character weight, again via Cauchy–Schwarz.

Load-bearing premise

The proof assumes that the optimal measurement for the uniform orbit ensemble can be taken covariant (of the form M_a = T_a M_0 T_a†) without a proof of that reduction.

What would settle it

Find a finite Abelian group A and a response state for which a non-covariant POVM achieves success probability strictly larger than (1/d)(Σ_χ √p_χ)². If such a measurement exists, the claimed exact optimality would fail; conversely, numerical search over POVMs for small d (e.g., d=2,3,4) confirming the bound would support the theorem.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The optimal single-query value-readout probability is now a closed-form function of the response state's Fourier-weight distribution, enabling direct comparisons of oracle performance across response states without numerical optimization.
  • For Grover-type oracles (A = Z₂²), the tradeoff shows that even a 5% phase infidelity can cut the achievable readout probability roughly in half (from 1 to about 0.464 for d=4), giving a concrete design constraint for oracle implementations.
  • The paper provides a complete achievable region: for any target phase fidelity Φ, the maximal Pval is attained by spreading the remaining Fourier weight uniformly over non-target characters, so any protocol aiming for intermediate performance should use that saturating family.
  • The result gives the Rényi-1/2 entropy a direct operational interpretation in the oracle setting, connecting information-theoretic measures of Fourier delocalization to a concrete quantum information-processing task.
  • The framework extends the known optimality of the square-root measurement for Abelian geometrically uniform ensembles to a form tailored to response-register oracles, making the dependency on Fourier weights explicit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the optimal Pval depends only on the Fourier weights and not on phases, a designer can choose response-state phases freely (e.g., to satisfy other constraints like computational-basis encoding) without sacrificing optimal value-readout performance – an observation the paper states implicitly but does not exploit.
  • The phase–value tradeoff suggests a resource-theoretic reading: S_eff = d·Pval behaves like a coherence measure under translations, and the monotonicity under majorization (Proposition B.1) implies that any mixing of Fourier distributions only increases value-readout capability, which could be connected to coherence under group actions.
  • The non-Abelian extension remains open; one could test the conjectured matrix-valued analogue by computing optimal discrimination probabilities for small non-Abelian groups and checking whether a matrix-valued Rényi-type expression emerges.
  • In multi-query settings, a natural further step would be to ask whether the single-query bound composes (e.g., whether Pval for k queries is governed by a k-fold convolution of the Fourier distribution), which is not addressed in the paper.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies response-register quantum oracles for finite Abelian response groups. For a pure response state with Fourier-weight distribution p, it claims (Theorem 1) that the optimal single-query probability of correctly identifying the oracle value, under a uniform prior over the translation orbit, is Pval = 2^{H_{1/2}(p)}/d = (1/d)(\sum_\chi \sqrt{p_\chi})^2. The proof proceeds through the orbit Gram matrix, a covariant-POVM upper bound, and an explicit rank-one attainment POVM, and it is followed by a phase-value complementarity corollary obtained by maximizing the Rényi-1/2 effective support at fixed target-character weight. The paper also provides explicit saturating response states and interprets the resulting quantity as an effective Fourier support.

Significance. If the proof is completed, the result is a clean and useful exact characterization: it gives the optimal value-readout probability an operational interpretation as the normalized Rényi-1/2 effective Fourier support, and it yields a tight phase-value tradeoff with explicit saturating states. The derivation is parameter-free and self-contained given standard Fourier analysis and known SRM optimality results; the paper honestly acknowledges that it does not prove a new optimality theorem. The main contribution is the oracle-value framing and the entropy/complementarity packaging of known geometrically-uniform-state discrimination results. The claim is plausible and the mathematical core is essentially correct, but two load-bearing proof gaps must be fixed.

major comments (3)
  1. [Sec. IV.C, Eq. (26)] The proof restricts to covariant POVMs M_a = T_a M_0 T_a^† with the sentence 'we may restrict...' but gives no proof or citation. Since the upper bound (33) and hence the tightness claim are derived only within this class, Theorem 1's optimality is not established as written. The reduction is in fact valid -- averaging any POVM {M_a} as \tilde{M}_b = (1/d)\sum_h T_h M_{b-h} T_h^† preserves the success probability and yields a covariant POVM -- but this argument must appear in the paper, or the authors should invoke a specific theorem from Refs. [12,15,17] that already proves optimality of SRM/covariant measurements for Abelian geometrically uniform ensembles.
  2. [Eq. (28), Eq. (38), and Appendix A, Eq. (A7)] The orthogonality relation is missing complex conjugation. For finite Abelian group characters, \sum_a \chi(a)\overline{\psi(a)} = d\delta_{\chi\psi}, not \sum_a \chi(a)\psi(a) = d\delta_{\chi\psi}. As written, Eq. (28) does not imply the diagonal condition m_{\chi\chi}=1/d, and Eq. (38) would not sum to I for the proposed attainment POVM. With the conjugate inserted, both steps go through; the same correction is needed in Appendix A.
  3. [Sec. II.B and Sec. IV.C] The paper states in Sec. II.B that SRM optimality for Abelian geometrically uniform ensembles is well established and that no new optimality theorem is provided, yet Sec. IV.C presents a self-contained derivation. The status is currently inconsistent: if Theorem 1 is meant to follow from the cited literature, the authors should state the precise theorem and verify its hypotheses; if it is meant to be a self-contained proof, the covariant-reduction gap in Eq. (26) must be filled. Please make this logical status explicit.
minor comments (3)
  1. [Abstract and Introduction] The 'optimal single-query value-readout probability' should be explicitly qualified as being for the uniform-prior orbit ensemble defined in Eq. (3); otherwise the abstract could be read as applying to arbitrary prior distributions over oracle values.
  2. [Eq. (18)] The sentence 'leaving only the Fourier weights p_\chi remain' is ungrammatical; it should be 'leaving only the Fourier weights p_\chi.'
  3. [Eq. (33)] There is a stray comma in the displayed expression after '\sqrt{p_\chi p_\psi}'; it should read '\sqrt{p_\chi p_\psi} e^{-i\theta_\chi} e^{i\theta_\psi} m_{\chi\psi}.'

Circularity Check

0 steps flagged

No significant circularity: the central theorem is derived from standard Fourier analysis and a direct Cauchy–Schwarz optimization; the proof has fixable gaps but no step reduces to its inputs by definition.

full rationale

The paper's claimed derivation of Theorem 1 is self-contained in the relevant sense: no step renames a fitted parameter as a prediction, no central premise is justified solely by the authors' own prior work, and the main formula (Eq. 4) is obtained by an explicit Cauchy–Schwarz maximization over a covariant POVM class followed by an explicit achievement construction (Eqs. 34–40). The optimality of the square-root measurement is invoked from external, well-established literature ([12,15,17,18]) and the paper explicitly disclaims providing a new optimality theorem, so this is independent support rather than a self-citation chain. The phase–value tradeoff (Corollary 1.1) is a direct optimization of the Rényi-1/2 quantity over probability distributions with a fixed target weight, again not circular. Two issues are worth flagging but they are mathematical gaps, not circularity: (i) Sec. IV.C asserts “we may restrict to covariant positive operator-valued measures (POVM)s of the form M_a = T_a M_0 T_a†” without proof; this reduction is in fact valid (twirling preserves the success probability) but the proof is absent, so the bound in Eq. (33) is strictly speaking derived only for the covariant class. (ii) Eq. (28) and Appendix A use Σ_a χ(a)ψ(a) = dδ_{χ,ψ} without the complex conjugate; the correct relation is Σ_a χ(a)\overline{ψ(a)} = dδ_{χ,ψ}. Both are repairable and do not make the theorem's content equivalent to its assumptions. The uniform-prior orbit ensemble (Eq. 3) is an explicit modeling choice. Hence no circularity; score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No fitted parameters and no new physical entities. The proof relies on standard Fourier analysis and standard measurement-theoretic facts; the effective Fourier support is just a named rewriting of an entropy expression. The main implicit assumptions are the uniform-prior model, the covariance reduction, and ideal translations.

axioms (4)
  • standard math Finite Abelian group Fourier analysis: characters form an orthonormal basis and satisfy ∑_a χ(a) ψ(a)* = d δ_{χψ}.
    Used in Eqs. (18), (28), and (A7) to diagonalize the translation representation and the Gram matrix.
  • domain assumption The optimal POVM for a uniform group-covariant pure-state ensemble can be chosen covariant of the form M_a = T_a M_0 T_a†.
    Invoked in Sec. IV.C ('we may restrict to covariant positive operator-valued measures') without proof; standard via symmetrization but not stated as an assumption.
  • domain assumption Oracle values are modeled as uniformly distributed group elements, so success probability is the average over the uniform orbit ensemble (Eq. 3).
    Defines what 'optimal single-query value-readout probability' means; nonuniform priors or worst-case inputs would change the quantity.
  • domain assumption The response register is a d-dimensional Hilbert space carrying the regular representation of the finite Abelian group A, with ideal noiseless translations.
    The oracle model in Sec. II A and Fig. 1; error/noise models are not considered.

pith-pipeline@v1.3.0-alltime-deepseek · 12486 in / 20056 out tokens · 176461 ms · 2026-08-02T05:54:59.904133+00:00 · methodology

0 comments
read the original abstract

Response-register quantum oracles admit two complementary operational interpretations: value readout and phase kickback. Although their computational equivalence is well understood, the quantitative relation between these two operational interpretations has lacked an exact characterization. We prove that, for finite Abelian response groups, the optimal single-query value-readout probability is exactly the normalized R\'enyi-1/2 effective Fourier support of the response state. This identity provides an exact information-theoretic characterization of optimal value-readout capability, gives the R\'enyi-1/2 effective Fourier support a direct operational interpretation in the oracle setting, and yields a tight phase--value complementarity theorem together with an explicit family of saturating response states.

Figures

Figures reproduced from arXiv: 2607.13198 by Hesam Soltanpanahi, Milad Ghadimi, Vahid Salari.

Figure 1
Figure 1. Figure 1: FIG. 1: Response-register oracle and its two operational interpretations. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Tight phase–value tradeoff, Eq. (6), for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Effective Fourier-support landscape for a [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Phase–value capability map for a [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 7 canonical work pages

  1. [1]

    Rapid solution of problems by quantum computation.Proceedings of the Royal Society of London

    David Deutsch and Richard Jozsa. Rapid solution of problems by quantum computation.Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907):553–558, 1992

    1907

  2. [2]

    Quantum com- plexity theory

    Ethan Bernstein and Umesh Vazirani. Quantum com- plexity theory. InProceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 11–20, 1993

    1993

  3. [3]

    A fast quantum mechanical algorithm for database search

    Lov K Grover. A fast quantum mechanical algorithm for database search. InProceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219, 1996

    1996

  4. [4]

    Quantum amplitude amplification and estimation

    Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. arXiv preprint quant-ph/0005055, 2000

    Pith/arXiv arXiv 2000

  5. [5]

    Quantum algorithms revis- ited.Proc

    Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca. Quantum algorithms revis- ited.Proc. Roy. Soc. Lond. A, 454:339, 1998. doi: 10.1098/rspa.1998.0164

    arXiv 1998

  6. [6]

    Cambridge university press, 2010

    Michael A Nielsen and Isaac L Chuang.Quantum compu- tation and quantum information. Cambridge university press, 2010

    2010

  7. [7]

    A. Yu. Kitaev. Quantum measurements and the Abelian stabilizer problem. 11 1995

    1995

  8. [8]

    Generalized deutsch- 10 jozsa algorithm for applications in data classification, lo- gistic regression, and quantum key distribution.Physical Review A, 113(1):012609, 2026

    M Ghadimi, V Salari, S Bakrani, M Zomorodi, N Gohari- Kamel, S Moradi, and D Oblak. Generalized deutsch- 10 jozsa algorithm for applications in data classification, lo- gistic regression, and quantum key distribution.Physical Review A, 113(1):012609, 2026

    2026

  9. [9]

    Lower Bounds on Quan- tum Query Complexity

    Peter Hoyer and Robert Spalek. Lower Bounds on Quan- tum Query Complexity. 9 2005

    2005

  10. [10]

    Dan Boneh and Richard J. Lipton. Quantum crypt- analysis of hidden linear functions (extended abstract). In Don Coppersmith, editor,Advances in Cryptology — CRYPTO ’95, volume 963 ofLecture Notes in Com- puter Science, pages 424–437. Springer-Verlag, 1995. doi: 10.1007/3-540-44750-4˙34

    work page doi:10.1007/3-540-44750-4 1995

  11. [11]

    On the power of quantum computation

    Daniel R Simon. On the power of quantum computation. SIAM journal on computing, 26(5):1474–1483, 1997

    1997

  12. [12]

    Eldar and G

    Yonina C. Eldar and G. David Forney. On quan- tum detection and the square-root measurement.IEEE Trans. Info. Theor., 47(3):858–872, 2001. doi: 10.1109/18.915636

    work page doi:10.1109/18.915636 2001

  13. [13]

    Statistical decision theory for quan- tum systems.Journal of multivariate analysis, 3(4):337– 394, 1973

    Alexander S Holevo. Statistical decision theory for quan- tum systems.Journal of multivariate analysis, 3(4):337– 394, 1973

    1973

  14. [14]

    H. Yuen, R. Kennedy, and M. Lax. Optimum test- ing of multiple hypotheses in quantum detection the- ory.IEEE Trans. Info. Theor., 21(2):125–134, 1975. doi: 10.1109/TIT.1975.1055351

    arXiv 1975

  15. [15]

    Optimum measurements for discrimination among symmetric quantum states and parameter estima- tion.International Journal of Theoretical Physics, 36(6): 1269–1288, 1997

    Masashi Ban, Keiko Kurokawa, Rei Momose, and Os- amu Hirota. Optimum measurements for discrimination among symmetric quantum states and parameter estima- tion.International Journal of Theoretical Physics, 36(6): 1269–1288, 1997

    1997

  16. [16]

    Wootters

    Paul Hausladen and William K. Wootters. A ‘pretty good’ measurement for distinguishing quantum states. Journal of Modern Optics, 41(12):2385–2390, 1994. doi: 10.1080/09500349414552221

    doi:10.1080/09500349414552221 1994

  17. [17]

    da Silva

    Hari Krovi, Saikat Guha, Zachary Dutton, and Mar- cus P. da Silva. Optimal Measurements for Symmet- ric Quantum States with Applications to Optical Com- munication.Phys. Rev. A, 92:062333, 2015. doi: 10.1103/PhysRevA.92.062333

    work page doi:10.1103/physreva.92.062333 2015

  18. [18]

    On the distinguishability of geometrically uni- form quantum states.J

    Juntai Zhou, Stefano Chessa, Eric Chitambar, and Felix Leditzky. On the distinguishability of geometrically uni- form quantum states.J. Phys. A, 58(41):415303, 2025. doi:10.1088/1751-8121/ae0a95

    work page doi:10.1088/1751-8121/ae0a95 2025

  19. [19]

    Bromley, Marco Cian- ciaruso, Marco Piani, Nathaniel Johnston, and Ger- ardo Adesso

    Carmine Napoli, Thomas R. Bromley, Marco Cian- ciaruso, Marco Piani, Nathaniel Johnston, and Ger- ardo Adesso. Robustness of Coherence: An Oper- ational and Observable Measure of Quantum Coher- ence.Phys. Rev. Lett., 116(15):150502, 2016. doi: 10.1103/PhysRevLett.116.150502

    doi:10.1103/physrevlett.116.150502 2016

  20. [20]

    Brom- ley, Carmine Napoli, Nathaniel Johnston, and Gerardo Adesso

    Marco Piani, Marco Cianciaruso, Thomas R. Brom- ley, Carmine Napoli, Nathaniel Johnston, and Gerardo Adesso. Robustness of asymmetry and coherence of quan- tum states.Phys. Rev. A, 93(4):042107, 2016. doi: 10.1103/PhysRevA.93.042107

    doi:10.1103/physreva.93.042107 2016

  21. [21]

    Quantum query complexity of symmetric oracle problems.Quan- tum, 5:403, 2021

    Daniel Copeland and Jamie Pommersheim. Quantum query complexity of symmetric oracle problems.Quan- tum, 5:403, 2021. doi:10.22331/q-2021-03-07-403

    doi:10.22331/q-2021-03-07-403 2021

  22. [22]

    Bergou, and Mark Hillery

    Emilio Bagan, John Calsamiglia, Janos A. Bergou, and Mark Hillery. A generalized wave-particle duality relation for finite groups. 3 2018. doi:10.1088/1751-8121/aabb21

    work page doi:10.1088/1751-8121/aabb21 2018

  23. [23]

    Fringe Visibility and Which- Way Information: An Inequality.Phys

    Berthold-Georg Englert. Fringe Visibility and Which- Way Information: An Inequality.Phys. Rev. Lett., 77: 2154–2157, 1996. doi:10.1103/PhysRevLett.77.2154

    doi:10.1103/physrevlett.77.2154 1996

  24. [24]

    Two interferometric complementarities.Phys

    Gregg Jaeger, Abner Shimony, and Lev Vaidman. Two interferometric complementarities.Phys. Rev. A, 51:54– 67, Jan 1995. doi:10.1103/PhysRevA.51.54

    work page doi:10.1103/physreva.51.54 1995

  25. [25]

    Springer-Verlag, New York, 1977

    Jean-Pierre Serre.Linear Representations of Finite Groups, volume 42 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1977. ISBN 978-0-387-90190- 9

    1977

  26. [26]

    London Mathematical Society Student Texts

    Audrey Terras.Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts. Cambridge University Press, 1999

    1999

  27. [27]

    Holevo.Probabilistic and Statistical Aspects of Quantum Theory

    A.S. Holevo.Probabilistic and Statistical Aspects of Quantum Theory. Publications of the Scuola Nor- male Superiore. Scuola Normale Superiore, 2011. ISBN 9788876423789

    2011

  28. [28]

    Holevo.Statistical Structure of Quantum Theory, volume 67 ofLecture Notes in Physics Mono- graphs

    Alexander S. Holevo.Statistical Structure of Quantum Theory, volume 67 ofLecture Notes in Physics Mono- graphs. Springer Berlin Heidelberg, 2001. ISBN 978-3- 540-42082-8. doi:10.1007/978-3-642-08728-8

    work page doi:10.1007/978-3-642-08728-8 2001

  29. [29]

    Barnett and Sarah Croke

    Stephen M. Barnett and Sarah Croke. Quantum state discrimination.Adv. Opt. Photon., 1(2):238–278, 2009. doi:10.1364/AOP.1.000238

    doi:10.1364/aop.1.000238 2009

  30. [30]

    On measures of entropy and informa- tion

    Alfr´ ed R´ enyi. On measures of entropy and informa- tion. InProceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pages 547–561, Berkeley, Calif., 1961. University of California Press

    1961

  31. [31]

    Mathematical Foundations, volume 5 ofSpringerBriefs in Mathematical Physics

    Marco Tomamichel.Quantum Information Process- ing with Finite Resources. Mathematical Foundations, volume 5 ofSpringerBriefs in Mathematical Physics. Springer, 2016. ISBN 978-3-319-21890-8, 978-3-319- 21891-5. doi:10.1007/978-3-319-21891-5

    doi:10.1007/978-3-319-21891-5 2016

  32. [32]

    Institute of Mathematical Statistics, Hayward, CA, 1988

    Persi Diaconis.Group Representations in Probability and Statistics, volume 11 ofLecture Notes–Monograph Series. Institute of Mathematical Statistics, Hayward, CA, 1988. ISBN 978-0-940600-14-0. Appendix A: Diagonalization of Gram matrix For every characterψ∈ bA, define a vector vψ := (ψ(a))a∈A.(A1) In coordinates, (vψ)a = ψ(a).(A2) We show thatv ψ is an ei...

    1988

  33. [33]

    more spread-out

    1≤S eff (p)≤d; 2.S eff (p) = 1 if and only ifpis a point mass; 3.S eff (p) =dif and only ifpis the uniform distribu- tion; 4.S eff (p) is monotone under mixing: ifp≻qin the majorization order, then Seff (p)≤S eff (q).(B4) Proof. Bounds.Since all probabilities are nonnegative, X χ∈ bA √pχ ≥ sX χ∈ bA pχ = 1,(B5) which gives Seff (p)≥1.(B6) For the upper bou...