Optimal quantum measurements for additive information and disturbance measures

Hiroaki Terashima

REVIEW 4 major objections 4 minor 1 cited by

Optimal quantum measurements for additive information and disturbance measures

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Logarithmic information and disturbance measures change which quantum measurement is optimal.

desk verdict Solid extension of Terashima's own geometric framework; the I-DR result is likely right, but the paper hands the load-bearing convex-hull and curvature arguments to earlier papers and needs to close that gap for DR unbounded. read the letter →

arxiv 2507.03346 v1 pith:UCTZNPWU submitted 2025-07-04 quant-ph

Hiroaki Terashima This is my paper · ORCID

classification quant-ph

keywords quantummeasurementinformation–disturbancetradeoffadditiveinformationmeasureestimationfidelityoperationphysicalreversibilityphysicallyallowedregionoptimal

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 derives additive information and disturbance measures from the standard multiplicative ones—estimation fidelity, operation fidelity, and physical reversibility—by taking logarithms, so that independent measurements on separable systems add. It then determines which measurements provide a given amount of information with the least disturbance for four pairs: $I_G$–$D_F$, $I_G$–$D_R$, $I$–$D_F$, and $I$–$D_R$. The central claim is that taking a logarithm changes the curvature of the lower boundary of the physically allowed region, and the sign of that curvature dictates the form of the optimal measurement. Concretely, the estimation-fidelity pairs keep the same $d$-outcome optimal family as the fidelity tradeoffs, while the entropy-reduction pairs switch to a $d+1$-outcome mixture of rank-one and identity operators for small information. The result matters because additivity is natural for independent measurements, and it genuinely changes which operations are optimal, not merely how the tradeoff is labeled.

What carries the argument

The load-bearing machinery is the physically allowed region for a single outcome together with the center-of-mass picture for averaging. A measurement is a set of operators, each represented by a point on the information–disturbance plane with weight equal to its outcome probability; the average values are the convex hull of those points, and the lower boundary of that hull is the information–disturbance tradeoff. Optimal measurements are configurations whose center of mass lies on the lower boundary. The second derivative of the single-outcome lower boundary $(1,d-1)$ decides whether that center of mass sits at one point on a convex curve or at the ends of a straight hull segment, and this curvature sign is the mechanism behind all the paper's optimality results.

What would settle it

Compute the average $I$ and $D_R$ directly from Eqs. (7) and (21) for a concrete two-outcome measurement in $d=4$ whose single-outcome points lie far into the unbounded $D_R$ region, and check whether the average point lies on or above the claimed lower boundary of the blend region; a point below that boundary would falsify the center-of-mass step.

Watch

Extended reading notes

Core claim

On the information–disturbance plane, each single-outcome measurement operator lands on a point, and averaging over outcomes is represented by the convex hull of those points. The paper's discovery is that the lower boundary $(1,d-1)$ of this region has a curvature whose sign—$+$, $-$, $0$, or $\mp$—completely determines the optimal measurement form for the pair. For $I_G$–$D_F$ and $I_G$–$D_R$ the boundary is convex, so an optimal measurement must consist of measurement operators all corresponding to the same point on $(1,d-1)$; the explicit $d$-outcome family in Eq. (32) saturates both tradeoffs. For $I$–$D_F$ and $I$–$D_R$ the boundary is inverted S-shaped, so below a threshold information $I_T$ the optimum is a mixture of rank-one operators at the tangency point $T$ and the identity, the $d+1$-outcome measurement in Eq. (34). The paper concludes that the optimal measurements for $I_G$–$D_R$ coincide with those for $G$–$F$ rather than $G$–$R$, while the optimal measurements for $I$–$D_R$ are completely different from those for $I$–$R$ yet are always optimal for $I$–$D_F$.

Load-bearing premise

The load-bearing premise is that averaging over outcomes is exactly the center-of-mass blend of single-outcome points, including when $D_R(m)$ is infinite and many operators sit at infinite disturbance; if that step fails, the claimed optimal measurement forms in Eqs. (32) and (34) do not follow.

Editorial extensions

If this is right

The $d$-outcome measurement family in Eq. (32) simultaneously saturates the $I_G$–$D_F$ and $I_G$–$D_R$ tradeoffs for any chosen information value, so a single physical construction achieves both additive tradeoffs.
For entropy-reduction information below $I_T$, the $d+1$-outcome mixture in Eq. (34) lowers the disturbance compared with the $d$-outcome family; the paper reports reductions of about 0.8% for $D_F$ and 30% for $D_R$ at $d=4$ and $I=0.05$.
The curvature sign acts as a classification: pairs with the same sign have optimal measurements of the same form, and the four signs $+$, $-$, $0$, and $\mp$ organize all additive and original information–disturbance pairs.
The known failure of $G$–$R$ optima to be $G$–$F$ optimal does not extend to the additive measures: every $I$–$D_R$ optimal measurement is also $I$–$D_F$ optimal, even though $I$–$D_R$ and $I$–$R$ optima are completely different.
Because the method only requires single-outcome multiplicative measures, it carries over to any other multiplicative information or disturbance quantity and to triplewise tradeoffs via two curvature signs.

Reading between the lines

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

A natural testable extension, not reported in the paper, is to realize Eq. (34) for $d=4$ and check whether the measured $D_R$ drop below $I_T$ follows the predicted roughly 30% reduction.
The curvature-sign rule suggests a general screening principle: applying any monotone function to an information or disturbance measure changes optimal measurements only if it changes the sign of the second derivative of the boundary; logarithms are one such case.
Because $D_R(m)$ is unbounded, the center-of-mass step could be stress-tested numerically by sampling measurements with outcomes that have very large $D_R(m)$ and checking whether the average ever falls below the claimed blend region; if it does, $I$–$D_R$ optimality needs a separate argument.

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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

The genuinely new thing here is the observation that taking logarithms of per-outcome fidelities—defining IG, DF, and DR—changes which measurements are minimally disturbing for a given information gain, and in particular that I–DR has a completely different optimal family from I–R, while I–DR optimals are always optimal for I–DF. That is a real conceptual point, not a trivial restatement of known tradeoffs. The curvature-sign classification in Table 1 and Fig. 4 is a clean way to organize the cases, and the explicit measurement forms in Eqs. (32) and (34) are concrete and checkable.

The paper is clearly written and the geometric picture is appealing. The author has built this framework over several prior papers, and the new measures fit naturally into it. The tradeoff inequalities in Section 4 are plausible and consistent with the figures. For a specialist reader, this is a useful contribution to the information–disturbance literature.

The soft spots are real but patchable. The center-of-mass/convex-hull step is asserted, not proved, and for DR, where the single-outcome region is unbounded (DR can be infinite), the standard convex hull is not even formally defined. That is the load-bearing step for Eq. (34). I think the conclusion is probably salvageable because the optimal measurements actually use only finite-DR points (Pd and T), so the infinite-disturbance part of the region can be excised without affecting the lower boundary of the convex hull. But the paper should state this explicitly and give a proof that the tangent line from Pd to T is the correct lower envelope for the finite-DR subset. As written, a skeptical reader cannot verify this without reconstructing the argument from Refs. [33] and [36].

Also, Table 1's curvature signs are simply imported from [36] with no derivation. Since the entire classification rests on those signs, a referee should ask for a derivation or at least a clear pointer to where each sign is computed. The heavy reliance on self-citations is not by itself a flaw, but it does mean the paper is not self-contained on its central claims.

This is not a fatal problem. The core idea is sound, and the unbounded-DR issue looks like a rigor gap rather than a counterexample. The paper deserves a serious referee: I would send it to review, with the request that the author tighten the convex-hull argument for the unbounded case and show the curvature signs explicitly.

Referee Report

4 major / 4 minor

Summary. This paper defines additive information and disturbance measures IG, DF, and DR as per-outcome logarithms of estimation fidelity, operation fidelity, and physical reversibility, and pairs them with the inherently additive entropy reduction I. Using the physically allowed single-outcome region from the author's earlier work, the paper argues that averaging over outcomes corresponds to taking the convex hull of that region, and then reads off optimal measurements from the curvature of the lower boundary. The central claims are that for IG–DR the optimal measurements coincide with those for G–F rather than G–R, while for I–DR the optimal measurements form a new family obtained by mixing the identity operation with the tangent point T.

Significance. If the results are correct, the paper establishes an interesting qualitative phenomenon: switching from multiplicative to logarithmic measures changes which measurements achieve the optimal information–disturbance tradeoff, even though the separate measures are monotonically related. The classification by curvature signs is simple and potentially useful, and the explicit constructions in Eqs. (32) and (34) are concrete and falsifiable. The paper is not fully self-contained, relying on Refs. [32,33,36] for the allowed regions, formulas, and curvature signs; however, those are published parameter-free derivations, so the main risk is not circularity but missing rigor in the convex-hull step for the unbounded DR region.

major comments (4)

[Section 4] Eq. (23) and Fig. 1(b),(d): the DR-based single-outcome region is unbounded, with DR(m)=∞ for rank-deficient operators, but the convex-hull step for averaging over outcomes is asserted rather than proved. The center-of-mass analogy requires a precise treatment for unbounded sets: one must show that any measurement with finite average DR uses only finite-DR outcomes, that the average pair lies in the convex hull of the finite-DR part of the single-outcome region, and that the lower boundary of that convex hull is the curve (1,d−1) for IG–DR or the tangent line from P_d to T for I–DR. Since Eqs. (32) and (34) are derived from this lower boundary, this gap is load-bearing; the manuscript currently offers no argument that the point P_1, which lies at infinite DR, cannot affect the finite-DR boundary.
[Section 5 / Table 1] The curvature signs in Table 1 and in Eqs. (26)–(27) are the entire basis for the classification of optimal measurements, but they are only cited to Ref. [36] and stated locally 'near P_d.' Please derive the second derivatives of D_F(m) and D_R(m) with respect to I(m) along the boundary (1,d−1), or quote the exact formulas from Ref. [36] and show explicitly that they imply the signs in Table 1 for all relevant d. In particular, reconcile the local negative sign in Eq. (27) with the ∓ entry for I–DR in Table 1, and state where the inflection point occurs.
[Section 5] The argument that 'for the center of mass to be at a point on a convex boundary, all particles must be at that point' presupposes strict convexity and excludes endpoints. The paper does not prove that (1,d−1) is strictly convex on the relevant interval for IG–DF and IG–DR, and it does not separately treat the endpoint cases P_d and P_1. Since Eq. (32) follows directly from this step, a rigorous proof should state the strictness condition and handle zero-curvature or endpoint cases.
[Section 5, after Eq. (31)] The statement 'the optimal measurements for I–DR are always optimal for I–DF' appears to contradict the earlier statement in the same section that the two inequalities are not necessarily saturated at the same time because I_T is not equal between I–DF and I–DR. For I smaller than both threshold values, the optimal I–DR measurement is a mixture of P_d and T_DR, whereas the optimal I–DF measurement is a mixture of P_d and T_DF; unless T_DR = T_DF, these are different physical measurements. Please clarify whether 'always optimal' means 'has the same form' or 'belongs to the optimal set,' and correct the claim if it is meant as a statement about the same measurement.

minor comments (4)

[Equations (28)–(29)] Please use explicit superscript notation such as 2^{I_G}, 2^{D_F}, and 2^{D_R} instead of the ambiguous '2IG', '1/2DF', and '1/2DR' in these inequalities.
[Figure 1] For panels (b) and (d), state in the caption that the DR axis is truncated and that P_1 and other rank-deficient points with DR=∞ are not drawn, so the blue shaded region is only a finite portion of the true single-outcome region.
[Table 1] The symbols +, −, 0, and ∓ should be defined explicitly in the caption or in the text; in particular, ∓ should be described as a sign change along the boundary, with the first symbol referring to the behavior near P_d.
[Section 3] The sentence 'the net amount of information should be G − (1/d)' is not used in the derivation of I_G; either connect it to Eq. (11) or remove it to avoid confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the additive-measure optimality claims are new consequences of parameter-free prior derivations; the unbounded-region convex-hull step is an omitted proof (correctness risk), not circularity.

full rationale

The derivation chain is: (1) single-outcome formulas for G(m), F(m), R(m) from Ref. [32]; (2) new additive measures IG, DF, DR defined in Eqs. (11)-(21); (3) single-outcome allowed regions plus the convex-hull/center-of-mass step for averaged values (Sec. 4, Ref. [33]); (4) curvature signs of (1,d-1) in Table 1, computed from derivative formulas of Ref. [36], with Eqs. (26)-(27) computed in-text; (5) optimal forms (32) and (34) read off from the convex-hull lower boundary. No step equates a prediction with an input by construction. The central claims ('IG-DR optimal measurements are those of G-F, not G-R' and 'I-DR optimal measurements are completely different from those for I-R') are new consequences of the computed curvature signs; they are not assumed by the definitions, and no parameter is fitted to the target result. The tangency point T, lambda_T, and c = sqrt(I/I_T) are determined by geometry and normalization, not by data. Self-citations [32], [33], [36] are load-bearing, but they are published parameter-free derivations whose assumptions (singular-value parametrization of measurement operators; outcome averaging as convex combination) do not include the additive-measure conclusions, so under the hard rules they are real evidence and do not raise the circularity score. Per the reviewing rule I flag one omitted proof, weighed in the verdict but not itself circular: Sec. 4 asserts 'the region for the average values is given by the convex hull of the region of a single outcome' without proof for the unbounded case, since Eq. (23) allows DR(m) = infinity and points Pr (r != d) and (k,l) (k+l != d) lie at DR = infinity; if the convex-hull identification fails on unbounded sets, the lower-boundary lines behind Eqs. (32) and (34) would not follow. That is a correctness/rigor risk, not a self-referential reduction, and is placed under correctness per the rules.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters. Its central claims depend on the standard measurement formalism, the pure-state unknown-input assumption inherited from Ref. [32], the convex-hull equivalence for outcome averaging from Ref. [33], the curvature signs from Ref. [36], and the paper's postulate that information and disturbance should be additive. No new physical entities are postulated.

assumptions (5)

standard math Standard quantum measurement formalism: measurement operators satisfy the completeness relation, with Born-rule probabilities and state update.
Used throughout Sections 2 and 5 as the model of a measurement.
domain assumption The system is initially in a completely unknown pure state with a uniform prior.
The allowed regions and fidelity formulas are taken from Ref. [32] under this assumption; stated in Section 4.
domain assumption Averaging over outcomes can be represented as the convex hull (center of mass) of the single-outcome allowed region.
Used to derive average tradeoffs and optimal measurements in Sections 4 and 5; taken from the author's prior work [33].
domain assumption The curvature signs of the lower boundary (1,d-1) summarized in Table 1 are correct for both original and additive measures.
Section 4 uses these signs and the cited derivatives from [36] to construct tangent lines and classify optimal measurements.
ad hoc to paper Information and disturbance should be additive under independent measurements on separable systems.
This postulate motivates the logarithmic definitions in Section 3 and is not derived from the underlying physics.

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Cite this review

Pith. "Pith review of Optimal quantum measurements for additive information and disturbance measures." pith.science (2026). https://pith.science/paper/UCTZNPWU

@misc{pith2026250703346,
  author       = {Pith},
  title        = {Pith review of: Optimal quantum measurements for additive information and disturbance measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UCTZNPWU}},
  note         = {Machine review of arXiv:2507.03346}
}

read the original abstract

Additive measures for information and disturbance in quantum measurements of a system are defined from well-known multiplicative measures such as estimation and operation fidelities using a logarithm. This is motivated by the fact that information and disturbance are naturally assumed to be additive while performing independent measurements on separable systems. Although the additivity makes no remarkable difference when information and disturbance are separately considered, it can change measurements that only introduce minimal disturbance relative to the amount of information. Such optimal measurements are shown for additive information and disturbance measures with a tradeoff relationship.

Figures

Figures reproduced from arXiv: 2507.03346 by the authors.

**Figure 2.** Amount and rate of decrease in DF from (1, d − 1) to the tangent line between Pd and T when d = 4, 8, 12. The solid and dashed lines denote the amount and rate, respectively. 0.05 0.10 0.15 0.20 0.25 0 0.05 0.1 0.15 0.2 0.25 15 30 45 60 75 Amount Rate (%) I(m) DR d=4 d=8 d=12 d=4 d=8 d=12 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Amount and rate of decrease in DR from (1, d − 1) to the tangent line between Pd and T when d = 4, 8, 12. The solid and dashed lines denote the amount and rate, respectively. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Relationships between the sets of optimal measurements [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The statistical disturbance bound of quantum measurements
quant-ph 2026-07 conditional novelty 8.0 of 10

A quantum measurement's statistics alone determine the minimum disturbance any implementation of it must cause, and this minimum is computable and experimentally estimable.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [33]

Terashima, Allowed region and optimal measurement for infor mation versus disturbance in quantum measurements, Quantum Inf

H. Terashima, Allowed region and optimal measurement for infor mation versus disturbance in quantum measurements, Quantum Inf. Pro cess. 16, 250 (2017)

work page 2017

[2] [36]

Terashima, Derivative of the disturbance with respect to inf ormation from quantum measurements, Quantum Inf

H. Terashima, Derivative of the disturbance with respect to inf ormation from quantum measurements, Quantum Inf. Process. 18, 63 (2019). 18

work page 2019

[3] [1]

C. A. Fuchs and A. Peres, Quantum-state disturbance versus informa- tion gain: Uncertainty relations for quantum information, Phys. Re v. A 53, 2038 (1996)

work page 1996

[4] [2]

Banaszek, Fidelity balance in quantum operations, Phys

K. Banaszek, Fidelity balance in quantum operations, Phys. Rev. Lett. 86, 1366 (2001)

work page 2001

[5] [3]

C. A. Fuchs and K. Jacobs, Information-tradeoﬀ relations for ﬁnite- strength quantum measurements, Phys. Rev. A 63, 062305 (2001)

work page 2001

[6] [4]

Banaszek and I

K. Banaszek and I. Devetak, Fidelity trade-oﬀ for ﬁnite ensemb les of identically prepared qubits, Phys. Rev. A 64, 052307 (2001)

work page 2001

[7] [5]

G. M. D’Ariano, On the Heisenberg principle, namely on the information–disturbance trade-oﬀ in a quantum measurement, Fo rtschr. Phys. 51, 318 (2003)

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M. Ozawa, Uncertainty relations for noise and disturbance in gen eralized quantum measurements, Ann. Phys. (NY) 311, 350 (2004). 15

work page 2004