{"id":"1da31a59-257f-4a6b-a468-f115d7b158fa","arxiv_id":"2507.03346","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Logarithmically defined additive information and disturbance measures change the curvature of the allowed tradeoff region, which alters the form of optimal minimal-disturbance measurements for some pairs.","lead":"This paper defines new additive measures of information and disturbance for quantum measurements by taking logarithms of standard fidelity measures, and works out which measurements achieve the best tradeoff. It shows that switching to additive measures changes the optimal measurement form for some information-disturbance pairs, which matters for designing minimally disturbing quantum measurements.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convex-hull step for averaging outcomes is not justified when DR(m)=∞; if it fails, the claimed optimal forms in Eqs. (32) and (34) do not follow.","rationale":"I read the paper as a derivation of optimal measurement forms for four additive information–disturbance pairs, with the main novelty being that logarithmically defined additive measures change which measurements are minimally disturbing. The argument's key logical bridge is the claim that averaging outcomes exactly corresponds to taking the convex hull of the single-outcome allowed region. That claim is already nontrivial for finite-bounded measures, and the paper defers it to Ref. [33]. The extension to DR(m)∈[0,∞) is not derived, even though many physically allowed single-outcome operators are at infinite DR. Without that extension, the center-of-mass arguments in Section 5 do not rigorously imply that optimal measurements have all operators at one point (Eq. 32) or at the endpoints of the tangent segment (Eq. 34). I do not see an internal inconsistency in the finite-DR cases, and the one-outcome plots and explicit optimal examples are credible. The correct disposition is therefore the same as the Reader's: conditional acceptance, pending a proof or explicit verification of the convex-hull construction for unbounded DR. No change to the Reader's verdict is needed.","tokens_in":9905,"tokens_out":10385,"duration_ms":136505,"concrete_test":"For d=4, compute the exact lower convex hull of the single-outcome I–DR region S. Parameterize all finite-DR single-outcome operators (including the family \\hat M^{(d)}_{1,d-1}(λ) and generic singular-value configurations); compute the tangent point T and the line from P_d to T; then verify by convex optimization over all mixtures of finite-DR points that no average point lies below that line for I<I_T. If any mixture (including limits of infinite-DR points) falls below the line, Eq. (34) is not the general optimal measurement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 asserts that the physically allowed region for an averaged measurement is the convex hull of the single-outcome region, via the center-of-mass analogy, and Section 5 uses this to conclude that optimal measurements must place all particles at one point (Eq. 32) or at the two endpoints of a straight section (Eq. 34). For IG–DF and I–DF the set is bounded, so the convex-hull step is standard. But for IG–DR and I–DR the single-outcome disturbance DR(m) is unbounded (Eq. 23): rank-deficient operators such as P_r (r<d) and (k,l) with k+l≠d have DR(m)=∞. The paper does not define an extended convex hull, does not prove that the finite-DR points are closed, and does not show that the lower boundary of conv(S) is exactly the tangent line from P_d to T. Since the I-DR optimal family in Eq. (34) is obtained from that line, this is the load-bearing step. The curvature signs in Table 1 are also taken from Ref. [36], but the convex-hull bridge is what turns those signs into statements about optimal measurements.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10143,"tokens_out":18415,"duration_ms":215402,"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":[{"comment":"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":"Section 4"},{"comment":"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":"Section 5 / Table 1"},{"comment":"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":"Section 5"},{"comment":"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.","section":"Section 5, after Eq. (31)"}],"minor_comments":[{"comment":"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.","section":"Equations (28)–(29)"},{"comment":"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.","section":"Figure 1"},{"comment":"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":"Table 1"},{"comment":"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.","section":"Section 3"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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].\n\nAlso, 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.\n\nThis 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.","headline":"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.","tokens_in":10619,"tokens_out":7610,"would_cite":true,"duration_ms":101563,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Logarithmic information and disturbance measures change which quantum measurement is optimal.","keywords":["quantum measurement","information–disturbance tradeoff","additive information measure","estimation fidelity","operation fidelity","physical reversibility","physically allowed region","optimal measurement"],"falsifier":"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.","tokens_in":9713,"feed_emoji":"⚛️","tokens_out":12157,"duration_ms":113341,"temperature":0.7,"pith_summary":"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.","feed_headline":"Taking logs changes the least-disturbing quantum measurement","feed_subtitle":"For low entropy gain, rank-one/identity mixtures give less disturbance than the d-outcome family.","key_machinery":"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.","core_discovery":"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 $I$–$D_F$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines estimation fidelity G and operation fidelity F and the G–F tradeoff whose optimal family the additive IG–DF pair inherits.","marker":"[2]"},{"why":"Defines entropy reduction I as an inherently additive information measure and supplies the single-outcome form I(m).","marker":"[5]"},{"why":"Establishes the G–R tradeoff, the comparison target that shows the IG–DR optimum differs from the G–R optimum.","marker":"[14]"},{"why":"Shows G–R optimal measurements need not be G–F optimal, the contrast with the claim that I–DR optima are always I–DF optimal.","marker":"[30]"},{"why":"Supplies the single-outcome definitions G(m), F(m), R(m), their bounds, and the rescaling invariance used to express the optimal measurement operators.","marker":"[32]"},{"why":"Provides the physically allowed region and the center-of-mass convex-hull argument that the paper's optimality analysis is built on.","marker":"[33]"},{"why":"Gives the second-derivative signs of the lower boundary (1,d-1) that underlie the curvature classification in Table 1.","marker":"[36]"}],"fun_headline_variants":["Logs bend the boundary: new least-disturbing measurements","Additive measure flips optimal measurement form","Curvature of info-disturbance boundary decides optimal measurement","Log-scale tradeoff: rank-one/identity mixtures beat d-outcome","Additive info-disturbance: curvature rules optimal measurements"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Logs bend the boundary: new least-disturbing measurements","Additive measure flips optimal measurement form","Curvature of info-disturbance boundary decides optimal measurement","Log-scale tradeoff: rank-one/identity mixtures beat d-outcome","Additive info-disturbance: curvature rules optimal measurements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000754,"raw_usage":{"total_tokens":3334,"prompt_tokens":906,"completion_tokens":2428,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":2345}},"tokens_in":522,"tokens_out":2428,"duration_ms":17965,"temperature":1.0,"reasoning_tokens":2345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:13:14.734362+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Banaszek, Fidelity balance in quantum operations, Phys","cited_arxiv_id":null,"evidence_quote":"Defines estimation fidelity G and operation fidelity F and the G–F tradeoff whose optimal family the additive IG–DF pair inherits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines entropy reduction I as an inherently additive information measure and supplies the single-outcome form I(m)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the G–R tradeoff, the comparison target that shows the IG–DR optimum differs from the G–R optimum."},{"cited_title":"Lim, Y.-S","cited_arxiv_id":null,"evidence_quote":"Shows G–R optimal measurements need not be G–F optimal, the contrast with the claim that I–DR optima are always I–DF optimal."},{"cited_title":"Terashima, Information, fidelity, and reversibility in general quantum measurements, Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the single-outcome definitions G(m), F(m), R(m), their bounds, and the rescaling invariance used to express the optimal measurement operators."},{"cited_title":"Terashima, Allowed region and optimal measurement for infor mation versus disturbance in quantum measurements, Quantum Inf","cited_arxiv_id":null,"evidence_quote":"Provides the physically allowed region and the center-of-mass convex-hull argument that the paper's optimality analysis is built on."},{"cited_title":"Terashima, Derivative of the disturbance with respect to inf ormation from quantum measurements, Quantum Inf","cited_arxiv_id":null,"evidence_quote":"Gives the second-derivative signs of the lower boundary (1,d-1) that underlie the curvature classification in Table 1."}],"review_version":1}