{"id":"0db226b5-40f3-4657-869c-6b48db29d755","arxiv_id":"1908.03863","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The average Wigner-Yanase coherence for MUMs and general SIC measurements is a constant multiple of the maximal coherence, and for SIC-POVMs it equals 1/d of the MUB average.","lead":"This paper calculates how much quantum coherence a quantum state keeps on average when measured with special families of complementary measurements. It finds simple formulas for mutually unbiased measurements and symmetric informationally complete measurements, and an exact relation between the two families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorems 1 and 2 are algebraically sound, and the imported trace identities from [19] hold under direct verification.","rationale":"The paper aims to compute the average Wigner-Yanase coherence over a complete set of MUMs and over a general SIC measurement, and to relate these quantities to maximal coherence. The central conditions are the two trace identities fixing the prefactors. The reader flagged these as the weakest assumption. Reading the full derivations, the algebra after those identities is straightforward and internally consistent. The critical question is whether the identities are true. They are. The MUM identity follows from cancellation of the A_b^2 and cross terms in construction (8) and completeness of the F_{n,b} basis; the SIC identity follows similarly from (26) and the parameter relation (27). Equations (16)-(19) rely on the known WYD trace identity for any orthonormal traceless basis, which is standard and correctly applied; Eq. (30) is a clean expansion with cross terms canceling. The specialization α=1/2 is valid because WYD entropy reduces to skew information. The comparison relations (23)-(25) and (33)-(35) are arithmetic consequences. There is no circularity: [19] is cited for the trace identities, but the identities are independently derivable from the constructions given in this paper. The only weakness is that the proof is deferred to a submitted manuscript; that is a presentational issue, not a correctness issue. Therefore the reader's ACCEPT verdict should stand unchanged.","tokens_in":7253,"tokens_out":32590,"duration_ms":323861,"concrete_test":"Perform the omitted algebraic verification: expand S_MUM = Σ_{b,n}(F^(b)_n)^2 using (8) and S_SIC = Σ_k P_k^2 using (26), with Σ_{b,n} F_{n,b}^2 = (d^2-1)/d I (and the analogous complete-orthonormal-basis identity for the SIC F_k). Confirm that all A_b^2 and cross terms cancel and the resulting operators are (1+√d)^2(d^2-1)I and adI, respectively. If these operator identities hold, the prefactors in Theorems 1 and 2 are fixed without reliance on [19].","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivations in §2 and §3 are sound. The only genuinely load-bearing unproved ingredient is the pair of trace identities imported from the authors' submitted paper [19]: at Eq. (15), Σ_{b,n} Tr[(F^(b)_n)^2ρ]=(1+√d)^2(d^2-1), and at Eq. (29), Σ_k Tr[(P_k)^2ρ]=ad. These fix the constant prefactors in Theorems 1 and 2, and neither is proved in the manuscript. However, direct expansion from the explicit constructions (8) and (26) verifies both. For MUMs, writing F^(b)_n = A_b - (d+√d)F_{n,b} (n<d) and F^(b)_d=(1+√d)A_b with A_b=Σ_n F_{n,b}, the A_b^2 and cross terms cancel exactly, leaving (d+√d)^2 Σ_{b,n}F_{n,b}^2 = (d+√d)^2 (d^2-1)/d I = (1+√d)^2(d^2-1)I. For SIC, the identity and cross terms from (26) cancel, and the t^2 term reduces to d^2(d+1)^2 Σ_{k=1}^{d^2-1}F_k^2 = d(d-1)(d+1)^3I, so S_SIC = 1/d^3 I + t^2 d(d-1)(d+1)^3 I = ad I by (27). The remaining steps (16)-(19) and (30) are standard WYD superoperator identities and check out, and the d=2 numerics are consistent. Thus no correctness risk remains; the only weakness is presentational: [19] is submitted, not published, so the reader cannot see the proof in the manuscript itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the measurement-based coherence Q(ρ,M)=Σ_i I(ρ,M_i) built from Wigner-Yanase skew information and evaluates its average with respect to complete sets of mutually unbiased measurements (MUMs) and general symmetric informationally complete measurements (SIC POVMs). The main results are Theorem 1, C(ρ,P_MUM) = (κd−1)/(d^2−1)[d−(Tr√ρ)^2] for MUMs with parameter κ, and Theorem 2, C(ρ,P_GSM) = (ad^3−1)/(d(d^2−1))[d−(Tr√ρ)^2] for general SIC POVMs with efficiency parameter a. Specializing to SIC-POVMs (a=1/d^2) gives C_SIC(ρ)=1/[d(d+1)][d−(Tr√ρ)^2] and the relation C_MUB(ρ)=d C_SIC(ρ). The proofs expand the Wigner-Yanase-Dyson quantity I_α around P=I/d+tF, use sum-of-squares identities for the traceless generators, and import a known WYD identity. The paper closes with asymptotic d→∞ comparisons and a pure-state example.","tokens_in":7649,"tokens_out":12202,"duration_ms":116018,"significance":"Theorems 1 and 2 are clean, non-obvious closed forms: the average coherence is always a fixed multiple of the maximal coherence d−(Tr√ρ)^2, with the constant determined solely by the measurement parameters. The MUB/SIC duality C_MUB = d C_SIC is particularly appealing and gives an operational link between two fundamental structures. The derivation is algebraic and the final expressions are explicit and falsifiable, with no free parameters beyond κ and a. The main limitation is self-containedness: the two trace identities imported from [19] are not proved here; however, they follow from direct expansion of the constructions (8) and (26), so the central claims are sound.","major_comments":[],"minor_comments":[{"comment":"The identities Σ_{b,n} Tr[(F_n^{(b)})^2ρ]=(1+√d)^2(d^2−1) and Σ_k Tr[(P_k)^2ρ]=ad are cited to the submitted manuscript [19]. They set the prefactors in Theorems 1 and 2, so the manuscript should either prove them (a few lines from (8)–(10) and (26)–(27)) or cite a published version. I verified both by direct expansion, so this does not affect the correctness of the theorems.","section":"Eqs. (15), (29)"},{"comment":"In the expansion leading to Eq. (30), the constant and linear terms are omitted; the cancellation follows from Σ_k P_k = I and Σ F_k = F together with tracelessness of the F_k. A one-sentence explanation would make the step easier to follow.","section":"Eq. (30)"},{"comment":"Reference [12] lacks a volume and page number, and reference [19] is marked 'submitted'; please update if available.","section":"References"},{"comment":"For a SIC POVM, which is a single d^2-element measurement, the phrase 'average coherence with respect to a general SIC measurement' might be clarified as an average over the elements of the POVM rather than over a set of measurements.","section":"Terminology"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper does exactly what it claims: it evaluates the average Wigner-Yanase coherence for a complete set of MUMs and for general SIC measurements, giving closed forms (κd−1)/(d²−1)[d−(Tr√ρ)²] and (ad³−1)/(d(d²−1))[d−(Tr√ρ)²], and the relation C_MUB = d·C_SIC. The math is sound. I went through the cancellations; the stress-test verification of the two imported trace identities holds up by direct expansion from the explicit constructions. The d=2 numerics are consistent.\n\nWhat's actually new: the MUM and general-SIC formulas are not in the prior literature, and the SIC-POVM relation C_MUB = d·C_SIC is an original observation. The technique is a direct extension of Luo and Su's approach, so this is a within-subfield contribution with moderate novelty, not a breakthrough. That's fine; it's a tidy, complete result.\n\nCredit: derivations are explicit and readable. Working with WYD information at general α and then specializing α=1/2 is a nice device. The ordering relations and asymptotics are clearly laid out. The citation pattern is mostly appropriate; [19] is the one self-citation used for a pair of trace identities, and since those identities are directly checkable, this is not a circularity problem.\n\nSoft spots: the main weakness is presentational. The two trace identities imported from [19] are load-bearing — they fix the prefactors in Theorems 1 and 2 — and neither is proved in this manuscript. The stress-test confirms they follow from the explicit constructions, so correctness is not at risk. But a referee will want those proofs in an appendix or the companion paper published. Eq. (30) also hides a cancellation of linear terms; it follows from Σ P_k = I, but an explicit line would help. These are minor.\n\nWho it's for: people working on coherence quantifiers or complementary measurements. It deserves a serious referee. I'd send it to review and expect acceptance after minor revision. I'd cite it if I were writing in this area.","headline":"Clean, checkable closed forms for average Wigner-Yanase coherence under MUMs and general SIC measurements, with a neat MUB/SIC relation; the only real weakness is importing two trace identities from an unpublished companion paper.","tokens_in":8147,"tokens_out":3033,"would_cite":true,"duration_ms":28542,"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":"For any quantum state, the average coherence over complete sets of mutually unbiased measurements or general symmetric informationally complete measurements is a fixed constant times a single state-dependent quantity.","keywords":["quantum coherence","Wigner-Yanase skew information","mutually unbiased measurements","symmetric informationally complete measurements","average coherence","MUB","SIC-POVM","Wigner-Yanase-Dyson information"],"falsifier":"In dimension $d=3$, construct the explicit complete MUM set, choose a mixed state $\\rho$ with distinct eigenvalues, numerically evaluate $Q(\\rho,P_{\\mathrm{MUM}})$ term by term, and compare with $(\\kappa d-1)/(d^2-1)[d-(\\operatorname{Tr}\\sqrt{\\rho})^2]$; equality at numerical precision supports the theorem, and any mismatch localizes an error in the trace identities. Repeating for a general SIC measurement checks the second theorem independently.","tokens_in":7073,"feed_emoji":"⚛️","tokens_out":16177,"duration_ms":153094,"temperature":0.7,"pith_summary":"This paper proves closed-form expressions for the average Wigner-Yanase coherence of a quantum state with respect to two families of complementary measurements: complete sets of mutually unbiased measurements (MUMs) and general symmetric informationally complete (SIC) measurements. In both cases the average is a constant multiple of $d-(\\operatorname{Tr}\\sqrt{\\rho})^2$, the same state-dependent quantity that appears in the maximal coherence, where $d$ is the Hilbert-space dimension. The multiplying constant is fixed by one measurement parameter: $\\kappa$ for MUMs and $a$ for general SIC measurements, and the formulas reproduce the MUB and SIC-POVM averages as special cases. Because explicit MUMs and general SIC measurements exist in every dimension, the MUM and general-SIC formulas are unconditional, while the SIC-POVM special case inherits the open existence question for SIC-POVMs. The paper also derives exact proportionality relations, notably that the SIC-POVM average equals $1/d$ times the MUB average.","feed_headline":"One parameter fixes average coherence for complementary measurements","feed_subtitle":"MUMs and general SIC measurements yield coherence averages that depend on the state only through one scalar.","key_machinery":"The central object is the Wigner-Yanase skew information $I(\\rho,M_i)=-\\frac{1}{2}\\operatorname{Tr}[\\sqrt{\\rho},M_i]^2$, summed over the operators of a measurement to define the coherence $Q(\\rho,\\mathcal{M})$. The computation also uses the more general Wigner-Yanase-Dyson information $I_\\alpha(\\rho,X)=-\\frac{1}{2}\\operatorname{Tr}([\\rho^\\alpha,X][\\rho^{1-\\alpha},X])$ to reach the average. The argument is carried by explicit constructions of MUMs and general SIC measurements from orthonormal traceless operators $F_{n,b}$, together with trace-sum identities: $\\sum_{b,n}\\operatorname{Tr}[(F_n^{(b)})^2\\rho]=(1+\\sqrt{d})^2(d^2-1)$, $\\sum_k\\operatorname{Tr}(P_k^2\\rho)=ad$, $\\sum_{b,n}F_{n,b}^2=(d-1/d)I$, and the averaged WYD identity $\\sum_{b,n}I_\\alpha(\\rho,F_{n,b})=d-\\operatorname{Tr}(\\rho^\\alpha)\\operatorname{Tr}(\\rho^{1-\\alpha})$. These identities make the summed state dependence factor out, yielding $Q_\\alpha(\\rho,P_{\\mathrm{MUM}})=\\frac{\\kappa d-1}{d-1}\\,Q_\\alpha(\\rho)$ and $Q_\\alpha(\\rho,P_{\\mathrm{GSM}})=\\frac{a d^3-1}{d(d^2-1)}\\,Q_\\alpha(\\rho)$ with $Q_\\alpha(\\rho)=d-\\operatorname{Tr}(\\rho^\\alpha)\\operatorname{Tr}(\\rho^{1-\\alpha})$; setting $\\alpha=1/2$ gives the coherence theorems.","core_discovery":"For any state $\\rho$ and any complete set of mutually unbiased measurements with parameter $\\kappa$, the average measurement-based coherence is $$C(\\rho,P_{\\mathrm{MUM}})=\\frac{\\kappa d-1}{$d^{2}$-1}\\,[d-(\\operatorname{Tr}\\sqrt{\\rho})^2].$$ For any general SIC measurement with parameter $a$, it is $$C(\\rho,P_{\\mathrm{GSM}})=\\frac{a $d^{3}$-1}{d($d^{2}$-1)}\\,[d-(\\operatorname{Tr}\\sqrt{\\rho})^2].$$ When $a=1/d^2$ the latter reduces to $C_{\\mathrm{SIC}}(\\rho)=\\frac{1}{d(d+1)}[d-(\\operatorname{Tr}\\sqrt{\\rho})^2]=\\frac{1}{d}C_{\\mathrm{MUB}}(\\rho)$. The paper further shows $C(\\rho,P_{\\mathrm{MUM}})\\le C_{\\mathrm{MUB}}(\\rho)=C_U(\\rho)<C_{\\max}(\\rho)$, with equality in the first bound only at $\\kappa=1$, and that $C(\\rho,P_{\\mathrm{MUM}})/C_{\\max}(\\rho)\\to\\kappa$ as $d\\to\\infty$ while $C_{\\mathrm{SIC}}(\\rho)/C_{\\max}(\\rho)=1/(d+1)\\to0$. Thus the average coherence of a state over these complementary measurement families factorizes into a state-dependent number and a coefficient that depends only on the measurement's overlap parameter.","pith_inferences":["The intermediate calculation actually supplies an $\\alpha$-family of identities: for $0<\\alpha<1$, the same factorization holds with $(\\operatorname{Tr}\\sqrt{\\rho})^2$ replaced by $\\operatorname{Tr}(\\rho^\\alpha)\\operatorname{Tr}(\\rho^{1-\\alpha})$, extending the coherence result to a family of Wigner-Yanase-Dyson uncertainty witnesses.","If one wants to measure these average coherences, the factorization suggests estimating only the single state number $d-(\\operatorname{Tr}\\sqrt{\\rho})^2$ and the measurement parameter, rather than evaluating the full operator sum; this estimation shortcut is an inference from the formulas, not a proposal in the paper.","The pattern that SIC-type averages die out while MUB-type averages saturate in large dimension hints that the number of measurement operators, or their frame potential, may control average coherence more than the detailed geometry; testing other informationally complete POVMs with different frame potentials would show whether this tradeoff is generic."],"forward_implications":["For SIC-POVMs, $C_{\\mathrm{SIC}}(\\rho)=\\frac{1}{d}C_{\\mathrm{MUB}}(\\rho)$, so in the same dimension a SIC measurement gives exactly a factor $1/d$ less average coherence than a complete set of MUBs.","For MUMs, $C(\\rho,P_{\\mathrm{MUM}})\\le C_{\\mathrm{MUB}}(\\rho)=C_U(\\rho)<C_{\\max}(\\rho)$, with equality only when $\\kappa=1$; in high dimensions $C(\\rho,P_{\\mathrm{MUM}})/C_{\\max}(\\rho)\\to\\kappa$.","For general SIC measurements, $C(\\rho,P_{\\mathrm{GSM}})\\le C_{\\mathrm{SIC}}(\\rho)$, and $C_{\\mathrm{SIC}}(\\rho)/C_{\\max}(\\rho)=1/(d+1)\\to0$ as $d\\to\\infty$, so SIC-type averages fade relative to the maximal coherence.","Because explicit complete MUMs and general SIC measurements exist for every dimension, the corresponding closed forms are valid in arbitrary dimension without resolving the open existence problems for complete MUBs or SIC-POVMs.","For any pure state the averages become $C_{\\mathrm{MUB}}(\\rho)=\\frac{d-1}{d+1}$, $C_{\\max}(\\rho)=\\frac{d-1}{d}$, and $C_{\\mathrm{SIC}}(\\rho)=\\frac{d-1}{d(d+1)}$, making the large-dimension contrast between MUB and SIC averages explicit."],"supporting_citations":[{"why":"Supplies the Wigner-Yanase skew-information coherence measure $Q(\\rho,\\mathcal{M})$ whose averages are computed.","marker":"[8]"},{"why":"Establishes the MUB and uniform-basis average formulas that the MUM result reduces to at $\\kappa=1$.","marker":"[12]"},{"why":"Provides the explicit construction of complete sets of mutually unbiased measurements and the parameter $\\kappa$.","marker":"[13]"},{"why":"Provides the construction of general symmetric informationally complete measurements and the parameter $a$.","marker":"[14]"},{"why":"Proves the averaged Wigner-Yanase-Dyson summation identity used to collapse the double sums.","marker":"[18]"},{"why":"Supplies the trace identities fixing the prefactors in both closed-form theorems.","marker":"[19]"},{"why":"Gives the normalization $\\sum_{b,n}F_{n,b}^2=(d-1/d)I$ used in the MUM calculation.","marker":"[20]"}],"fun_headline_variants":["One scalar decides average coherence over MUMs and SICs","Coherence average factorizes: state part and measurement part","Complementary measurements: coherence average follows a single parameter","MUM and SIC coherence averages collapse to one state quantity","Average coherence over complementary measurements: one number suffices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the squared operators in the explicit MUM and general SIC constructions satisfy the trace-sum identities cited from [19], together with the cited WYD summation identity from [18]; if any of these identities fails, the constant prefactors in the two theorems would not follow.","fun_headline_variants_meta":{"raw":{"variants":["One scalar decides average coherence over MUMs and SICs","Coherence average factorizes: state part and measurement part","Complementary measurements: coherence average follows a single parameter","MUM and SIC coherence averages collapse to one state quantity","Average coherence over complementary measurements: one number suffices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001072,"raw_usage":{"total_tokens":4468,"prompt_tokens":905,"completion_tokens":3563,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":3482}},"tokens_in":521,"tokens_out":3563,"duration_ms":21883,"temperature":1.0,"reasoning_tokens":3482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:00:14.433186+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In dimension $d=3$, construct the explicit complete MUM set, choose a mixed state $\\rho$ with distinct eigenvalues, numerically evaluate $Q(\\rho,P_{\\mathrm{MUM}})$ term by term, and compare with $(\\kappa d-1)/(d^2-1)[d-(\\operatorname{Tr}\\sqrt{\\rho})^2]$; equality at numerical precision supports the theorem, and any mismatch localizes an error in the trace identities. Repeating for a general SIC measurement checks the second theorem independently.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Wigner-Yanase skew-information coherence measure $Q(\\rho,\\mathcal{M})$ whose averages are computed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the MUB and uniform-basis average formulas that the MUM result reduces to at $\\kappa=1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit construction of complete sets of mutually unbiased measurements and the parameter $\\kappa$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the construction of general symmetric informationally complete measurements and the parameter $a$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the averaged Wigner-Yanase-Dyson summation identity used to collapse the double sums."},{"cited_title":"submitted","cited_arxiv_id":null,"evidence_quote":"Supplies the trace identities fixing the prefactors in both closed-form theorems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the normalization $\\sum_{b,n}F_{n,b}^2=(d-1/d)I$ used in the MUM calculation."}],"review_version":1}