Average coherence with respect to complementary measurements

Bin Chen; Shao-Ming Fei

Average coherence with respect to complementary measurements

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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SIC measurements, with a neat MUB/SIC relation; the only real weakness is importing two trace identities from an unpublished companion paper. read the letter →

arxiv 1908.03863 v1 pith:2VASON7M submitted 2019-08-11 quant-ph

Bin Chen , Shao-Ming Fei This is my paper · ORCID

classification quant-ph

keywords quantumcoherenceWigner-YanaseskewinformationmutuallyunbiasedmeasurementssymmetricinformationallycompleteaverageMUBSIC-POVMWigner-Yanase-Dyson

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

What'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.

Credit: 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.

Soft 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.

Who 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.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)

[Eqs. (15), (29)] 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.
[Eq. (30)] 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.
[References] Reference [12] lacks a volume and page number, and reference [19] is marked 'submitted'; please update if available.
[Terminology] 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.

Circularity Check

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No circularity: The average-coherence formulas in Theorems 1 and 2 are derived by direct algebra; the two self-cited trace identities in Eq. (15) and Eq. (29) are parameter-free and immediately verifiable from the paper's own explicit MUM and SIC constructions.

full rationale

The derivation is self-contained apart from two trace identities imported from the authors' own submitted paper [19] at Eq. (15), sum_{b,n} Tr[(F_n^{(b)})^2rho] = (1+sqrt(d))^2(d^2-1), and at Eq. (29), sum_k Tr[(P_k)^2rho] = a d. I flag these as an omitted proof because [19] is not published and the identities are not derived in-line; however, this is not circularity. Neither identity is the target theorem, and both follow directly from the explicit constructions already stated in the manuscript. Expanding the MUM operators (8) makes the F^{(b)} cross terms cancel, leaving (d+sqrt(d))^2 sum_{b,n} F_{n,b}^2 = (1+sqrt(d))^2(d^2-1)I by the standard orthonormality trace identity used in Eq. (17). The SIC identity follows similarly from (26) and the parameter relation (27), since the F cross terms cancel and the remaining square reduces to a d I. The rest of the proof, Eqs. (16)-(19) and (30), consists of standard WYD-superoperator manipulations using the external results [18] and [20]. Theorems 1 and 2 are exact evaluations of an independently defined average coherence; no parameter is fitted to the final expression, and no conclusion is assumed by definition. The only self-citations in the paper, [4] and [19], are contextual or supply a directly checkable algebraic lemma, so they do not make the derivation circular. The paper is self-contained against external benchmarks; the sole presentational weakness is that the reader must verify the two [19] identities from the construction, but that verification is straightforward and does not affect the circularity verdict.

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

The paper introduces no new physical entities or fitted parameters. Its results depend on the standard constructions of MUMs and general SICs, on a published WYD identity, and on two trace identities cited from the authors' own submitted manuscript. Those identities are verifiable directly from the explicit constructions.

assumptions (5)

domain assumption Existence of complete sets of MUMs for any dimension d
Used to define C(rho,P_MUM) in Eq. (11); constructed in Ref. [13].
domain assumption Existence of general SIC measurements for any dimension d
Used to define C(rho,P_GSM) in Eq. (28); constructed in Ref. [14].
standard math Trace identity sum_{b,n} I_alpha(rho,F_{n,b}) = d - Tr(rho^alpha)Tr(rho^{1-alpha}) from Ref. [18]
Used in Eq. (18) to evaluate the sum of WYD information over the traceless basis operators.
domain assumption Sum-of-squares identities from Ref. [19] (authors' own submitted work)
sum_{b,n} Tr[(F^(b)_n)^2 rho] = (1+sqrt(d))^2(d^2-1) and sum_k Tr[(P_k)^2 rho] = a*d are used in Eqs. (15) and (29) to fix the constant prefactors.
domain assumption The property sum_k P_k = I for the MUM and general SIC sets
Implied by the POVM completeness condition; used implicitly to cancel linear terms in Eq. (30).

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Cite this review

Pith. "Pith review of Average coherence with respect to complementary measurements." pith.science (2026). https://pith.science/paper/2VASON7M

@misc{pith2026190803863,
  author       = {Pith},
  title        = {Pith review of: Average coherence with respect to complementary measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VASON7M}},
  note         = {Machine review of arXiv:1908.03863}
}

read the original abstract

We investigate the average coherence with respect to a complete set of complementary measurements. By using a Wigner-Yanase skew information-based coherence measure introduced in [Phys. Rev. A \textbf{96}, 022130, 2017], we evaluate the average coherence of a state with respect to any complete set of mutually unbiased measurements and general symmetric informationally complete measurements, respectively. We also establish analytically the relations among these average coherences.

Figures

Figures reproduced from arXiv: 1908.03863 by the authors.

**Figure 1.** The blue solid line is Cmax(ρ), the dashed line is CMUB(ρ), and the dot-dashed line is CSIC(ρ). 4 Conclusion In summary, we have studied the average coherence with respect to complementary measurements. By evaluating the average coherence associated with MUMs and general SIC measurements, respectively, we have also established the relations among these quantities and the maximal coherence of quantum states. It has… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[19]

submitted

Chen, B., Fei, S.-M.: Total variance and invariant information in co mplementary measurements. submitted

work page
[1]

Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. P hys. Rev. Lett. 113, 140401 (2014) 10

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Yu, X.-D., Zhang, D.-J., Xu, G.-F., Tong, D.-M.: Alternative framewo rk for quanti- fying coherence. Phys. Rev. A 94, 060302(R) (2016)

work page 2016
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Quan- tum Inf

Chen, B., Fei, S.-M.: Notes on modiﬁed trace distance measure of c oherence. Quan- tum Inf. Process. 17, 107 (2018)

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Rastegin, A.E.: Quantum-coherence quantiﬁers based on the Ts allis relative α en- tropies. Phys. Rev. A 93, 032136 (2016)

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [19]

submitted

Chen, B., Fei, S.-M.: Total variance and invariant information in co mplementary measurements. submitted

work page

[2] [1]

Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. P hys. Rev. Lett. 113, 140401 (2014) 10

work page 2014

[3] [2]

Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measur ing quantum coherence with entanglement. Phys. Rev. Lett. 115, 020403 (2015)

work page 2015

[4] [3]

Yu, X.-D., Zhang, D.-J., Xu, G.-F., Tong, D.-M.: Alternative framewo rk for quanti- fying coherence. Phys. Rev. A 94, 060302(R) (2016)

work page 2016

[5] [4]

Quan- tum Inf

Chen, B., Fei, S.-M.: Notes on modiﬁed trace distance measure of c oherence. Quan- tum Inf. Process. 17, 107 (2018)

work page 2018

[6] [5]

Rastegin, A.E.: Quantum-coherence quantiﬁers based on the Ts allis relative α en- tropies. Phys. Rev. A 93, 032136 (2016)

work page 2016

[7] [6]

Winter, A., Yang, D.: Operational resource theory of coherenc e. Phys. Rev. Lett. 116, 120404 (2016)

work page 2016

[8] [7]

Napoli, C., Bromley, T.R., Cianciaruso, M., Piani, M., Johnston, N., Ade sso, G.: Robustness of coherence: An operational and observable measu re of quantum coher- ence. Phys. Rev. Lett. 116, 150502 (2016)

work page 2016