Pith. sign in

REVIEW 6 minor 39 references

Maximizing measurement coherence recovers sharpness

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 · glm-5.2

2026-07-08 22:36 UTC pith:R7KI4LJ5

load-bearing objection Clean equivalence result between measurement coherence and sharpness for structured POVMs; proofs are correct and limitations are honestly scoped.

arxiv 2607.05847 v1 pith:R7KI4LJ5 submitted 2026-07-07 quant-ph

Maximal coherence of quantum measurement and the resource theory of sharpness

Kyunghyun Baek , Yonggi Jo , Hyunchul Nha This is my paper
classification quant-ph
keywords coherencemaximalmeasurementsharpnessquantumbasiselementsmeasurements
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.

The paper proves that for a broad class of quantum measurements—those whose elements share a common eigenbasis—the maximum coherence achievable by rotating the reference basis exactly equals the measurement's sharpness, defined as its distance from trivial, uninformative measurements. This extends a known state-level identity (maximal coherence equals purity) to the measurement level. The central object is the maximal coherence of measurement, C_D^max(A) = max_U C_D(U†AU), which optimizes any distance-based coherence monotone over all unitary basis changes. The proof hinges on a dephasing map in a mutually unbiased basis that sends incoherent measurements to trivial ones, allowing a matching lower bound. The equivalence also extends, for element-additive distances, to POVMs admitting a common mutually unbiased basis structure. Outside these structural conditions, the equality can fail. The paper further establishes that maximal coherence is faithful (vanishes only on trivial measurements) and monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant channels for general measurements. Three physical examples confirm the scope and boundaries of the result.

Core claim

The maximal coherence of measurement C_D^max(A) coincides exactly with distance-based sharpness S_D(A) for POVMs whose elements share a common eigenbasis (Theorem 2), and for element-additive distances this extends to POVMs with a common mutually unbiased basis structure (Corollary 3). The proof works by choosing a unitary that maps the common eigenbasis to a mutually unbiased basis relative to the incoherent basis, then applying a dephasing map in that MUB which collapses any incoherent measurement to a trivial one, yielding a lower bound that matches the upper bound from the inclusion of trivial measurements within incoherent ones.

What carries the argument

Maximal coherence of measurement C_D^max(A) = max_U C_D(U†AU); distance-based sharpness S_D(A) = min_{B in T(d,n)} D(A,B); dephasing map E†+ in a mutually unbiased basis; fuzzifying operations F(A)_x = αE†(A_x) + (1-α)q_x I; element-additive Schatten-p norm distances

Load-bearing premise

The equivalence between maximal coherence and sharpness holds only for POVMs whose elements share a common eigenbasis or a common mutually unbiased basis structure; for general POVMs lacking this structure, the proof's key dephasing argument does not apply and the equality can fail.

What would settle it

Find a POVM with a common eigenbasis for which C_D^max(A) ≠ S_D(A) for some contractive, jointly convex, unitarily invariant distance D, or show that the dephasing-map lower bound does not hold for some distance in this class.

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

If this is right

  • For dichotomic POVMs in any dimension, maximal coherence is a full sharpness monotone under all fuzzifying operations, since two-outcome POVM elements always commute and thus share a common eigenbasis.
  • In single-photon Mach-Zehnder interferometry with an on-off detector, the maximal Fisher information equals the squared maximal coherence of the measurement, giving a direct metrological interpretation of the resource quantity.
  • In noisy photon-number-resolving detectors, maximal coherence is monotonic in quantum efficiency and vanishes at zero efficiency, unlike PVM-based unsharpness measures which cannot distinguish the trivial measurement at zero efficiency from the ideal detector.
  • For general POVMs lacking common-eigenbasis or common-MUB structure (e.g., SIC POVMs), the gap between maximal coherence and sharpness is bounded but generally nonzero, and monotonicity under arbitrary fuzzifying operations remains open.
  • The result provides a measurement-theoretic analogue of the maximal-coherence/purity correspondence for quantum states, bridging two previously separate resource theories.

Where Pith is reading between the lines

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

  • The structural limitation to common-eigenbasis or common-MUB POVMs suggests that the gap C_D^max < S_D for general POVMs quantifies a form of basis-incompatibility among the POVM elements—a resource distinct from either coherence or sharpness individually.
  • If monotonicity under general fuzzifying operations were eventually established for all POVMs, maximal coherence would serve as a universal sharpness monotone computable from coherence alone, potentially simplifying resource-theoretic analysis of measurement quality.
  • The Fisher information connection in the Mach-Zehnder example hints that maximal coherence might bound metrological performance for broader classes of sensing protocols beyond single-photon phase estimation, though this would require extension beyond dichotomic measurements.
  • The failure of the minimax interchange (max over unitaries vs. min over incoherent measurements) is reminiscent of similar obstacles in channel resource theories where free operations do not form a convex set, suggesting that techniques from those settings might be portable here.

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

0 major / 6 minor

Summary. The manuscript establishes a quantitative connection between two resource theories of quantum measurement—coherence (basis-dependent off-diagonal structure of POVM elements) and sharpness (deviation from trivial, uninformative measurements). The central result (Theorem 2) shows that for POVMs whose elements share a common eigenbasis, the maximal distance-based coherence $C^{max}_D(A) = max_U C_D(U^† A U)$ coincides exactly with the distance-based sharpness $S_D(A) = min_{B in T} D(A,B)$, for any distance $D$ that is contractive, jointly convex, and unitarily invariant. This equivalence extends to POVMs with a common MUB structure for element-additive distances (Corollary 3). The paper also establishes faithfulness (Theorem 1), monotonicity under fuzzifying operations for dichotomic POVMs (Corollary 4) and under mixed-unitary/unitarily covariant channels for general POVMs (Theorem 5, Corollary 6), and illustrates the results via qubit POVMs, Mach-Zehnder phase sensing, and noisy photon-number-resolving detection.

Significance. The paper provides a clean, parameter-free derivation: $C^{max}_D$ is defined as a basis-optimized distance (Eq. 15) and $S_D$ as a distance to the trivial set (Eq. 13), with the proof of Theorem 2 using only contractivity, unitary invariance, and a MUB-dephasing map. The analogy to the Streltsov et al. maximal-coherence/purity correspondence for states is well-motivated. The Mach-Zehnder example (Eq. 46) gives a concrete operational link to Fisher information, and the PNRD example demonstrates that $C^{max}_D$ behaves as a proper sharpness monotone where PVM-based measures fail. The structural limitations (common eigenbasis / common MUB) are honestly scoped, and the monotonicity table (Table I) clearly delineates what is proven versus what remains open.

minor comments (6)
  1. §II.A, Eq. (5) and surrounding text: the reference [?] appears in the PNRD POVM definition (Eq. 47) and should be replaced with the proper citation.
  2. §II.A, paragraph containing Eq. (3): the text reads 'itscoherence[15]' and 'itssharpness[16, 17, 23–30]' — missing spaces before bracketed citations. Similar formatting issues appear elsewhere (e.g., 'fuzzifyng' in Table I caption and §III.C).
  3. §IV.A, Fig. 1(c): the caption states the horizontal axis is the mixing parameter 'p', but the text uses 'q' for the convex combination $M_q = q M_{XY} + (1-q) M_{SIC}$. The axis label and text should be consistent.
  4. §III.B, Corollary 3: the proof applies a different dephasing map $E^†_x$ for each element $A_x$. It would help the reader to explicitly note that element-additivity of $D$ is what allows the per-element application of contractivity to be summed, since this is the load-bearing property for the extension beyond common-eigenbasis POVMs.
  5. §III.C, Corollary 6: the condition that the maximizing unitary $U^*$ can be chosen within the covariance group $G$ is somewhat restrictive. A brief remark on how restrictive this is in practice (or a concrete example beyond Werner-Holevo) would strengthen the result's accessibility.
  6. §IV.B, Eq. (45): the computation yields $C^{max}_{D_1}(E(phi)) = nu/2$, independent of $phi$. The text then invokes Appendix I of [16] to equate $C^{max}_{D_1} = C^{max}_{D_diamond} = C^{max}_{D_infty}$ for dichotomic POVMs. A one-sentence clarification of why this equivalence holds specifically for dichotomic POVMs would aid readers unfamiliar with [16].

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and accurate reading of the manuscript and for the recommendation of minor revision. The referee's summary correctly captures the main results, their scope, and the structural limitations. As the referee did not raise any major comments requiring substantive changes, our response is brief: we note a few minor typographical corrections to be made in the next version and otherwise confirm that the manuscript stands as written.

read point-by-point responses

  1. Referee: No major comments were raised by the referee.

    Authors: We note that the referee's report contains no major comments. We address minor points below. revision: partial

  2. Referee: Typographical issues in the manuscript (e.g., 'measuremenet' in the Introduction, 'relavent' in the same paragraph, 'fuzzifyng' missing the 'i' in several instances in Section III.C and Table I, 'distanc' in Section II.A, and a missing reference for the PNRD POVM elements in Eq. 47).

    Authors: We thank the referee for the careful reading that allowed these to be identified. We will correct all typographical errors in the revised manuscript: 'measuremenet' → 'measurement', 'relavent' → 'relevant', 'fuzzifyng' → 'fuzzifying' (throughout Section III.C and Table I), 'distanc' → 'distance', and 'itscoherence'/'itssharpness' (missing spaces in the Introduction). We will also supply the missing citation for the PNRD POVM elements in Eq. (47). These are purely cosmetic corrections that do not affect any results or arguments. revision: yes

Circularity Check

0 steps flagged

No circularity found. The proof of Theorem 2 is parameter-free and self-contained.

full rationale

The central result (Theorem 2, Eq. 17) establishes C^max_D(A) = S_D(A) for common-eigenbasis POVMs via a two-sided proof that uses only the independent definitions of C^max_D (Eq. 15: basis-optimized coherence) and S_D (Eq. 13: distance to trivial measurements), plus general properties of D (contractivity under CP unital maps, unitary invariance) and the MUB structure. The upper bound (Eqs. 19–21) is a clean relaxation using T(d,n) ⊂ I(d,n) and unitary invariance of D. The lower bound (Eqs. 22–25) constructs a specific unitary V mapping the common eigenbasis to a MUB, applies a dephasing CP unital map E†+, and uses contractivity — all standard mathematical steps with no fitted parameters or definitional equivalences. No step reduces to its input by construction. Self-citations [15, 16, 17] provide background definitions (coherence of measurements, sharpness monotones, fuzzifying operations) but are not load-bearing for the proof of Theorem 2 itself; the proof relies on externally verifiable mathematical properties (MUB orthogonality relations, CP unitality, contractivity) rather than on unverified claims from the authors' prior work. The examples (Mach-Zehnder, PNRD) are straightforward computations applying Theorem 2 to specific POVMs, with no circular reasoning. The paper honestly acknowledges where the equivalence fails (SIC POVMs) and where monotonicity remains open (general fuzzifying operations on non-dichotomic POVMs). This is a self-contained derivation with no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters are introduced or fitted. No new physical entities are postulated. The axioms are standard mathematical properties (contractivity, joint convexity, MUB existence) and domain assumptions (fuzzifying operations as free operations, common-eigenbasis structure) drawn from the established resource theory framework of [16, 17].

axioms (4)
  • standard math Distance D between POVMs is contractive under CP unital maps and jointly convex (Sec II.A, Eq. 5-7)
    Standard property of channel divergences; used to establish monotonicity of C_D and S_D.
  • standard math Existence of a mutually unbiased basis (MUB) relative to any fixed incoherent basis in dimension d (Theorem 1 proof, Theorem 2 lower bound)
    MUBs are known to exist in all prime-power dimensions; the paper uses the Fourier basis construction which is standard.
  • domain assumption Fuzzifying operations (Eq. 12) are the free operations in the resource theory of sharpness (Sec II.B)
    Adopted from Buscemi et al. [17]; defines the class of operations under which sharpness monotones must not increase.
  • domain assumption The POVM elements of A share a common eigenbasis (Theorem 2) or admit a common MUB structure (Corollary 3)
    This is the structural condition restricting the scope of the equivalence result. It is not ad hoc but limits generality.

pith-pipeline@v1.1.0-glm · 18844 in / 2257 out tokens · 176100 ms · 2026-07-08T22:36:49.588015+00:00 · methodology

0 comments
read the original abstract

A resource theory of quantum measurement can be addressed in terms of quantum coherence and measurement sharpness, respectively. The former analyzes the off-diagonal structure of POVM elements in a predetermined basis while the latter analyzes the deviation from trivial, state-independent, measurements. We establish a direct connection between the two resource theories by identifying measurement sharpness as the maximal coherence that is achievable under all possible unitary changes of the reference basis. For a broad class of POVMs whose elements share a common eigenbasis, we show that the maximal distance-based coherence of measurement coincides exactly with the corresponding distance-based sharpness monotone. We further extend this equivalence, with element-additive distances, to POVMs whose elements admit a common mutually unbiased basis structure. These results provide a measurement-theoretic analogue of the maximal-coherence \& purity correspondence for quantum states. We also show that the maximal coherence of measurement is faithful with respect to trivial measurements and is monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant preprocessing channels. Finally, we illustrate the operational meaning and limitations of the equivalence through qubit POVMs, single-photon phase sensing, and noisy photon-number resolving detection. In particular, the maximal Fisher information in a Mach-Zehnder interferometer is shown to be determined by the squared maximal coherence of the measurement, while in an imperfect photon-number resolving detector the maximal coherence behaves as a proper sharpness monotone, unlike conventional PVM-based unsharpness measures.

Figures

Figures reproduced from arXiv: 2607.05847 by Hyunchul Nha, Kyunghyun Baek, Yonggi Jo.

Figure 1
Figure 1. Figure 1: Comparison of the sharpness SD and the maximal coherence of measurement C max D . (a) Bloch-sphere representation of the joint X–Y POVM MXY ; all four rank-one elements have eigenbases lying in the xy-plane of the Bloch sphere; the incoherent basis {|0⟩, |1⟩} along z-axis (orange arrows) is mutually unbiased to every such eigenbasis, so the common-MUB condition of Corollary 3 is satisfied. (b) Bloch-sphere… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of a single-photon Mach–Zehnder phase [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of sharpness measures for the imperfect [PITH_FULL_IMAGE:figures/full_fig_p009_3.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

39 extracted references · 39 canonical work pages · 1 internal anchor

  1. [1]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion(Cambridge University Press, 2010). 11

    work page 2010

  2. [2]

    Quantum computing 40 years later

    J. Preskill, (2021), 10.48550/arXiv.2106.10522, arXiv:2106.10522 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2106.10522 2021

  3. [3]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Nature Pho- tonics5, 222 (2011)

    work page 2011

  4. [4]

    M. G. A. Paris, International Journal of Quantum Information07, 125 (2009), https://doi.org/10.1142/S0219749909004839

    work page doi:10.1142/s0219749909004839 2009

  5. [5]

    Pironio, A

    S. Pironio, A. Ac´ ın, N. Brunner, N. Gisin, S. Massar, and V. Scarani, New Journal of Physics11, 045021 (2009)

    work page 2009

  6. [6]

    Chitambar and G

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

    work page 2019

  7. [7]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys.81, 865 (2009)

    work page 2009

  8. [8]

    Baumgratz, M

    T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett.113, 140401 (2014)

    work page 2014

  9. [9]

    Streltsov, G

    A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys.89, 041003 (2017)

    work page 2017

  10. [10]

    Theurer, D

    T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio, Phys. Rev. Lett.122, 190405 (2019)

    work page 2019

  11. [12]

    Gour and C

    G. Gour and C. M. Scandolo, Phys. Rev. Lett.125, 180505 (2020)

    work page 2020

  12. [13]

    Gour and C

    G. Gour and C. M. Scandolo, Phys. Rev. A103, 062422 (2021)

    work page 2021

  13. [14]

    Xu, Phys

    J. Xu, Phys. Rev. A100, 052311 (2019)

    work page 2019

  14. [15]

    K. Baek, A. Sohbi, J. Lee, J. Kim, and H. Nha, New J. Phys.22, 093019 (2020)

    work page 2020

  15. [16]

    Tendick, M

    L. Tendick, M. Kliesch, H. Kampermann, and D. Bruß, Quantum7, 1003 (2023)

    work page 2023

  16. [17]

    Buscemi, K

    F. Buscemi, K. Kobayashi, and S. Minagawa, Quantum 8, 1235 (2024)

    work page 2024

  17. [18]

    Takagi and B

    R. Takagi and B. Regula, Phys. Rev. X9, 031053 (2019)

    work page 2019

  18. [19]

    Streltsov, U

    A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Phys. Rev. Lett.115, 020403 (2015)

    work page 2015

  19. [20]

    H. Kwon, K. C. Tan, T. Volkoff, and H. Jeong, Phys. Rev. Lett.122, 040503 (2019)

    work page 2019

  20. [21]

    Theurer, S

    T. Theurer, S. Satyajit, and M. B. Plenio, Phys. Rev. Lett.125, 130401 (2020)

    work page 2020

  21. [22]

    Ahnefeld, T

    F. Ahnefeld, T. Theurer, D. Egloff, J. M. Matera, and M. B. Plenio, Phys. Rev. Lett.129, 120501 (2022)

    work page 2022

  22. [23]

    Busch, T

    P. Busch, T. Heinonen, and P. Lahti, Physics Letters A 320, 261 (2004)

    work page 2004

  23. [24]

    Carmeli, T

    C. Carmeli, T. Heinonen, and A. Toigo, Journal of Physics A: Mathematical and Theoretical40, 1303 (2007)

    work page 2007

  24. [25]

    Heinosaari, D

    T. Heinosaari, D. Reitzner, and P. Stano, Foundations of Physics38, 1133 (2008)

    work page 2008

  25. [26]

    Busch, Foundations of Physics39, 712 (2009)

    P. Busch, Foundations of Physics39, 712 (2009)

    work page 2009

  26. [27]

    Baek and W

    K. Baek and W. Son, Scientific Reports6, 30228 (2016)

    work page 2016

  27. [28]

    Liu and S

    Y. Liu and S. Luo, Phys. Rev. A104, 052227 (2021)

    work page 2021

  28. [29]

    Mitra, International Journal of Theoretical Physics 61, 236 (2022)

    A. Mitra, International Journal of Theoretical Physics 61, 236 (2022)

    work page 2022

  29. [30]

    Mitra, S

    A. Mitra, S. Mukherjee, and C. Lee, Phys. Rev. A112, 062233 (2025)

    work page 2025

  30. [31]

    Streltsov, H

    A. Streltsov, H. Kampermann, S. W¨ olk, M. Gessner, and D. Bruß, New Journal of Physics20, 053058 (2018)

    work page 2018

  31. [32]

    Takagi, B

    R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Phys. Rev. Lett.122, 140402 (2019)

    work page 2019

  32. [33]

    Cooney, M

    T. Cooney, M. Mosonyi, and M. M. Wilde, Communi- cations in Mathematical Physics344, 797 (2016)

    work page 2016

  33. [34]

    Leditzky, E

    F. Leditzky, E. Kaur, N. Datta, and M. M. Wilde, Phys. Rev. A97, 012332 (2018)

    work page 2018

  34. [35]

    Gour, IEEE Transactions on Information Theory65, 5880 (2019)

    G. Gour, IEEE Transactions on Information Theory65, 5880 (2019)

    work page 2019

  35. [36]

    Gour and A

    G. Gour and A. Winter, Phys. Rev. Lett.123, 150401 (2019)

    work page 2019

  36. [37]

    Skrzypczyk and N

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

    work page 2019

  37. [38]

    R. F. Werner and A. S. Holevo, Journal of Mathematical Physics43, 4353 (2002)

    work page 2002

  38. [39]

    X. Yuan, H. Zhou, Z. Cao, and X. Ma, Phys. Rev. A92, 022124 (2015)

    work page 2015

  39. [40]

    J. Ma, Y. Zhou, X. Yuan, and X. Ma, Phys. Rev. A99, 062325 (2019)

    work page 2019